Modulus and Phase

Complex numbers are very abstract in themselves, and to better understand their properties, we need to draw parallels between them and other less abstract quantities. One way of doing this is treating a complex number as a vector and finding similar properties, such as the magnitude (also known as the modulus) of a complex number, its direction, and phase. It is suggested that you go over the article on Complex Numbers here before continuing with this article! 

Modulus and Phase Modulus and Phase

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Contents
Table of contents

    A Quick Review of Complex Numbers - the Formula and Symbol

    It is a very fruitful idea to represent complex numbers as points in the XY plane. But this plane is not a regular Cartesian plane with normal cartesian coordinates, but we shall call it the complex plane. Also sometimes known as the Argand plane, after the mathematician Jean-Robert Argand.

    Complex numbers are plotted very naturally in this plane. Let there be a complex plane of the form z=a+ib{"x":[[147,146,145,145,146,147,149,151,157,165,172,178,183,185,188,191,192,192,191,188,184,180,175,168,163,158,154,149,145,141,140,139,137,137,136,136,137,138,141,143,151,157,163,169,174,179,183,185,187,185,184],[133,127,126,128,130,136,139,143,151,161,171,179,187,192,195,199],[264,261,263,265,269,275,278,284,291,298,300],[270,262,264,267,272,278,285,297,302,313,318,321,322],[396,397,397,397,395,392,389,386,382,380,375,372,369,367,366,366,367,369,372,376,378,380,386,390,393,396,398,399,399,399,399,400,400,400,400,400,400,400,400,400,401,403,405,406,408,409],[470,470,470,470,470,470,470,470,470,469,468,467,466,465],[448,445,446,448,453,459,465,473,482,485,486,489,490],[529,529,529,529,529,529,529,528,528,528,527,527,526,526,529,532,536,538],[531,527,528,529,530,531],[569,569,569,569,569,569,569,569,569,569,568,567,565,563,561,560,560,560,560,561,564,566,569,571,573,576,579,582,584,585,586,587,587,587,586,584,582,579,576,573,569,566,563,561]],"y":[[191,189,189,188,188,188,188,188,186,185,183,182,181,181,180,180,180,181,182,185,190,196,203,213,220,228,233,239,243,247,249,250,252,253,253,254,254,254,254,254,252,251,249,248,247,247,247,247,247,247,247],[223,221,220,220,220,220,220,220,220,219,218,217,215,213,212,212],[208,207,207,208,208,208,208,208,208,208,207],[232,234,234,234,234,233,232,230,229,227,226,226,225],[210,207,204,203,201,197,196,195,195,195,198,201,205,210,215,219,223,224,225,224,223,222,218,214,210,205,202,199,198,197,199,203,205,207,212,215,217,222,225,229,232,234,235,235,234,233],[199,197,196,197,201,203,208,214,220,226,232,237,241,243],[221,221,222,222,223,223,223,222,220,220,220,220,220],[204,203,202,201,203,204,205,209,211,214,218,221,225,229,231,231,229,228],[182,179,179,179,180,182],[171,168,167,166,167,169,172,174,179,185,194,198,208,213,217,220,223,224,223,221,216,212,208,207,204,202,202,202,204,207,209,212,214,218,221,224,226,228,229,229,229,227,225,222]],"t":[[0,7,23,32,458,466,473,474,483,492,499,509,516,525,531,542,549,574,582,592,598,606,615,625,633,641,649,658,666,677,682,683,692,699,700,708,716,717,725,731,742,749,758,766,775,782,791,799,808,842,846],[1058,1066,1073,1099,1108,1115,1115,1125,1132,1142,1148,1158,1165,1175,1182,1192],[1634,1646,1682,1684,1692,1700,1709,1715,1725,1732,1735],[1862,1873,1899,1908,1915,1926,1932,1942,1949,1959,1966,1976,1983],[2367,2376,2382,2391,2399,2409,2416,2427,2433,2442,2449,2459,2465,2476,2482,2492,2499,2509,2515,2524,2524,2532,2543,2550,2561,2566,2576,2583,2592,2599,2615,2626,2633,2634,2643,2650,2651,2659,2666,2676,2682,2692,2699,2709,2716,2719],[5612,5620,5627,5651,5660,5667,5677,5683,5694,5700,5710,5717,5727,5733],[5854,5865,5870,5884,5894,5900,5911,5917,5927,5934,5935,5945,5952],[6310,6317,6322,6334,6359,6368,6369,6378,6385,6385,6394,6400,6411,6417,6428,6435,6444,6453],[6556,6564,6593,6601,6611,6618],[6782,6789,6797,6802,6826,6835,6844,6850,6861,6867,6878,6884,6894,6902,6911,6918,6928,6935,6959,6967,6978,6985,6995,7001,7011,7017,7028,7034,7044,7052,7062,7068,7078,7084,7095,7100,7111,7117,7128,7134,7144,7151,7161,7167]],"version":"2.0.0"} where a,b{"x":[[191,188,187,186,183,181,177,173,168,162,156,150,143,137,131,126,122,119,117,117,117,117,118,122,127,134,142,150,158,167,175,183,190,196,200,203,204,204,203,202,200,198,195,193,190,188,186,185,184,184,184,185,186,188,191,195,199,204,209,215,220,226,231],[267,265,265,265,265,265,264,264,264,264,264],[345,344,344,344,344,344,343,342,340,338,336,334,332,331,330,329,328,328,328,328,329,331,335,338,342,346,350,355,359,364,368,371,375,378,380,383,385,386,386,386,386,384,380,375,369,363,355,348,341,336,332,330,329,329,328,328,329,332],[534,526,520,513,505,498,490,481,473,465,456,449,443,437,432,429,427,426,426,426,426,430,435,441,449,458,467,477,488,496,504,511,517,521,524,526,527],[430,434,441,451,462,474,487,498,507,514,518,522,524,524,524],[666,664,664,664,663,661,658,655,652,649,645,642,640,637,635,633,631,630,629,628,628,628,629,630,634],[658,656,656,657,660,666,675,686,699,711,722,731,737,741,743,745,746,746,746,744,739,733,724,714,703,693,682,674,669,665,664,666,668,671,676,681,687,693,700,705,710,716,720,724,728,732,736,739,743,746,749,752,755],[694,691,691,688,684,683,680,677,674,671,668,666,664,662,661,660,660,660,660,660,660,663]],"y":[[239,230,228,226,224,222,222,222,222,224,228,233,239,246,252,259,266,272,277,280,282,284,284,284,284,280,274,268,261,253,244,237,229,224,220,218,218,220,224,229,234,241,247,254,262,271,279,287,294,300,305,307,309,310,310,310,310,309,307,305,302,299,297],[307,313,317,322,328,334,339,343,346,348,347],[156,156,162,169,177,186,196,207,217,226,236,244,252,260,266,271,275,278,277,275,272,267,261,255,249,244,239,235,233,232,232,232,234,237,240,245,249,253,258,263,268,273,278,283,288,293,297,301,303,305,305,305,305,302,298,291,285,279],[186,181,181,183,186,190,195,201,207,214,222,229,236,243,250,256,262,267,272,277,280,283,286,289,292,294,294,294,294,294,294,291,289,287,285,283,282],[253,248,248,246,244,241,239,237,235,234,234,234,234,236,238],[152,151,155,162,171,184,197,210,224,234,246,256,266,276,286,295,304,312,320,326,330,333,335,336,336],[178,169,166,163,159,154,150,146,142,141,140,140,141,144,148,152,157,163,169,176,184,191,199,207,214,221,226,230,233,234,234,234,234,234,236,240,244,248,254,260,267,274,281,288,295,301,306,312,316,320,322,324,325],[144,145,151,171,198,212,226,241,255,270,282,295,308,320,331,341,350,358,365,370,375,380]],"t":[[0,17,28,36,45,50,61,67,78,84,95,100,112,117,128,134,145,150,161,167,178,184,200,211,217,228,234,245,250,262,267,278,284,295,300,311,317,351,362,367,378,384,395,401,412,417,428,434,445,450,462,472,478,484,495,500,511,517,527,534,544,551,561],[815,874,877,884,896,901,912,918,929,934,983],[1256,1299,1309,1317,1328,1334,1345,1351,1362,1368,1379,1384,1396,1401,1413,1418,1429,1434,1484,1493,1501,1511,1519,1528,1534,1545,1551,1562,1568,1578,1584,1595,1602,1612,1618,1628,1634,1645,1651,1662,1668,1678,1684,1695,1701,1712,1718,1729,1734,1745,1751,1762,1768,1779,1784,1795,1801,1812],[2220,2242,2251,2263,2268,2280,2285,2297,2301,2313,2318,2330,2335,2346,2351,2362,2370,2379,2385,2396,2401,2412,2421,2429,2434,2445,2451,2462,2468,2479,2485,2495,2501,2512,2518,2529,2535],[2740,2792,2801,2813,2818,2830,2835,2847,2852,2863,2868,2880,2885,2910,2918],[3210,3251,3261,3273,3278,3286,3297,3302,3314,3318,3329,3335,3347,3352,3364,3369,3380,3385,3397,3402,3413,3418,3429,3435,3446],[3650,3668,3677,3685,3693,3702,3713,3718,3729,3735,3746,3752,3763,3769,3779,3785,3796,3802,3813,3819,3829,3835,3846,3852,3863,3868,3879,3885,3896,3902,3913,3968,3977,3985,3994,4002,4013,4018,4030,4035,4046,4052,4063,4068,4080,4085,4096,4102,4113,4119,4130,4135,4147],[4445,4463,4469,4487,4509,4513,4519,4529,4535,4546,4552,4563,4569,4579,4586,4596,4602,4612,4619,4629,4635,4646]],"version":"2.0.0"}. Here a is called the real part, and b is called the imaginary part. The coordinates of z on the complex plane will be (a, b). The y-axis is called the imaginary axis, and the x-axis is the real axis. The line 'r' is referred to as the terminal arm in this article.

    Modulus and Phase, Complex numbers made of  'real' and 'imaginary' parts, StudySmarterComplex numbers are made up of a real and an imaginary part, StudySmarter Originals

    The Modulus of a Complex Number

    Let us denote the distance of the point from the origin as r, and we can use the Pythagorean theorem to calculate its distance from the origin, which is

    r2=a2+b2

    This distance r is known as the Modulus of z hence we get,

    z=r=a2+b2{"x":[[178,179,176,176,175,175,174,174,173,173,173,173,173,173,173,174,175,176,177,177,178],[288,287,287,287,287,287,287,287,287,287,287,287,287,287,287,287,287,287,287,287,287,288,288,289],[348,343,344,348,352,357,362,366,371,375,378,380,381],[344,343,341,342,345,346,349,353,356,361,367,372,377,381,383,384,383,383],[209,207,205,204,204,204,205,206,209,211,212,215,216,218,223,228,230,232,234,236,237,238,238,238,238,238,236,235,234,232,229,227,226,224,221,218,216,214,212,212,214,217,219,220,222,227,233,238,240,243,247,248,250,251,252],[213,209,210,214,219,226,234,242,250,256,259,261,260,259,257,255],[451,449,448,447,448,450,451,454,456,459,463,467,471,474,477,479,479,479,477,476,474,472,471,469,469,468,468,468,469,470,472,474,477,481,484,488,490,491,492,492,492,491,489,487,485,483,480,479,478,478,479,481,484,486,490,493],[536,537,539,541,542,547,550,555,560,564,567,570,571,571],[533,532,534,536,538,540,545,549,555,562,567,571,574,576],[628,628,628,630,632,633,635,637,638,639,640,640,640,640,641,642,643,644,645,646,647,647,648,649,650,651,652,653,655,656,658,660,661,663,664,666,668,669,670,672,673,674,676,677,678,679,680,680,680,681,681,681,681,681,680],[679,679,679,680,681,683,686,691,699,710,716,731,738,764,783,802,822,843,862,872,892,919,935,949,955,960,968,971,974,977,978,979,978,977],[715,720,721,722,723,724,724,724,725,725,724,724,723,722,720,718,716,714,711,709,708,707,705,705,704,704,705,706,708,710,713,716,719,722,724,726,728,730,731,732,733,733,734,734,735,735,736],[749,749,749,749,749,750,752,754,757,761,764,766,768,768,768,767,765,763,762,761,760,760,760,760,761,764,767,772,777,783,787,794,797],[806,807,808,812,814,817,823,826,829,834,836,838,839,840],[826,826,826,826,826,826,825,824,823,822,822,822,822],[894,896,896,897,897,897,898,897,897,897,897,896,896,895,895,894,893,893,893,892,891,891,890,890,890,890,891,893,895,897,900,903,906,909,911,913,914,915,915,915,913,911,908,904,900,895,891,888,886,885,886,887,888],[928,928,927,927,927,927,928,930,934,937,939,942,944,945,945,945,944,943,941,938,935,933,932,935,939,946,954,964,974]],"y":[[106,106,147,153,166,170,180,188,199,204,206,209,211,212,213,213,211,208,206,204,202],[110,108,107,106,107,108,110,112,117,121,129,139,150,160,168,175,181,185,189,192,194,194,192,189],[135,133,133,133,133,133,133,133,133,133,133,134,134],[156,156,157,157,157,156,156,155,155,154,152,151,150,149,149,149,149,150],[136,136,136,136,135,134,134,133,131,131,130,130,130,129,129,129,129,130,130,131,132,133,134,135,136,141,145,147,150,153,157,159,161,163,166,169,170,173,175,177,177,177,177,177,177,176,175,174,174,174,174,174,174,174,174],[161,161,160,159,158,156,155,153,152,152,151,151,151,152,153,154],[174,175,175,175,175,175,173,172,169,167,163,157,150,144,139,136,134,132,132,132,132,132,133,134,135,136,138,139,140,140,142,143,144,144,144,144,144,145,145,146,148,149,152,156,161,165,170,174,177,180,181,182,182,182,181,179],[142,142,142,142,142,141,141,140,139,139,139,139,139,140],[162,163,163,162,162,162,161,160,159,158,157,156,155,155],[194,193,192,190,189,188,187,186,185,185,185,186,188,189,191,194,197,200,203,206,209,211,213,214,215,216,216,215,214,212,209,206,202,198,192,185,176,167,159,151,143,135,128,122,116,110,107,104,101,100,101,102,104,105,106],[98,97,96,96,95,95,96,96,97,97,97,97,97,96,96,95,95,95,96,96,97,99,101,103,103,104,106,106,107,108,108,108,108,108],[181,178,178,178,177,176,175,174,174,172,171,170,169,167,166,165,165,166,167,171,173,175,179,181,185,187,189,190,190,189,188,186,183,181,179,177,176,176,177,179,182,184,187,189,191,193,193],[135,133,131,130,129,128,127,126,125,125,125,126,129,132,136,140,143,147,150,152,154,155,157,158,159,159,159,157,155,153,152,150,149],[174,174,174,174,174,173,172,172,171,170,170,169,168,168],[159,160,162,165,169,171,177,182,187,190,193,194,195],[148,142,140,139,138,137,136,136,137,138,139,142,145,150,153,156,162,165,169,172,178,181,185,186,188,189,189,187,186,183,181,179,178,178,178,180,182,185,188,191,193,196,198,199,200,200,200,200,198,197,193,191,189],[141,140,137,136,135,134,134,133,132,132,132,133,133,135,138,139,142,144,147,150,152,154,155,155,154,152,149,147,144]],"t":[[0,2,6,8,16,24,36,41,50,58,66,67,75,83,83,100,109,116,125,125,133],[1672,1681,1689,1709,1718,1726,1734,1735,1743,1750,1758,1767,1775,1785,1791,1804,1808,1818,1825,1835,1841,1859,1867,1875],[2324,2331,2367,2375,2386,2392,2404,2408,2421,2425,2435,2442,2445],[2586,2591,2595,2601,2609,2618,2618,2627,2634,2642,2651,2658,2669,2675,2687,2692,2718,2722],[8622,8628,8633,8635,8638,8652,8663,8670,8677,8688,8689,8702,8704,8711,8720,8727,8740,8741,8753,8755,8763,8764,8778,8779,8791,8794,8803,8808,8813,8820,8827,8829,8838,8840,8850,8853,8862,8870,8878,8894,8920,8928,8938,8940,8950,8952,8969,8977,8989,8994,9004,9013,9020,9027,9030],[9310,9317,9328,9337,9344,9354,9361,9370,9378,9386,9394,9403,9428,9437,9445,9454],[14373,14384,14389,14404,14438,14455,14463,14473,14480,14490,14496,14508,14513,14525,14529,14544,14547,14556,14572,14580,14590,14596,14608,14613,14614,14623,14632,14641,14648,14649,14658,14665,14673,14682,14689,14698,14710,14714,14723,14724,14731,14740,14746,14756,14763,14773,14780,14789,14796,14806,14813,14823,14830,14838,14846,14856],[15279,15323,15333,15339,15341,15348,15356,15366,15375,15380,15393,15397,15407,15418],[15596,15604,15609,15615,15623,15623,15631,15640,15647,15657,15663,15673,15680,15692],[16504,16510,16531,16547,16556,16564,16576,16580,16592,16597,16614,16625,16631,16642,16649,16657,16668,16674,16680,16693,16698,16708,16714,16725,16731,16743,16747,16759,16764,16774,16781,16789,16797,16807,16814,16823,16830,16841,16847,16857,16864,16874,16881,16891,16897,16907,16914,16924,16931,16941,16989,16997,17007,17014,17024],[17265,17275,17297,17306,17326,17331,17341,17347,17357,17365,17366,17377,17381,17392,17398,17412,17415,17424,17431,17445,17447,17458,17465,17477,17482,17482,17492,17498,17499,17511,17516,17524,17531,17541],[18022,18027,18034,18040,18048,18058,18064,18074,18081,18095,18098,18099,18109,18116,18127,18132,18145,18149,18158,18166,18174,18175,18183,18192,18198,18208,18214,18227,18231,18243,18248,18259,18264,18274,18281,18293,18301,18308,18314,18325,18331,18344,18348,18360,18364,18376,18381],[18554,18559,18566,18574,18581,18592,18598,18608,18615,18625,18631,18641,18648,18658,18664,18678,18681,18692,18698,18699,18709,18715,18728,18733,18742,18750,18758,18766,18775,18783,18792,18798,18799],[19081,19116,19117,19124,19133,19133,19143,19149,19150,19159,19165,19175,19181,19185],[19307,19341,19349,19358,19366,19375,19381,19392,19398,19409,19415,19425,19426],[19764,19770,19775,19779,19783,19792,19800,19823,19832,19833,19840,19848,19849,19859,19866,19866,19876,19883,19884,19894,19899,19900,19910,19915,19926,19932,19948,19963,19967,19975,19982,19992,19998,20009,20015,20026,20032,20043,20048,20059,20065,20073,20082,20092,20098,20111,20115,20127,20133,20134,20143,20149,20150],[20404,20410,20415,20417,20423,20433,20442,20449,20459,20466,20466,20477,20483,20493,20499,20500,20510,20515,20526,20532,20543,20549,20560,20583,20592,20599,20609,20615,20626]],"version":"2.0.0"}

    where z{"x":[[385,384,384,384,384,384,384,384,384,384,385,385,386,387,387,388,389,390,390,390,391,391,392,392,393,394,395,397],[446,446,451,458,469,480,492,505,515,525,536,544,553,560,565,569,572,573,573,573,573,571,566,561,554,546,539,531,522,513,504,495,488,481,476,473,472,474,476,481,487,495,503,512,523,533,542,552,560,566,572,576,578,580,581,582,582,582],[494,501,512,526,541,554,566,574,580,584,587,589,589,587,584],[651,652,652,652,652,652,650,648,646,643,641,639,637,636,635,634,634,634,634,634,634,634,635,635,636,637,638,639,641]],"y":[[191,202,208,217,227,237,248,260,271,283,296,308,320,333,346,358,370,380,388,395,400,404,406,407,407,407,406,404],[257,258,258,258,256,254,253,251,250,249,248,248,248,248,248,249,251,253,256,259,263,267,272,277,282,288,293,298,303,308,313,318,322,326,330,332,333,333,333,333,332,331,329,327,325,324,323,322,322,322,322,322,322,322,322,323,324,325],[289,289,289,289,288,286,284,283,282,281,280,280,281,282,285],[172,174,180,188,198,208,221,232,245,258,270,283,297,309,322,334,346,356,365,373,380,385,388,391,392,392,392,392,392]],"t":[[0,22,30,40,46,57,63,73,80,90,96,107,113,123,130,140,146,157,163,173,180,190,197,207,213,230,240,246],[550,614,623,631,640,647,657,663,673,680,689,697,707,714,724,730,740,747,758,764,774,780,790,797,807,814,824,830,840,847,856,863,874,880,890,897,907,955,964,974,981,991,997,1007,1014,1024,1030,1040,1047,1057,1065,1074,1081,1090,1097,1107,1114,1124],[1338,1366,1373,1381,1390,1397,1407,1414,1424,1430,1441,1447,1488,1497,1507],[1700,1739,1748,1757,1764,1774,1780,1791,1797,1808,1814,1824,1831,1841,1847,1858,1864,1874,1880,1890,1897,1908,1914,1925,1931,1941,1947,1958,1967]],"version":"2.0.0"} denotes the modulus of z.

    The modulus of a complex number is the distance of that point from the origin on a complex plane. Alternatively, the square of the modulus of a complex number is equal to the sum of the squares of its real and imaginary parts. (z2=a2+b2{"x":[[176,176,175,174,174,175,176,178,180,183,187,190,195,200,205,209,213,217,219,222,223,224,225,225,224,222,220,217,213,209,205,199,194,188,183,178,174,171,168,166,164,163,163,163,163,164,166,169,172,176,181,186,191,196,201,205,209,213,216,219,221,222,223],[175,174,171,169,168,169,170,172,176,180,184,188,193,199,205,210,215,220,223,226,227,228],[138,138,139,139,140,140,138,135,131,127,123,119,115,112,110,110,110,110,111,112,113],[244,245,245,245,245,245,245,244,242,241,240,239,239,238,238,238,238,238,238,238,238,238],[262,259,259,259,260,261,263,266,268,271,274,276,278,278,279,279,277,275,272,270,268,267,267,269,271,273,277,281,285,289,293,297,299],[298,295,294,294,296,298,302,306,310,315,319,323,326,329,330,332,332],[292,291,287,287,288,289,293,298,304,311,319,326,332,337,339,341,343],[422,425,426,426,426,425,423,421,418,414,410,405,400,395,390,386,384,383,383,384,387,389,392,396,400,405,409,414,419,423,426,428,429,429,429,429,428,428,428,428,430,431,433,435,437,438,439],[447,446,446,446,446,446,446,447,448,449,451,453,455,456,458,459,459,459,458,457,455,452,449,447,445,445,447,450,454,459,464,468,473,476],[487,486,487,490,493,496,500,504,507,509,510],[492,492,492,493,494,496,497,497,497,496,495,494,494,493,493,494],[562,562,563,563,564,564,565,566,566,566,566,564,563,561,560,559,558,557,557,557,557,558,558,559,560,562,564,566,570,572,575,577,578,579,579,579,577,575,572,568,564,561,558,558,557,558],[598,597,596,596,595,595,595,595,595,595,595,596,598,599,602,605,608,612,615,617,618,619,618,617,615,612,610,607,606,608,611,617,623,630,638,643,648]],"y":[[155,154,154,154,153,153,153,153,153,153,153,153,152,152,152,151,151,151,151,151,151,152,153,154,155,157,159,162,165,168,172,176,180,185,191,196,200,204,208,211,214,216,219,220,222,223,224,224,224,224,223,223,222,222,222,221,221,221,222,222,222,223,223],[189,189,189,188,188,188,188,188,188,188,188,188,188,188,187,186,185,185,184,184,184,184],[143,142,142,143,146,151,158,169,181,193,205,218,230,242,253,260,265,270,273,275,276],[149,149,147,146,145,148,152,158,166,176,187,199,209,218,227,234,240,246,250,254,258,259],[120,116,115,114,114,114,114,114,114,114,114,115,116,117,120,122,126,129,133,137,139,142,143,144,144,143,142,140,138,136,134,132,131],[192,191,191,190,190,190,189,188,187,186,184,183,183,182,182,182,183],[215,215,214,213,213,213,213,212,210,208,206,203,201,199,198,198,198],[192,186,185,184,183,182,181,180,180,179,179,180,182,185,190,194,199,204,207,209,211,211,211,210,208,206,203,199,195,192,188,186,185,187,189,193,197,201,205,209,212,215,218,219,220,220,219],[158,152,150,149,147,146,145,144,143,143,143,142,142,143,144,146,148,151,155,158,162,166,169,172,174,175,175,174,172,170,167,165,163,162],[196,196,195,195,194,193,192,191,190,189,189],[183,182,181,181,181,183,185,188,192,197,201,206,210,212,215,215],[155,154,153,151,149,148,148,148,150,154,159,165,172,179,186,192,197,200,203,205,206,206,205,204,201,199,197,194,192,192,192,194,197,201,204,207,211,214,217,219,221,221,222,220,218,215],[158,156,154,153,152,150,148,146,145,143,141,139,138,136,136,135,135,135,137,138,141,144,147,150,153,156,158,160,161,161,161,159,157,155,153,151,149]],"t":[[0,6,13,33,41,50,57,67,74,83,90,100,107,117,124,133,140,150,157,167,174,183,191,200,207,217,224,234,240,250,257,267,274,284,290,300,307,320,325,334,341,350,357,370,374,385,391,403,407,419,424,434,440,452,457,469,474,484,491,500,507,518,524],[796,802,806,808,824,833,841,850,857,866,874,883,891,900,907,917,924,935,942,951,958,969],[1312,1321,1357,1366,1376,1383,1391,1401,1408,1417,1425,1434,1446,1450,1458,1467,1474,1484,1491,1502,1508],[1997,2002,2007,2008,2016,2049,2058,2067,2074,2084,2091,2100,2108,2117,2124,2137,2141,2154,2158,2168,2175,2184],[2508,2516,2521,2541,2550,2558,2567,2575,2584,2591,2601,2608,2618,2628,2634,2648,2650,2658,2667,2675,2685,2691,2701,2717,2725,2735,2741,2753,2759,2771,2775,2786,2791],[3159,3168,3175,3192,3201,3208,3218,3225,3235,3242,3251,3258,3268,3275,3285,3292,3301],[3459,3468,3473,3475,3487,3492,3501,3508,3518,3525,3535,3542,3551,3558,3568,3575,3585],[4023,4031,4035,4042,4051,4059,4068,4075,4085,4092,4102,4109,4118,4125,4135,4142,4152,4159,4168,4176,4185,4192,4200,4209,4218,4225,4236,4242,4254,4259,4269,4275,4285,4309,4320,4325,4337,4342,4354,4359,4368,4375,4385,4392,4402,4409,4416],[4641,4653,4659,4667,4675,4686,4692,4709,4718,4726,4735,4742,4752,4759,4768,4775,4785,4792,4801,4809,4818,4825,4837,4842,4856,4859,4876,4889,4892,4906,4909,4920,4926,4938],[5182,5195,5225,5234,5242,5252,5259,5271,5276,5288,5292],[5450,5460,5464,5476,5485,5492,5502,5509,5519,5526,5536,5543,5553,5559,5571,5575],[5942,5947,5952,5959,5962,5968,5976,5993,6002,6009,6018,6026,6036,6043,6053,6059,6070,6076,6089,6093,6109,6122,6140,6143,6156,6159,6169,6176,6193,6202,6209,6222,6226,6236,6243,6253,6259,6270,6276,6286,6293,6303,6309,6319,6326,6336],[6587,6592,6596,6601,6609,6619,6626,6637,6643,6653,6660,6670,6676,6686,6693,6703,6710,6720,6726,6738,6743,6757,6760,6770,6776,6791,6795,6803,6810,6835,6843,6853,6862,6870,6876,6887,6893]],"version":"2.0.0"})

    Calculate the modulus of the complex number z=3+4i

    Solution:

    Step 1: From the definition of the modulus of a complex number,

    z2=(real part)2+(imaginary part)2

    Step 2: The number beside the i is always the imaginary part. From the question, 3 and 4 are the real and imaginary parts, respectively:

    z=32+42z=5{"x":[[181,180,179,178,179,181,183,187,190,194,198,203,207,210,213,215,217,218,218,218,218,216,214,211,207,203,196,193,190,189,186,184,182,181,181,182,183,184,187,190,195,201,207,211,217,219,222,227,230,232,233,234,234,232,230],[179,177,175,177,181,186,196,204,208,212,219,228,230,233,235,235,234,233],[259,259,259,258,258,259,259,260,260,261,262,263,264,265,265,265,265,266,266,266,266,266,266,267],[148,148,148,149,149,150,150,151,151,152,152,153,153,153,153,153,152,152,153,154,156,156,157,158,159,160],[329,327,325,324,323,324,327,328,330,336,338,344,350,355,360,363],[332,328,330,333,335,337,342,344,347,353,359,364,368],[413,412,411,410,410,411,412,413,415,416,418,420,421,422,423,424,425,425,426,427,428,429,430,430,431,432,432,433,435,436,436,437,437,438,439,440,441,442,443,444,445,446,447,448,448,449,450,450,451,452,453,455,455,456,458,458,459,459,460,460,460,460,459,459,457,456,455,454,454,453,452,452,452,452,453,454,454,455,455,456,457,460,463,468,474,478,489,501,516,541,550,559,578,589,599,620,631,664,676,711,733,754,775,793,811,827,839,851,861,868,874,878,880,879],[518,519,519,520,520,521,521,522,523,523,524,526,528,529,533,535,539,542,543,544,544,543,541,539,536,535,533,532,531,530,530,531,533,534,535,538,540,543,546,547,549,550,550,549,547,545,542,539,536,535,533,531,530,529,527,527,526,526,526],[566,565,564,565,566,566,567,569,571,574,576,579,581,582,582,582,580,578,576,574,572,570,568,567,568,571,575,577,583,590,600,603],[640,637,636,637,641,645,651,657,660,664,670,673,675,679,681],[664,664,664,664,664,664,663,663,662,661,661,661,662,662],[724,724,724,725,725,725,725,725,725,725,723,722,720,719,718,717,716,715,715,716,718,722,724,726,732,735,739,744,747,749,754,755,756,758,759,759],[738,738,738,739,740,740,739,739,738,737,737,737,737,737,738,739,739,740,741],[766,766,766,767,768,770,771,772,775,777,778,781,782,784,785,785,785,784,782,781,779,776,775,774,772,772,773,774,775,780,782,791,793,801,805],[106,105,106,108,111,119,124,137,158,163,169,179,184,188,198,207,215,218,223,224,224,224,223,222],[111,110,111,113,118,121,136,142,161,173,183,193,202,209,211,215,217,218,216,214,211],[204,204,202,202,201,201,200,201,203,205,208,212,214,215,220,222,226,229,232,232,231,229,226,223,219,217,215,211,208,206,206,206,206,206,206,208,211,213,214],[354,353,352,351,350,350,353,357,362,372,379,387,393,396,401,404,405,405,405,404,403,399,397,392,386,381,375,368,364,362,361,361,365,369,375,378,381,388,392,396,402,405,412,413,414,412,410,408],[372,371,373,377,382,388,396,401,404,405,404,404,403,402],[319,318,317,317,318,319,320,321,323,324,325,326,326,326,326,325,325,325,325,326,328,330,333,335],[433,433,433,433,433,433,433,433,433,433,433,433,434,434,435,435,436,436],[502,501,497,496,497,499,503,505,510,513,515,520,522,525,529,530,532],[501,500,500,499,498,499,500,504,506,513,516,523,530,537,542,545],[657,657,658,658,657,655,654,652,648,646,639,637,633,629,626,625,624,624,625,625,625,625,625,624,624,623,623,623,623,624,625,626,628,629,630,632,634,635,636,638,640,644,647,651,654,658,660,662,663,663,663,660,658,657,653,651,649,645,643,642,640,638,637]],"y":[[149,149,148,148,148,148,149,149,149,149,148,147,146,145,145,144,144,143,144,145,146,149,153,158,163,169,179,185,191,193,198,203,206,207,208,209,209,209,208,208,206,204,202,200,199,198,198,197,197,197,197,197,198,198,199],[177,177,177,177,176,174,171,168,167,165,163,162,161,161,160,161,162,162],[134,132,131,131,132,135,137,140,147,151,160,169,177,184,191,194,199,203,206,208,209,210,209,207],[148,147,145,143,142,140,139,140,141,143,145,153,163,175,182,196,208,216,226,230,233,235,236,237,237,237],[161,161,161,161,161,161,161,160,160,158,158,156,155,154,154,154],[186,186,186,185,185,184,183,182,181,180,179,178,178],[208,208,207,207,206,206,205,205,204,203,202,202,202,201,201,202,202,203,205,208,212,216,218,220,224,226,228,231,234,236,237,239,240,240,240,240,240,240,238,235,234,230,228,222,216,208,204,199,190,185,181,171,167,158,148,140,136,127,119,113,110,107,102,101,98,97,96,96,95,95,96,97,98,99,99,98,97,96,95,94,94,94,94,94,94,94,94,93,92,90,89,89,87,87,86,85,85,84,84,84,84,84,85,85,86,87,88,89,90,91,92,93,93,93],[176,175,174,173,171,170,169,168,167,166,165,163,163,162,161,160,160,160,160,162,164,166,169,173,176,178,180,182,183,184,186,186,187,187,188,188,189,190,192,193,196,199,203,207,211,215,218,220,222,222,223,223,223,223,221,220,219,216,214],[141,140,136,133,132,131,130,129,127,126,126,126,126,129,132,136,139,142,146,149,151,154,155,156,156,156,155,154,152,150,147,147],[184,183,183,183,183,183,183,182,182,182,181,181,181,181,180],[172,171,170,169,170,176,181,187,193,199,204,207,210,211],[162,160,159,159,160,162,163,167,169,172,177,180,187,190,192,196,198,199,200,200,200,199,198,197,196,195,194,192,191,190,189,188,188,187,187,186],[179,176,175,175,175,178,183,188,193,200,206,209,211,216,219,221,222,222,222],[146,144,142,141,140,138,137,136,134,133,132,131,131,131,132,134,136,139,143,144,147,150,152,154,156,157,157,157,157,157,157,156,155,154,152],[351,351,351,351,350,350,349,348,347,347,347,347,347,347,347,347,347,347,348,349,350,351,352,352],[390,390,390,390,390,390,390,389,386,384,382,381,380,380,380,379,380,381,383,384,386],[329,328,328,327,327,326,326,327,329,331,333,336,337,338,341,342,345,348,354,358,363,370,376,384,392,397,401,409,417,423,426,428,431,432,433,433,432,430,429],[337,337,337,336,336,335,335,335,334,334,334,335,336,336,337,338,340,341,342,343,345,348,350,355,360,365,369,376,379,381,382,383,382,380,378,378,377,375,374,373,373,372,372,372,373,374,374,375],[356,356,356,356,356,355,353,352,351,351,351,352,352,352],[321,320,320,319,320,324,326,328,335,340,351,364,376,389,395,405,414,421,426,428,431,432,433,432],[317,316,317,320,327,338,352,365,376,387,395,406,409,411,414,416,417,418],[343,343,343,343,343,343,343,343,344,344,344,344,345,345,345,345,345],[365,365,366,366,366,366,366,366,366,364,364,362,361,359,357,357],[314,313,312,311,311,312,312,313,314,315,317,318,319,319,320,320,320,319,319,320,323,326,329,335,341,351,357,362,366,369,370,371,371,370,368,366,364,362,361,360,359,357,357,357,357,360,362,366,371,374,380,389,392,395,401,403,406,410,412,414,416,417,415]],"t":[[0,10,14,16,56,63,73,80,89,96,106,113,123,130,139,146,156,163,188,197,205,213,222,230,240,246,255,265,272,280,289,299,308,314,322,330,339,339,347,357,363,373,380,381,389,398,399,407,416,425,431,432,448,455,464],[733,740,743,747,756,763,772,780,789,789,797,810,814,825,830,855,866,873],[1256,1260,1267,1272,1298,1306,1314,1315,1323,1330,1339,1347,1356,1364,1375,1375,1381,1391,1398,1409,1415,1425,1432,1441],[1911,1914,1918,1920,1924,1938,1955,1963,1964,1973,1974,1981,1990,1998,2006,2014,2023,2030,2040,2049,2056,2064,2065,2073,2080,2090],[2676,2681,2685,2690,2698,2724,2731,2740,2740,2748,2757,2764,2773,2780,2790,2797],[2921,2927,2950,2956,2964,2964,2974,2981,2981,2991,2998,3007,3015],[3493,3501,3505,3515,3532,3548,3557,3565,3575,3581,3590,3598,3607,3615,3623,3631,3640,3641,3648,3657,3666,3675,3682,3682,3692,3698,3699,3711,3715,3725,3731,3745,3747,3748,3757,3765,3774,3781,3790,3798,3807,3814,3815,3824,3831,3840,3848,3849,3857,3864,3865,3874,3881,3889,3898,3906,3908,3915,3924,3932,3940,3942,3948,3957,3964,3974,3981,3982,3991,3998,4007,4025,4042,4049,4116,4134,4140,4157,4175,4189,4198,4210,4220,4224,4232,4241,4248,4257,4264,4274,4282,4284,4291,4299,4301,4308,4315,4324,4331,4341,4348,4357,4365,4374,4382,4391,4398,4408,4415,4424,4432,4441,4448,4464],[4965,4969,4975,4990,4999,5008,5016,5016,5025,5032,5033,5043,5049,5049,5060,5065,5077,5081,5082,5091,5099,5108,5121,5124,5132,5141,5148,5149,5159,5165,5175,5183,5191,5199,5199,5208,5215,5225,5231,5232,5242,5249,5260,5265,5278,5282,5294,5299,5309,5315,5316,5328,5332,5333,5342,5349,5350,5359,5365],[5604,5609,5613,5617,5624,5632,5633,5642,5649,5658,5665,5675,5682,5692,5699,5711,5721,5725,5733,5742,5749,5759,5766,5775,5791,5800,5810,5815,5826,5832,5843,5849],[6142,6146,6150,6175,6183,6192,6199,6212,6216,6217,6226,6233,6233,6247,6250],[6377,6382,6386,6391,6432,6442,6449,6462,6466,6476,6482,6496,6499,6511],[6788,6794,6801,6816,6825,6833,6833,6843,6849,6850,6860,6866,6876,6882,6883,6893,6900,6909,6921,6926,6933,6942,6950,6950,6960,6966,6966,6976,6983,6983,6993,6999,6999,7009,7015,7026],[7180,7187,7195,7200,7224,7233,7243,7250,7259,7266,7276,7283,7284,7294,7300,7309,7315,7316,7327],[7550,7554,7558,7563,7566,7576,7583,7584,7593,7600,7601,7611,7616,7626,7633,7643,7649,7660,7666,7666,7677,7683,7693,7699,7710,7717,7727,7733,7734,7744,7749,7760,7766,7777,7785],[45275,45322,45481,45489,45492,45498,45508,45513,45526,45531,45532,45542,45548,45548,45559,45568,45575,45580,45591,45597,45608,45614,45631,45640],[45989,45993,46014,46015,46025,46030,46042,46047,46058,46065,46075,46082,46091,46098,46108,46114,46125,46131,46142,46148,46158],[46506,46510,46514,46516,46523,46531,46542,46573,46582,46590,46598,46607,46615,46615,46626,46631,46642,46647,46658,46665,46675,46682,46691,46698,46708,46715,46715,46725,46732,46741,46748,46749,46759,46765,46765,46776,46791,46800,46808],[47331,47339,47343,47348,47357,47389,47401,47408,47415,47426,47432,47442,47449,47459,47464,47476,47481,47482,47493,47499,47500,47510,47515,47526,47531,47542,47548,47559,47565,47575,47582,47592,47608,47615,47626,47632,47635,47644,47649,47650,47660,47664,47675,47681,47691,47714,47725,47731],[47937,47941,47975,47981,47991,48002,48009,48015,48026,48032,48056,48059,48065,48066],[48347,48354,48358,48366,48383,48391,48399,48400,48411,48415,48426,48431,48443,48448,48449,48461,48466,48476,48482,48493,48498,48510,48515,48516],[48851,48877,48882,48892,48903,48910,48915,48926,48931,48943,48948,48960,48966,48967,48978,48983,48983,48994],[49344,49350,49354,49361,49383,49390,49398,49399,49411,49416,49417,49428,49433,49434,49444,49449,49450],[49564,49568,49572,49577,49583,49599,49600,49610,49615,49627,49632,49644,49649,49660,49665,49668],[49994,49998,50004,50016,50033,50041,50050,50051,50061,50065,50077,50083,50094,50103,50111,50116,50127,50143,50149,50191,50200,50209,50215,50227,50232,50244,50250,50260,50266,50277,50285,50294,50309,50317,50327,50333,50344,50350,50351,50362,50366,50378,50384,50394,50404,50411,50416,50427,50433,50444,50449,50461,50467,50468,50479,50483,50484,50495,50500,50501,50511,50525,50542]],"version":"2.0.0"}

    Thus, the modulus of z=3+4i is 5

    A relation between a complex number, its conjugate, and its modulus

    Let z=a+ib{"x":[[193,192,191,190,192,194,197,202,207,213,219,223,226,227,227,225,222,219,214,208,203,198,192,189,186,184,184,184,185,187,188,193,196,199,203,206,210,218,226,232,238,243,247,250,252,253,252],[177,174,177,181,187,194,202,212,220,226,232,235,237,239,240],[305,304,303,303,302,304,306,310,318,321,327,330,333,336,339,343,345],[315,314,315,317,319,320,324,329,335,341,346,350],[442,443,443,443,443,443,442,440,437,431,429,427,425,421,418,416,415,414,414,415,418,420,423,425,430,434,437,441,444,446,448,448,448,449,449,449,449,449,449,450,450,452,453,455],[485,484,486,490,493,499,503,506,513,521,524,527,530,531,530,528],[515,514,513,512,511,510,509,507,505,504,504,504,504,504,504,504,504,505,507,509],[599,600,600,599,598,597,597,596,596,596,597,599,601,602,606,608,611,614,617,619],[624,620,619,619,619,619,620],[670,670,671,673,673,673,672,670,668,667,665,664,664,663,663,663,663,663,664,667,669,671,673,676,679,681,682,684,686,688,689,689,689,687,685,683,680,677,673,669,665,662,659,657,656,655,654]],"y":[[172,172,172,172,172,172,172,172,171,170,168,167,167,167,168,170,174,178,183,189,195,201,208,213,218,222,225,226,227,228,228,228,228,228,227,227,226,224,223,222,222,222,222,222,223,224,225],[202,203,203,203,202,200,198,195,192,189,188,187,186,186,186],[182,182,182,181,181,180,179,178,176,175,174,173,173,172,172,171,171],[201,201,201,200,199,199,197,194,192,189,187,185],[176,174,173,172,170,169,167,166,165,165,166,168,169,173,178,183,187,189,193,194,195,195,194,194,190,188,185,182,179,177,177,179,183,189,194,196,200,201,203,205,207,207,207,206],[187,187,186,186,185,184,184,183,182,181,180,179,179,178,178,178],[170,170,169,169,169,170,172,177,185,193,196,202,205,208,212,215,218,220,220,221],[163,162,164,168,171,174,177,181,185,192,195,200,202,203,206,206,206,206,204,200],[148,145,145,146,147,148,149],[134,133,131,128,129,132,135,144,151,160,169,177,184,189,193,196,198,199,199,196,193,190,188,186,183,182,181,181,181,182,184,187,189,192,195,198,201,204,206,209,211,211,211,210,208,205,204]],"t":[[0,9,19,49,74,83,91,100,108,117,124,135,142,152,162,167,174,185,192,201,213,217,224,237,245,250,265,267,271,280,280,288,289,296,297,304,308,317,324,336,341,350,358,367,374,384,400],[672,678,699,708,717,725,734,741,752,758,770,775,785,791,802],[1144,1150,1155,1157,1165,1183,1191,1201,1208,1217,1225,1235,1235,1236,1243,1251,1258],[1398,1404,1417,1425,1435,1435,1442,1450,1458,1467,1475,1482],[1853,1859,1863,1872,1879,1883,1892,1901,1909,1918,1925,1925,1935,1942,1951,1960,1968,1975,1985,1992,2002,2008,2019,2025,2034,2042,2057,2059,2071,2075,2084,2101,2108,2118,2125,2135,2142,2142,2155,2158,2170,2176,2185,2193],[2449,2459,2469,2476,2485,2493,2502,2502,2508,2518,2526,2535,2542,2553,2576,2588],[2717,2723,2730,2737,2751,2759,2769,2775,2785,2793,2802,2809,2810,2819,2827,2835,2842,2852,2859,2870],[3240,3249,3276,3284,3292,3293,3302,3310,3310,3319,3320,3328,3337,3337,3347,3352,3359,3369,3376,3383],[3493,3500,3505,3526,3543,3559,3562],[3863,3877,3882,3884,3910,3919,3926,3935,3943,3952,3959,3969,3978,3986,3993,4002,4009,4019,4026,4043,4053,4059,4073,4076,4086,4093,4093,4103,4109,4119,4126,4135,4143,4153,4159,4169,4177,4186,4193,4203,4209,4219,4226,4236,4243,4253,4254]],"version":"2.0.0"} be a complex number, then the conjugate of it is z¯=a-ib{"x":[[244,245,248,252,258,264,271,278,284,290,296,300,304,308,311,313,315,315,315,313,311,308,304,299,294,289,283,277,271,266,260,255,251,247,244,241,239,238,238,238,240,242,246,250,255,260,266,273,280,287,293,300,305,310,314,317,319,321,321,320,318,316],[241,240,241,245,250,257,265,275,283,291,299,305,311,316,320,322,324],[245,244,247,251,258,266,274,282,289,296,301,305,310,313,315,317,318,319,318],[377,377,378,380,383,388,396,403,410,416,422,427,431,433,435,436,435,435],[374,375,378,384,392,402,409,416,423,427,430,432,431,430],[513,515,515,515,515,514,513,510,507,503,499,494,490,487,483,480,477,475,475,475,476,478,480,482,485,488,492,495,499,504,508,512,517,520,522,524,524,524,524,524,525,526,527,528,530,532,534,535,536,537],[570,569,572,575,580,586,594,602,608,615,619,623,625,626,627,627,626],[678,676,674,674,673,673,673,674,676,678,681,684,687,690,693,696,698,700],[690,690,689,691,693],[753,752,752,751,751,751,751,751,751,752,752,751,751,750,748,747,746,745,744,743,743,745,746,749,752,755,759,763,767,771,774,777,779,780,780,780,780,780,779,776,773,769,764,758,752,746,740,735,731,728]],"y":[[207,207,206,206,205,204,203,202,201,201,200,200,200,200,200,200,201,202,203,205,208,211,215,219,223,228,233,238,244,249,254,259,264,268,272,275,278,280,281,282,282,283,283,282,282,282,281,281,280,280,279,279,279,279,279,279,279,279,280,280,281,282],[249,249,248,248,247,246,245,245,244,243,243,242,241,241,241,241,241],[161,161,161,160,160,159,158,157,156,156,155,155,154,154,154,154,154,154,154],[244,243,243,242,241,240,238,237,235,234,233,232,231,230,230,230,231,232],[218,217,216,215,212,210,208,206,205,204,204,203,203,204],[227,222,220,219,217,215,213,212,211,210,210,211,212,215,218,222,226,231,234,237,239,240,241,241,241,240,238,237,234,231,229,226,223,221,219,219,220,222,224,226,229,231,233,235,236,238,239,239,239,239],[224,224,223,222,221,220,219,218,217,217,216,216,216,216,216,217,217],[213,216,219,221,225,228,232,236,239,242,244,246,247,248,247,246,244,243],[177,176,175,175,176],[168,168,169,169,172,176,182,188,194,200,206,212,218,223,228,232,236,238,241,242,241,239,236,233,229,225,222,219,216,215,215,216,218,221,225,229,233,238,242,246,249,252,254,255,256,255,253,250,247,242]],"t":[[0,30,39,48,55,65,78,81,89,98,105,115,122,131,139,148,155,165,173,182,189,199,206,217,223,232,240,248,255,268,272,281,289,298,305,317,326,331,339,348,357,365,380,380,389,397,406,415,422,432,439,448,455,465,473,483,489,501,506,517,522,533],[841,846,856,865,872,882,889,898,906,915,927,932,939,948,956,965,979],[1473,1480,1506,1515,1523,1532,1539,1549,1556,1565,1573,1582,1589,1600,1606,1615,1623,1632,1649],[2440,2459,2466,2473,2483,2489,2499,2506,2516,2523,2533,2539,2549,2556,2566,2583,2590,2597],[4784,4801,4807,4816,4824,4833,4840,4850,4857,4867,4874,4888,4907,4915],[5368,5374,5378,5383,5391,5400,5407,5417,5424,5434,5441,5450,5458,5467,5474,5488,5491,5500,5507,5517,5524,5534,5541,5551,5558,5567,5574,5585,5591,5603,5608,5618,5625,5637,5641,5651,5667,5675,5685,5691,5700,5707,5717,5724,5734,5742,5751,5758,5767,5773],[6116,6124,6157,6166,6174,6184,6191,6201,6208,6217,6224,6236,6241,6255,6257,6274,6285],[6645,6646,6651,6655,6661,6668,6675,6689,6692,6701,6708,6717,6725,6734,6741,6751,6758,6766],[6932,6938,6944,6958,6968],[7292,7298,7303,7307,7318,7325,7334,7341,7351,7358,7368,7375,7385,7392,7401,7408,7418,7425,7435,7441,7466,7475,7484,7492,7502,7509,7518,7525,7536,7546,7551,7558,7568,7575,7589,7593,7601,7608,7619,7625,7635,7642,7652,7658,7668,7675,7685,7692,7702,7708]],"version":"2.0.0"} (recall the Complex Number article section about conjugates)!

    Step 1: If we multiply the conjugates together, like so:

    zz-=a+iba-ib

    Step 2: And when we expand the brackets, we get

    zz_=a2-abi+abi-i2b2 =a2-i2b2

    And as i=-1{"x":[[290,290,290,289,289,288,287,286,286,286,287,288,289,290,291,292,294,296,299,302,307,312],[297,296,295,295,296,297,298],[360,357,357,359,364,369,376,385,393,401,407,413,418,421,425,426],[371,372,377,382,388,396,402,409,415,420,424,428,431,433,434,434],[499,500,501,502,503,505,507,509,510,511,511,511,511,511,511,512,512,513,514,514,515,516,517,518,519,520,521,522,523,524,526,527,528,530,532,534,536,538,540,542,543,545,546,547,547,547,548,548,548,547,547,547,546,546,545,544,544,543,543,542,542,543,543,544,545,546,548,552,556,561,568,576,584,592,601,610,620,630,639,648,657,664,670,676,680,682,684,685,686,686,685,684],[574,573,572,573,576,580,585,591,599,606,612,616,619,620,619],[645,646,646,646,646,645,645,644,644,644,644,644,644,643,641,640,638,636,635,634,634,634,635,635,636,638,639,640]],"y":[[240,237,238,240,244,248,254,260,266,272,278,283,288,291,294,296,297,297,296,294,289,285],[192,190,187,185,186,187,189],[247,247,246,246,245,244,243,242,241,240,239,238,238,237,237,237],[275,271,270,270,270,269,269,268,268,267,266,264,263,262,262,263],[293,294,294,294,294,294,294,296,298,300,304,307,310,313,315,317,320,321,323,324,325,325,325,325,325,324,322,320,317,314,310,306,301,295,289,282,275,269,262,255,249,242,236,230,225,219,215,210,206,203,200,197,194,192,190,189,188,187,186,186,187,187,188,188,188,189,189,189,188,188,187,187,187,187,187,188,188,189,189,189,189,190,190,191,192,192,193,194,194,195,195,196],[262,262,262,262,262,262,262,261,261,260,260,259,259,258,257],[225,224,222,223,226,228,232,235,239,243,248,252,257,263,269,275,281,286,291,296,299,303,306,308,310,312,312,312]],"t":[[0,8,31,41,48,57,69,75,87,91,102,108,115,125,132,142,148,158,165,173,181,191],[371,377,381,388,402,408,415],[926,930,940,948,957,965,974,982,994,998,1008,1015,1024,1032,1041,1048],[1266,1273,1282,1295,1299,1307,1315,1324,1332,1341,1349,1358,1365,1375,1382,1406],[1921,1926,1933,1942,1949,1958,1965,1975,1982,1991,1999,2008,2016,2025,2032,2041,2049,2058,2066,2075,2082,2092,2099,2108,2125,2132,2146,2149,2162,2165,2175,2183,2192,2199,2211,2216,2225,2232,2242,2249,2259,2266,2275,2282,2292,2299,2308,2316,2325,2332,2342,2349,2359,2366,2379,2382,2394,2399,2411,2416,2473,2483,2492,2510,2516,2525,2532,2541,2549,2557,2566,2575,2582,2592,2599,2609,2616,2626,2632,2642,2649,2659,2666,2675,2682,2692,2699,2709,2716,2726,2732,2740],[3206,3214,3241,3266,3275,3283,3292,3299,3309,3316,3328,3333,3344,3349,3366],[3583,3588,3592,3599,3609,3616,3626,3633,3643,3649,3659,3666,3676,3683,3693,3702,3712,3716,3725,3733,3743,3750,3759,3766,3776,3783,3793,3799]],"version":"2.0.0"}, therefore

    i2=-1{"x":[[267,267,267,267,267,266,266,265,265,265,264,264,264,263,263,263,263,263,264,265,267,268,270,279,282,286,289,291,297,299,302],[275],[323,327,329,332,335,339,343,346,349,351,354,355,355,355,355,354,352,350,348,345,343,341,339,338,338,339,342,346,351,357,363,369,377,383,390,395,400,402,404,405,406,406,406,404],[419,421,425,433,441,451,460,467,474,478,481,485,485,486,486],[436,445,452,460,480,488,496,502,505,507,506,503,500],[568,573,577,584,593,601,610,618,624,629,632,633,632,631,630,626],[680,682,682,682,681,679,677,675,674,672,671,670,669,668,667,666,666,666,666,666]],"y":[[235,243,247,251,258,265,273,282,290,298,305,313,319,325,330,334,338,341,342,343,344,344,344,342,339,337,334,331,310,306,297],[186],[175,173,173,173,174,177,180,183,186,190,198,202,206,210,214,219,223,226,229,233,235,237,239,240,239,237,235,233,230,227,225,222,220,218,217,217,216,216,216,216,216,217,218,220],[268,266,266,266,265,265,264,263,263,262,262,262,263,265,269],[308,304,304,303,299,298,296,296,295,295,295,298,301],[266,268,268,268,267,265,263,262,260,259,258,258,258,258,258,260],[203,208,215,224,234,244,255,265,275,284,292,299,306,310,314,318,320,322,324,325]],"t":[[0,58,69,76,86,92,103,109,119,125,136,142,153,159,169,175,186,192,204,209,221,225,237,260,269,275,289,292,320,325,357],[630],[1344,1399,1401,1410,1418,1427,1436,1446,1452,1459,1477,1484,1493,1502,1509,1519,1526,1536,1542,1552,1559,1569,1576,1586,1633,1643,1653,1659,1670,1676,1687,1696,1703,1714,1719,1726,1736,1747,1753,1759,1776,1784,1792,1802],[2233,2285,2296,2302,2314,2319,2326,2336,2349,2353,2360,2378,2386,2393,2402],[2608,2643,2653,2660,2680,2685,2693,2702,2710,2719,2760,2769,2776],[3036,3093,3102,3110,3120,3126,3137,3143,3154,3160,3171,3177,3210,3221,3227,3236],[3433,3469,3477,3488,3497,3504,3510,3520,3527,3537,3543,3553,3560,3568,3582,3586,3593,3603,3610,3620]],"version":"2.0.0"}

    Step 3: Substituting for i2=-1{"x":[[317,317,317,315,313,311,309,308,306,305,305,306,307,311,313,315,319,323,326,330,332,335,337,338],[351,349,349,349,349,349,350,351,351,352,354,355,356,357,358,358,359,360,361,361,361,361,361,361,359,358,357,358,359,363,368,375,383,390,398,404,410,414,416,418,419,419,419,419,419,418,417,416,414],[323],[412,411,413,416,422,428,434,440,445,450,453,457,459,461],[433,438,443,450,458,466,472,477,480,482,482,482,482,482,482,482],[542,548,554,563,573,583,591,597,600,601,601,600],[646,645,645,645,645,645,644,643,642,641,640,640,640,640,640,640,640]],"y":[[224,226,230,236,244,253,261,269,283,294,302,303,303,303,301,298,294,290,286,282,278,274,271,268],[170,162,161,160,159,157,154,153,151,150,149,147,147,147,147,150,153,157,162,169,175,181,188,193,197,200,202,202,200,198,195,192,188,184,181,178,176,175,174,174,174,175,176,177,178,180,182,185,187],[198],[238,237,237,237,237,237,237,237,237,238,239,241,243,246],[284,284,282,279,276,273,272,271,270,270,271,273,274,277,279,284],[254,251,251,251,251,250,249,248,248,248,249,249],[211,217,224,234,245,256,266,275,284,291,297,302,305,307,309,310,313]],"t":[[0,49,57,73,79,93,96,106,119,139,155,173,179,189,197,206,213,223,230,240,248,256,264,273],[839,887,897,908,913,930,939,946,956,963,975,979,988,999,1019,1022,1030,1042,1046,1054,1063,1072,1079,1089,1096,1106,1113,1178,1188,1196,1206,1213,1223,1230,1240,1246,1257,1263,1277,1280,1290,1306,1320,1322,1330,1342,1346,1356,1363],[1725],[2197,2246,2255,2264,2273,2281,2290,2297,2307,2313,2323,2330,2340,2347],[2538,2605,2614,2623,2630,2640,2647,2657,2663,2673,2697,2707,2713,2724,2730,2740],[3036,3081,3089,3097,3113,3115,3125,3131,3140,3147,3180,3190],[3404,3449,3456,3464,3473,3480,3490,3497,3507,3514,3524,3531,3541,3547,3555,3564,3580]],"version":"2.0.0"} into the equation in step 2, we get

    zz_=a2--1b2 zz_=a2+b2

    The right-hand side is just the square of the modulus! So, we have

    zz_=z2

    In other words,

    (complex number)×(it's conjugate) = modulus2

    Modulus and Phase Angle

    Let us go back to the geometric view of the complex number,

    Modulus and Phase, phase of a complex number, StudySmarter The phase of a complex number is the angle between the positive x-axis and the terminal arm, StudySmarter Originals

    Let θ{"x":[[427,422,421,420,418,415,412,410,408,406,405,404,404,404,404,404,404,404,406,408,410,413,416,419,422,425,428,431,435,438,441,445,448,451,454,457,459,461,463,464,464,464,464,463,454,451,448,444,441,438,435,431,427,423,419,414,410,407,403,400,398,397,397,397,397,397,398,400,404,409,414,420,427,434,441,447,453,459,464,469,474,479,484,488,492,496,498,500,503,504]],"y":[[206,210,214,219,226,234,242,251,261,269,277,283,290,297,302,308,313,317,320,323,325,327,328,328,328,328,326,323,318,314,308,301,294,286,277,269,260,251,243,235,227,221,215,210,195,193,191,190,190,190,190,191,194,199,204,210,216,223,229,236,241,246,250,253,255,259,262,265,267,270,271,272,273,273,273,273,272,270,268,266,263,260,258,255,253,252,251,250,250,250]],"t":[[0,43,55,61,69,78,85,95,102,111,118,128,135,145,152,162,168,179,185,196,202,212,219,233,243,244,253,262,269,278,285,295,302,312,319,329,335,345,356,362,368,379,385,396,424,427,435,446,452,462,469,479,485,496,502,512,518,532,543,544,553,561,569,578,585,595,602,611,619,628,635,645,656,661,669,678,685,695,702,712,719,728,735,745,752,762,769,778,785,796]],"version":"2.0.0"} be the angle made by z with the positive x-axis. We call this angle θ{"x":[[447,440,436,433,430,428,426,423,422,421,419,418,417,417,417,417,418,420,423,426,430,434,439,443,448,452,456,461,465,469,473,476,480,482,485,488,490,491,492,492,492,492,491,489,486,483,479,475,462,457,451,446,441,435,431,426,422,419,417,416,415,415,415,415,416,419,423,428,435,443,451,462,471,479,487,493,498,501,504]],"y":[[212,219,226,236,247,257,267,276,286,294,304,313,322,330,338,345,351,356,360,363,364,365,365,365,363,360,356,350,344,337,330,322,315,306,295,286,277,267,259,251,243,237,232,227,224,221,218,216,214,214,214,218,222,227,232,238,244,249,255,260,264,269,274,278,282,285,286,287,287,287,287,286,284,282,281,280,280,280,280]],"t":[[0,26,34,44,53,61,68,77,84,94,101,109,117,127,134,144,151,161,175,177,184,194,201,211,217,228,236,244,255,261,267,277,285,293,301,311,318,327,334,344,353,361,367,378,384,394,401,411,439,442,451,461,474,477,484,496,501,511,517,527,536,544,556,560,568,577,587,594,601,610,618,627,634,644,653,660,668,677,684]],"version":"2.0.0"} the phase angle of z, also known as the argument of z. In polar form, z is expressed as,

    z=rcosθ+isinθ

    and from the graph, we can use trigonometric ratios to say that,

    tanθ=ba{"x":[[219,217,216,216,216,216,214,212,211,208,207,205,202,200,199,198,197,197,197,197,197,197,198,198],[189,187,184,185,186,189,193,198,203,209,216,222,229,234,239,242,245,247,248],[261,262,263,264,264,265,265,265,266,266,266,266,266,265,263,261,259,254,251,248,244,241,238,236,236,236,238,239,241,243,245,247,250,252,255,257,260,261,262,263,263,264,265,266,266,268,269,270,272,273,274,276,277,279,280,282,283,285,287,288,288,289,289,289,288,288,288,287,287,286,286,286,286,287,287,288,289,290,292,294,297,299,302,304,309,310,312,313,315,316,316,317,317,317,318,318,319,321,323,326],[367,366,365,365,363,362,360,359,357,357,356,356,357,359,362,365,369,373,377,381,385,389,393,397,400,404,404,405,405,404,402,399,396,391,388,383,378,374,370,367,366,366,368,371,376,381,387,393,400,406],[464,463,460,459,462,464,468,471,475,479,483,487,490,493,495,496,497,497],[447,446,444,446,450,456,462,470,478,484,490,496,499,502,505,506],[646,646,646,646,646,646,646,646,646,645,644,642,641,640,640,639,639,638,638,638,638,639,640,640,641,641,641,643,644,648,653,657,660,663,667,670,672,674,675,675,674,672,668,664,660,656,650,645,638,631,626,621,618,617,618,619],[588,588,588,588,592,598,606,616,625,635,643,651,659,665,671,677,681,684,685,685,685,684,683],[634,635,636,637,638,639,640,641,641,641,639,636,633,629,625,621,617,614,610,607,605,604,604,605,607,611,614,619,623,628,633,637,642,645,648,651,653,655,655,656,656,657,657,657,657,657,657,659,662,666,672,679,686,690]],"y":[[168,166,165,167,172,179,189,202,215,230,244,257,272,283,294,304,312,318,323,326,329,330,330,329],[230,231,231,231,231,231,231,231,231,230,228,227,225,224,223,222,222,222,222],[286,286,285,285,284,283,282,280,278,275,272,269,266,262,259,256,255,253,255,259,264,270,276,282,286,290,291,291,291,291,290,288,286,284,280,277,275,273,272,272,274,276,279,281,284,286,288,289,291,291,291,291,289,288,286,283,279,275,271,266,263,261,260,261,262,264,268,272,276,280,285,288,291,293,294,294,294,293,290,287,282,278,274,270,266,265,265,267,270,274,279,284,289,294,298,301,303,304,304,303],[225,224,226,227,232,239,247,256,266,275,283,291,297,301,305,307,307,307,306,303,299,294,287,280,272,254,244,234,225,218,211,206,203,202,202,205,209,215,221,229,234,239,243,245,246,247,247,247,245,244],[238,237,235,234,234,234,235,235,236,236,237,238,238,239,239,240,241,242],[274,273,271,271,270,269,269,267,266,265,263,262,261,260,260,260],[161,159,154,153,150,149,150,152,156,162,169,177,185,195,204,213,221,228,233,239,243,247,250,251,252,253,251,246,243,237,232,230,230,230,232,234,237,240,244,248,252,257,261,265,268,270,271,271,270,268,265,262,258,254,250,247],[335,334,332,331,330,329,327,326,325,324,324,323,323,322,322,322,321,321,321,322,323,324,324],[408,408,407,405,404,402,399,396,393,390,384,382,380,379,379,380,383,386,391,396,402,407,410,413,415,415,415,414,412,410,407,403,400,396,393,390,388,387,386,386,388,390,393,397,402,411,415,417,419,419,418,415,411,409]],"t":[[0,13,19,34,41,51,57,67,74,84,91,101,107,117,124,134,142,151,163,167,183,191,201,207],[492,502,507,524,534,541,552,557,566,574,583,591,600,607,617,624,633,641,648],[935,946,952,958,967,977,984,991,1000,1008,1017,1024,1034,1044,1051,1066,1068,1086,1091,1104,1107,1117,1124,1133,1141,1150,1158,1167,1174,1184,1191,1200,1208,1217,1224,1234,1241,1250,1258,1280,1283,1291,1300,1308,1317,1324,1334,1343,1350,1365,1367,1385,1391,1404,1408,1417,1424,1434,1441,1451,1458,1467,1474,1491,1500,1508,1517,1524,1534,1541,1551,1558,1567,1576,1584,1591,1601,1617,1624,1634,1644,1651,1665,1666,1685,1692,1703,1708,1718,1724,1734,1741,1751,1758,1767,1774,1784,1791,1801,1808],[2334,2346,2370,2375,2384,2391,2401,2408,2417,2425,2434,2441,2451,2458,2468,2475,2484,2491,2501,2508,2518,2525,2535,2545,2551,2571,2580,2590,2592,2605,2608,2618,2625,2635,2642,2651,2658,2668,2675,2684,2692,2701,2708,2717,2725,2734,2742,2751,2758,2768],[3173,3174,3180,3194,3212,3217,3225,3235,3242,3251,3258,3268,3275,3285,3292,3302,3309,3317],[3477,3492,3497,3510,3518,3525,3535,3542,3552,3558,3568,3575,3585,3592,3602,3608],[4118,4121,4126,4130,4134,4142,4159,4168,4175,4185,4192,4202,4209,4219,4225,4235,4242,4252,4259,4269,4276,4285,4293,4302,4309,4319,4351,4369,4376,4395,4410,4418,4425,4435,4442,4452,4459,4469,4476,4486,4492,4502,4509,4520,4526,4535,4542,4552,4559,4569,4576,4586,4593,4602,4609,4616],[5044,5050,5057,5060,5069,5076,5086,5093,5102,5109,5119,5126,5137,5142,5152,5159,5169,5176,5184,5193,5202,5209,5217],[5504,5515,5534,5555,5560,5571,5576,5586,5598,5603,5619,5626,5635,5643,5653,5659,5669,5676,5686,5693,5703,5709,5720,5726,5736,5743,5753,5759,5769,5776,5787,5793,5803,5809,5820,5826,5837,5843,5855,5859,5870,5876,5886,5896,5903,5918,5926,5936,5943,5953,5959,5969,5976,5984]],"version":"2.0.0"}

    Solving for θ we get

    θ=tan-1ba{"x":[[194,194,193,191,190,189,186,184,181,179,177,174,172,170,168,167,166,166,165,166,167,169,172,175,179,183,189,194,200,206,212,217,223,228,233,239,243,247,250,252,254,254,253,251,248,244,240,236,231,226,220,215,209,202,197,191,186,182,179,176,173,173,174,176,180,185,191,198,206,214,223,232,242,252,260,268,274,277],[349,348,346,342,339,336,336,338,341,345,351,357,363,370,378,385,390,394,396,397,398,398,396],[338,338,340,341,344,350,357,364,371,378,385,392,398,402,406,408,408,407,406,403],[502,502,502,501,499,496,494,492,490,486,484,483,482,482,482,481,481,481],[467,467,467,467,468,471,476,483,498,513,519,525,529,531,533,533],[569,570,571,572,573,574,574,574,574,573,572,569,566,563,559,555,551,547,542,538,534,531,527,526,525,524,525,527,529,532,536,540,550,556,561,566,572,576,578,581,582,583,583,583,583,583,583,583,583,584,585,587,590,593,596,598,601,604,606,608,610,611,612,613,614,614,615,615,615,615,614,614,613,613,612,611,610,610,609,609,609,609,609,609,610,612,615,617,620,623,626,629,632,634,637,639,641,643,645,647,647,647,647,647,648,650,652,655],[628,628,629,631,634,638,643,649,655,662,667,672,676,679,680,681,680,678,677],[691,692,693,694,694,694,694,694,693,692,692,691,691,691,691,691,691,691,691],[785,785,785,785,785,784,783,782,780,778,777,775,774,773,772,771,771,770,770,770,770,772,773,776,778,781,783,786,788,790,793,795,797,799,800,800,799,797,794,790,787,784,781,779,778,777,776,775,775,775],[748,750,753,757,763,770,778,787,795,804,813,820,826,830,832,833,832,830,828],[793,793,794,796,798,799,800,800,800,799,797,795,791,787,782,777,771,766,762,758,756,754,754,754,756,758,760,764,768,771,776,780,786,791,795,799,802,804,805,806,806,806,807,807,807,807,807,808,809,810,812,814,815],[728,729,729,728,727,725,723,721,719,716,713,710,708,706,704,703,701,699,698,698,698,699,701,703,705],[868,869,870,872,873,875,875,877,877,878,879,879,880,880,879,878,878,875,874,869,865,861,859,857,853,850,847,845,843,841]],"y":[[238,239,240,243,245,250,256,262,269,276,283,290,298,306,314,321,329,335,342,348,352,356,359,361,363,363,363,361,358,354,349,344,337,330,322,313,305,296,287,280,272,265,258,252,246,242,238,235,233,232,232,234,237,241,245,250,256,262,268,274,285,290,295,298,301,304,305,307,308,308,308,308,307,305,304,303,303,303],[291,291,290,289,289,289,290,290,290,289,289,288,286,285,283,282,282,282,282,283,285,287,289],[324,322,321,321,320,319,319,319,318,317,317,316,315,314,314,314,315,316,318,320],[243,244,248,256,267,279,288,298,308,326,342,349,355,359,363,366,368,369],[305,303,302,300,299,298,298,297,294,291,290,290,290,290,291,292],[346,345,343,341,340,338,336,334,332,329,327,325,323,322,321,321,321,321,323,326,329,333,337,341,344,348,350,352,354,354,355,355,352,350,347,344,341,338,336,335,334,335,337,339,341,344,347,350,353,356,358,359,359,359,358,356,354,351,347,344,340,336,331,328,325,322,320,319,320,321,323,326,330,333,337,341,345,348,351,353,355,356,355,354,353,350,346,343,339,336,333,330,329,329,329,330,332,334,338,346,351,355,359,362,364,366,366,366],[268,267,266,266,265,265,265,264,264,263,262,262,261,261,261,261,261,261,261],[224,220,219,221,223,226,231,236,242,249,255,261,267,272,276,279,282,285,287],[241,240,237,236,238,241,245,250,255,261,267,273,279,284,289,293,297,300,303,305,306,305,304,302,299,297,295,294,293,293,294,296,299,303,307,311,314,317,320,322,324,326,326,326,325,323,320,316,312,309],[365,365,363,362,362,362,362,362,362,361,361,360,359,359,359,359,359,360,361],[425,424,423,420,419,417,415,413,411,409,406,405,403,403,402,403,405,407,411,414,417,421,424,427,430,432,433,433,433,432,431,428,425,422,418,415,412,409,408,407,408,409,412,415,419,424,428,433,437,440,443,445,446],[294,293,292,293,294,296,299,304,309,317,326,336,347,358,369,380,391,403,415,426,436,447,455,464,468],[282,282,282,282,283,287,290,296,300,304,313,317,326,341,346,351,356,367,372,389,401,411,416,420,430,437,444,447,450,452]],"t":[[0,9,15,19,23,30,39,48,55,65,72,82,89,98,106,115,123,132,143,148,163,165,178,181,196,198,205,215,222,231,239,248,256,265,272,282,289,298,306,315,322,331,339,348,356,365,372,382,389,398,406,415,422,432,443,448,461,465,479,482,498,506,515,522,532,539,548,556,565,572,582,589,598,606,615,622,632,639],[1034,1043,1049,1052,1066,1081,1100,1114,1123,1132,1139,1149,1156,1165,1173,1182,1189,1199,1206,1215,1223,1232,1239],[1438,1447,1452,1456,1465,1473,1482,1489,1499,1506,1515,1523,1532,1540,1549,1556,1573,1582,1590,1599],[1893,1908,1915,1925,1933,1940,1949,1956,1967,1982,2002,2007,2015,2023,2032,2040,2049,2055],[2221,2227,2232,2234,2240,2249,2256,2266,2285,2301,2308,2316,2323,2333,2340,2348],[2639,2646,2651,2653,2658,2666,2673,2683,2690,2700,2707,2717,2723,2733,2740,2750,2757,2767,2774,2783,2790,2800,2807,2817,2823,2833,2840,2848,2857,2868,2881,2881,2901,2906,2916,2923,2933,2940,2950,2957,2966,2983,2990,3000,3007,3016,3023,3033,3040,3050,3057,3066,3073,3083,3090,3100,3107,3117,3123,3133,3140,3149,3157,3167,3181,3183,3201,3207,3223,3233,3240,3250,3257,3266,3273,3283,3290,3300,3307,3316,3323,3333,3350,3358,3367,3373,3383,3390,3400,3407,3417,3423,3433,3440,3450,3457,3467,3480,3482,3501,3507,3516,3523,3533,3540,3550,3557,3567],[4012,4024,4033,4040,4052,4057,4069,4081,4083,4090,4102,4107,4116,4123,4133,4140,4157,4167,4173],[4586,4593,4601,4616,4624,4634,4641,4650,4657,4671,4677,4689,4692,4704,4707,4717,4724,4734,4741],[5211,5218,5224,5227,5241,5251,5258,5272,5279,5291,5291,5306,5308,5317,5324,5334,5341,5351,5357,5368,5374,5407,5417,5424,5435,5441,5451,5458,5468,5474,5485,5491,5501,5508,5518,5525,5535,5541,5551,5558,5572,5576,5588,5591,5606,5609,5617,5624,5634,5641],[6027,6034,6040,6043,6050,6058,6069,6075,6085,6091,6102,6108,6119,6125,6135,6142,6158,6168,6177],[6539,6544,6549,6551,6558,6568,6575,6585,6591,6602,6608,6618,6625,6635,6642,6652,6658,6669,6675,6686,6691,6702,6708,6719,6725,6736,6741,6752,6762,6769,6777,6789,6792,6807,6810,6818,6825,6835,6842,6852,6875,6885,6891,6902,6908,6919,6925,6933,6941,6952,6958,6967,6974],[7430,7443,7466,7491,7501,7508,7519,7525,7535,7541,7552,7558,7569,7575,7585,7592,7603,7608,7619,7625,7636,7644,7653,7664,7666],[8052,8059,8064,8066,8071,8078,8085,8096,8097,8107,8109,8119,8125,8136,8143,8143,8153,8159,8167,8175,8185,8192,8192,8203,8209,8220,8225,8226,8237,8241]],"version":"2.0.0"}

    IMPORTANT! For the purposes of this article, the -1 on tan is not an exponent (i.e., it is not the reciprocal of tangent - cotangent). This is used because it represents the inverse tangent function, arctangent, on most calculators.

    An alternative way to calculate the phase angle is using sine and cosine.

    sinθ = opposite sidehypotenuse

    Subbing in variables from figure 2 above, we get:

    sinθ = br

    We just learned that r is the modulus, a2+b2. Substituting this expression for r in the equation above, we get

    sinθ=ba2+b2{"x":[[142,142,147,148,143,138,133,128,124,121,120,121,125,130,143,149,153,155,155,154,151,147,142,137,132,127,123,121,120,120],[175,174,173,172,171,171,171,172,173,174,175,177,177,177,177,177,177,176],[172,173,174,174,175,176,176,176],[206,205,204,204,205,205,206,206,206,207,207,206,206,206,206,208,211,214,218,222,226,228,230,231,232,232,232,233,233,233,233,234,234,234,235],[271,270,270,270,270,270,270,270,269,269,269,271,273,274,277,280,285,289,292,294,296,297,297,297,295,292,288,284,279,276,272,270,269,270,271,274,278,283,287,293,298],[348,346,347,349,353,359,364,370,377,379],[334,333,338,339,344,351,355,358,365,368,370,375,376],[557,557,557,557,557,557,557,558,558,558,558,558,558,557,556,555,554,554,554,554,556,558,561,564,567,571,573,575,576,576,576,575,572,569,565,560,556,553,551,550,550],[433,432,430,429,430,439,445,449,455,468,476,485,504,522,541,561,580,598,633,648,655,661,672,677,681,688,691,696,697,698,698,696],[425,424,424,425,426,426,427,428,429,432,433,434,435,436,438,440,442,444,445,446,448,448,449,449,449,449,449,450,450,450,450,449,449,449,449,450,450,451,451,452,452,452,452,452,452,453,453,453,453],[449,448,447,446,447,448,451,455,468,473,479,492,506,513,522,537,545,554,572,580,589,608,618,646,656,701,717,731,743,753,762,768,770,774,776,774,772],[513,514,515,516,516,517,518,518,518,516,514,511,507,502,498,493,489,486,486,486,487,490,491,492,495,497,499,502,504,507,509,510,512,514,516,517,518,520,521,522,522,523,523,523,524,524],[544,542,542,542,542,542,542,543,546,548,550,553,555,556,556,557,556,556,555,553,552,551,549,549,548,548,549,551,554,561,567,572,575],[606,607,609,613,619,623,626,633,638,640],[620,619,619,621,623,624,625,627,628,630,630,631,630,629,628,628],[673,673,673,673,673,673,673,672,670,669,669,668,668,668,669,670,671,672,677,679,684,687,688,691,692,694,696,696,696,695,693,693,691,689,685,682,680,679,674,673,671,670],[718,717,716,715,714,713,712,712,712,713,714,717,720,723,725,726,728,728,729,728,725,722,719,716,715,716,718,722,724,727,733,736]],"y":[[188,187,179,178,173,173,176,179,184,188,191,194,196,197,200,201,203,205,207,210,213,216,219,221,223,224,224,223,222,221],[181,181,181,181,180,181,184,185,188,193,196,216,220,223,230,236,242,242],[162,160,160,159,159,159,160,162],[191,191,189,190,192,195,199,204,208,213,217,221,222,220,217,213,207,201,195,191,189,189,192,195,200,205,210,215,220,227,230,233,234,235,235],[179,178,176,175,176,180,186,192,199,207,214,219,223,225,225,225,222,218,213,210,203,196,187,178,171,164,160,158,158,160,163,168,173,178,183,187,190,193,195,195,194],[179,179,179,179,179,178,178,178,178,178],[205,205,205,204,203,201,200,199,199,198,198,198,198],[137,136,134,132,131,132,135,139,145,152,159,167,175,180,184,186,187,186,185,182,179,176,172,169,167,167,167,169,173,177,180,184,188,191,193,195,196,196,194,192,189],[242,242,242,241,241,241,241,240,240,240,239,239,238,237,236,236,235,235,235,235,235,235,235,235,235,235,234,234,234,234,233,233],[343,343,342,342,344,345,347,351,354,362,367,371,373,373,373,371,368,363,361,358,351,347,338,329,324,315,311,301,295,291,287,284,282,280,282,286,289,292,299,306,313,316,323,328,332,334,336,339,341],[275,275,275,275,275,275,275,275,274,273,272,271,270,269,269,268,268,267,267,266,266,265,265,264,264,263,263,263,263,263,263,263,263,263,264,264,265],[332,331,329,329,328,326,323,320,319,317,315,315,315,317,321,326,333,341,345,348,349,349,349,349,347,346,345,342,340,337,336,334,332,330,328,327,327,329,331,334,336,338,339,341,343,344],[289,286,284,283,282,281,280,280,279,278,278,280,282,285,287,290,296,298,300,303,305,306,308,309,309,310,310,310,310,308,306,304,303],[321,321,320,320,319,318,318,317,316,316],[310,309,308,308,308,309,310,312,313,319,321,329,334,339,342,344],[292,291,287,285,284,283,287,294,307,317,320,323,328,334,335,335,335,335,330,327,323,321,321,319,319,320,323,325,327,331,335,336,337,339,340,340,340,340,338,337,333,327],[287,286,285,283,282,280,278,276,275,275,275,276,278,279,280,282,284,286,290,292,297,299,301,302,303,302,302,301,301,301,300,299]],"t":[[0,0,11,13,56,58,66,75,85,91,102,108,118,125,149,150,158,168,175,186,191,201,208,217,225,234,242,251,258,261],[412,419,424,426,434,451,459,467,468,476,484,502,509,509,517,526,544,552],[684,694,698,701,708,720,742,745],[1122,1140,1147,1167,1175,1185,1192,1202,1208,1219,1225,1246,1253,1270,1275,1285,1292,1301,1309,1318,1325,1335,1342,1352,1359,1368,1375,1385,1392,1401,1409,1418,1426,1427,1435],[1646,1661,1666,1669,1700,1709,1718,1725,1738,1745,1752,1760,1768,1774,1776,1785,1794,1802,1810,1818,1825,1835,1842,1852,1859,1868,1875,1889,1892,1905,1909,1919,1925,1936,1942,1955,1959,1972,1976,1988,1992],[2323,2338,2367,2376,2385,2392,2404,2409,2419,2420],[2557,2573,2579,2585,2593,2602,2610,2610,2619,2627,2628,2636,2642],[3214,3222,3228,3230,3242,3268,3276,3286,3293,3302,3311,3319,3328,3336,3343,3353,3359,3376,3386,3393,3402,3409,3419,3426,3436,3443,3453,3459,3469,3476,3486,3493,3503,3509,3520,3526,3536,3543,3552,3559,3569],[4022,4033,4038,4053,4110,4128,4140,4143,4143,4153,4159,4160,4172,4176,4187,4193,4203,4210,4227,4238,4243,4244,4254,4260,4261,4268,4276,4286,4293,4294,4303,4310],[4977,4996,5007,5027,5035,5043,5044,5054,5060,5070,5077,5087,5093,5103,5110,5120,5127,5137,5144,5148,5154,5160,5170,5177,5177,5187,5193,5203,5211,5220,5227,5237,5244,5253,5278,5287,5294,5295,5306,5311,5322,5327,5338,5343,5353,5357,5361,5370,5377],[5623,5637,5643,5645,5653,5662,5670,5678,5697,5702,5703,5711,5719,5727,5728,5736,5744,5745,5755,5761,5762,5771,5777,5787,5793,5813,5822,5828,5838,5845,5859,5862,5870,5877,5887,5903,5911],[6620,6632,6638,6644,6644,6655,6662,6671,6677,6688,6694,6705,6711,6721,6727,6738,6744,6755,6762,6771,6778,6789,6795,6795,6809,6811,6812,6822,6827,6837,6843,6844,6845,6855,6862,6871,6878,6890,6895,6906,6912,6920,6928,6929,6938,6945],[7093,7108,7114,7115,7127,7128,7141,7144,7155,7162,7172,7178,7188,7195,7205,7211,7222,7229,7229,7236,7246,7247,7255,7262,7262,7272,7278,7288,7294,7305,7312,7321,7328],[7604,7636,7637,7644,7656,7662,7663,7673,7678,7688],[7809,7817,7821,7846,7854,7861,7862,7872,7878,7888,7894,7905,7912,7922,7928,7938],[8117,8126,8132,8134,8140,8149,8178,8200,8212,8222,8231,8232,8240,8255,8263,8272,8278,8295,8315,8323,8328,8339,8345,8358,8361,8371,8378,8388,8394,8405,8421,8428,8429,8440,8445,8455,8466,8466,8472,8480,8488,8495],[8649,8657,8663,8665,8671,8678,8689,8696,8705,8722,8729,8739,8748,8755,8762,8763,8773,8778,8789,8795,8805,8812,8822,8829,8839,8855,8862,8872,8879,8880,8889,8896]],"version":"2.0.0"}

    Using Pythagoras theorem, we can determine r in terms of a and b:

    r=a2+b2{"x":[[114,114,121,125,128,134,142,144,150,154,158,162,174,176,179,181,183,185,187,188,188,190,190,190,191,191,190,190,188,187,185,183,182,179,178,176,176,176,176,176,179,181,184,186,188,191,193,196,199,202,206,209,211,218,221,231,234,246,250,254,262,270,278,281,285,292,299,302,309],[324,325,331,333,336,346,353,360,368,371,382,385,394,396,398,403,404],[328,329,335,337,340,351,360,368,377,381,393,397,406,408],[448,449,450,452,454,456,457,466,468,474,476,481,483,486,487,489,491,492,493,495,496,498,499,500,501,503,504,505,507,508,509,513,517,520,524,525,530,532,535,537,538,539,540,540,540,540,539,538,537,536,534,532,530,528,526,526,525,525,527,530,534,537,540,548,568,574,580,587,603,611,619,637,646,667,689,712,737,762,775,815,829,869,883,918,930,940,960,968,984,1004,1009,1021,1024,1027,1028],[654,654,656,656,657,658,658,657,656,651,648,639,636,626,623,620,615,612,610,608,604,603,604,607,610,614,619,624,628,638,645,653,660,667,670,672,677,681,682,685,685,686,688,690,691,696,698,704,706],[699,698,697,698,701,705,710,715,721,726,728,731,734,736,736,735,734,730,727,720,718,712,711,711,712,714,718,721,724,735,744,754,762,767,775],[776,778,782,788,792,796,801,813,817,827,833,838,841],[805,805,807,808,810,814,815,817,818,818,819,820,821,822],[913,915,918,919,919,918,915,912,909,904,902,898,893,892,889,889,889,889,890,892,894,900,902,912,915,919,926,930,933,940,943,945,947,950,950,951,951,949,946,941,938,931,928,919,916,913,907,901,898,896,895],[984,983,979,978,977,977,977,977,978,981,983,988,991,994,995,996,997,997,997,997,995,994,992,992,992,994,996,1004,1007,1020,1024]],"y":[[268,267,262,259,258,253,250,249,249,252,255,261,290,295,305,316,321,327,336,341,347,357,361,366,370,379,383,390,395,399,401,401,401,398,396,386,383,370,359,354,335,328,312,304,296,281,273,260,246,233,221,210,204,191,187,177,174,167,165,164,162,161,161,161,161,161,161,161,161],[255,256,256,255,255,253,252,250,249,248,247,247,246,246,246,246,246],[316,316,316,315,314,311,309,306,303,302,300,299,298,298],[369,370,371,371,371,371,371,372,373,377,379,387,390,396,399,405,412,417,420,431,434,442,447,451,454,456,456,456,454,452,451,442,433,424,413,407,388,381,360,347,332,325,303,295,281,268,261,255,242,236,225,215,205,198,190,186,177,174,166,162,158,156,155,152,148,147,146,146,145,145,144,143,143,141,139,137,136,134,133,132,131,131,131,132,133,134,136,136,138,141,141,143,144,145,146],[354,353,350,348,344,339,337,330,327,321,320,319,320,328,332,336,345,351,356,363,382,388,400,401,401,399,395,390,386,374,365,355,348,344,343,342,344,348,351,358,363,367,382,390,393,398,398,395,393],[252,251,243,241,238,235,231,229,228,228,228,230,233,237,241,248,251,262,267,280,285,296,298,301,304,304,304,304,304,300,297,293,291,290,288],[352,352,351,349,348,347,345,342,341,338,336,335,333],[321,322,328,331,334,344,351,357,360,363,369,374,376,380],[253,251,244,244,245,249,257,265,276,291,299,315,339,345,360,367,371,373,375,375,375,372,370,363,361,358,356,355,355,355,356,358,359,364,366,370,374,379,383,387,389,394,395,397,397,396,394,391,386,378,373],[245,245,241,239,238,236,235,233,232,230,229,229,230,232,234,236,241,244,247,249,257,260,266,269,272,273,273,274,274,272,271]],"t":[[0,10,14,15,30,38,45,51,59,70,75,87,116,118,125,137,142,142,153,159,159,170,175,176,187,192,193,203,209,220,226,234,242,253,259,270,276,287,292,303,309,320,326,326,337,342,343,353,359,370,376,386,392,403,409,420,426,437,442,443,453,459,470,476,476,487,492,503,509],[928,940,943,944,951,959,968,976,986,992,1003,1009,1020,1026,1026,1036,1043],[1233,1242,1245,1247,1251,1259,1270,1276,1284,1293,1303,1309,1320,1326],[1598,1609,1618,1626,1636,1643,1643,1666,1668,1676,1685,1693,1703,1709,1710,1720,1727,1736,1743,1753,1760,1770,1776,1787,1793,1803,1809,1820,1826,1827,1834,1843,1853,1860,1870,1877,1887,1893,1903,1909,1920,1926,1936,1943,1954,1960,1960,1970,1976,1977,1987,1994,2003,2009,2020,2026,2037,2043,2053,2060,2070,2076,2077,2087,2105,2109,2110,2121,2126,2127,2137,2143,2144,2154,2160,2170,2177,2187,2193,2204,2211,2220,2226,2237,2243,2248,2251,2260,2270,2276,2287,2293,2304,2310,2314],[2733,2738,2745,2754,2760,2771,2777,2787,2793,2804,2810,2821,2826,2837,2843,2843,2854,2860,2860,2871,2876,2887,2893,2904,2910,2921,2927,2937,2943,2954,2960,2971,2977,2987,2993,2994,3004,3010,3021,3027,3027,3039,3043,3054,3060,3071,3077,3087,3094],[3275,3286,3289,3294,3304,3310,3321,3327,3338,3343,3344,3355,3360,3371,3377,3388,3394,3404,3411,3421,3427,3438,3443,3452,3460,3471,3477,3477,3488,3493,3504,3510,3521,3527,3536],[3692,3721,3728,3737,3744,3744,3755,3760,3769,3777,3787,3794,3804],[3991,3999,4002,4004,4010,4019,4027,4038,4044,4044,4055,4060,4071,4077],[4284,4294,4298,4303,4311,4322,4327,4338,4344,4355,4360,4372,4377,4388,4394,4402,4411,4421,4427,4438,4444,4455,4461,4472,4477,4478,4489,4494,4494,4505,4510,4511,4522,4527,4528,4539,4544,4555,4561,4571,4577,4586,4594,4604,4610,4611,4622,4627,4640,4644,4655],[4826,4836,4841,4846,4855,4861,4861,4872,4877,4888,4894,4905,4911,4922,4927,4928,4939,4944,4944,4956,4961,4972,4977,4989,4994,5006,5011,5022,5028,5039,5044]],"version":"2.0.0"}

    Substituting for r into cosine, we get (as we did for sine):

    cosθ=aa2+b2{"x":[[152,152,152,152,151,149,147,144,140,136,132,129,127,130,134,139,144,151,157,163,170,174,178,180,180,180,179,178,176,175,175,177,178,179,181,183,184,186,187,188,189,190,191,192,193,193,193,194,194,193,191,190,188,186,185,184,184,184,186,188,192,197,203,209,215,222,227,232,236,239,241,242,242,241,240,237,234,231,227,225,223,223,225,229,233,239,243,247,249,249,249,247,244,241,237,235,233,233],[291,290,290,289,289,288,287,286,284,283,283,283,284,286,289,292,296,299,302,305,307,308,309,308,305,302,299,295,292,289,286,284,282,282,283,287,291,298,307,315,324],[370,367,364,362,363,366,370,377,383,390,393],[359,356,354,356,359,364,371,379,389,399,408],[624,624,625,625,625,625,624,621,618,614,609,605,603,599,598,597,597,598,601,604,607,611,614,617,619,621,622,623,624,624,624,624,624,625,626,628,630,633,637,641],[508,507,504,505,508,513,520,524,529,541,548,571,587,606,624,645,667,687,708,729,746,762,775,785,794,800,804,806,807,807,805,801,800],[486,487,488,489,490,491,492,493,493,494,495,496,497,498,498,500,501,504,506,508,510,512,513,514,515,516,517,517,518,519,519,520,520,520,520,519,519,519,519,519,520,521,523,525,528,532,540,550,557,564,581,600,620,641,663,686,708,730,751,772,791,810,826,840,852,861,868,874,877,878,879,879],[599,599,599,599,599,599,598,597,596,595,593,592,588,583,578,573,569,565,563,562,562,562,562,563,566,567,569,573,575,578,583,585,593,598,602,604,606,606,606,606,606,606,606,606,606,606,606,606,607,607,610,611,613,615],[631,630,630,629,629,629,629,629,630,632,635,639,643,647,650,652,653,653,653,650,647,642,641,639,636,635,634,634,635,638,642,648,655,662,668,671],[693,692,690,690,692,695,700,705,711,716,721,725,728],[692,693,694,696,698,699,701,703,703,705,705,705,705,705,705,704,703,703,703],[765,765,764,764,764,764,764,764,764,764,764,764,764,763,762,762,761,762,763,765,766,768,770,773,775,777,778,782,785,788,789,790,790,788,786,782,779,775,772,769,768,767,766],[805,804,801,799,799,800,803,806,810,813,816,818,819,819,819,818,816,815,811,809,808,807,806,807,811,815,822,831,841]],"y":[[209,208,207,206,204,204,203,203,204,208,214,221,247,253,258,261,262,262,261,257,251,247,242,238,234,231,228,226,225,224,226,229,233,237,241,246,250,253,256,256,256,253,250,246,241,236,231,226,221,217,215,214,214,215,217,221,226,231,236,240,244,247,248,248,246,243,240,236,231,226,221,216,212,209,207,205,205,205,207,210,212,215,218,221,224,227,229,232,235,237,240,244,246,249,251,252,252,251],[211,211,210,209,208,208,210,214,221,228,236,245,252,257,260,262,262,261,257,252,245,237,229,221,213,208,204,202,201,201,204,207,212,216,220,223,225,226,226,225,223],[223,223,223,223,223,223,223,224,224,225,225],[246,247,248,248,249,249,248,246,244,243,241],[208,207,206,203,202,200,197,195,194,194,196,199,201,205,210,214,217,219,220,221,220,218,216,214,212,209,207,206,206,207,210,213,217,220,223,225,227,227,227,225],[281,281,280,280,279,279,279,279,279,280,281,282,282,282,282,282,282,282,282,282,282,283,284,285,286,287,288,289,290,291,291,292,292],[421,421,421,421,422,422,423,423,424,425,428,430,431,434,435,435,434,432,429,425,421,417,411,406,400,393,386,378,369,363,360,355,353,350,348,346,345,344,343,344,344,344,344,343,343,343,342,342,342,341,340,339,338,337,336,336,336,336,337,338,339,341,342,343,345,346,347,349,350,351,352,353],[409,408,406,405,404,401,400,398,397,395,394,393,392,392,393,396,400,406,411,413,416,420,422,423,425,425,425,425,424,423,421,419,413,409,405,403,401,403,405,407,409,413,415,416,420,422,423,426,427,428,429,430,430,430],[367,367,366,365,364,363,362,361,361,360,360,360,360,361,363,365,368,370,372,376,381,385,387,388,391,392,393,394,394,394,394,393,392,390,389,388],[411,411,411,410,408,407,405,403,402,401,401,401,401],[397,397,397,397,397,398,399,399,400,403,404,405,409,410,412,415,418,420,422],[373,372,370,369,368,370,374,380,387,390,401,405,408,414,418,421,423,421,419,416,415,414,411,410,409,409,409,410,413,416,417,422,424,427,428,430,430,431,431,430,429,428,427],[377,376,374,372,371,371,371,371,372,374,376,378,381,382,384,387,390,391,395,395,396,397,397,397,397,396,394,391,389]],"t":[[0,12,18,33,36,44,52,61,71,77,87,94,121,127,136,144,154,161,171,178,187,194,204,211,220,227,237,244,254,261,294,303,311,320,327,337,344,354,361,371,378,387,394,404,411,421,427,438,444,454,461,471,478,488,494,504,511,521,528,538,544,554,561,571,577,588,594,604,611,621,628,638,644,654,661,671,678,688,694,704,711,721,728,737,744,754,761,771,778,788,794,804,811,821,828,838,844,847],[1277,1281,1286,1295,1303,1321,1328,1338,1344,1355,1361,1372,1378,1388,1395,1405,1411,1421,1428,1438,1445,1455,1461,1472,1478,1488,1497,1505,1511,1521,1528,1538,1545,1555,1564,1571,1580,1588,1595,1605,1611],[1955,1962,1970,1978,2003,2012,2022,2028,2036,2045,2048],[2168,2171,2179,2195,2205,2212,2222,2228,2239,2245,2255],[2869,2877,2882,2884,2890,2896,2906,2915,2923,2930,2939,2946,2956,2962,2973,2978,2989,2995,3006,3012,3023,3029,3039,3045,3056,3062,3075,3078,3089,3106,3112,3123,3128,3139,3145,3156,3164,3172,3179,3189],[3567,3579,3586,3596,3604,3612,3622,3629,3629,3640,3645,3656,3662,3677,3679,3689,3695,3706,3712,3722,3729,3739,3745,3756,3765,3772,3779,3789,3795,3806,3812,3823,3829],[4421,4432,4446,4462,4473,4479,4487,4496,4497,4507,4512,4523,4529,4540,4546,4556,4562,4577,4579,4590,4596,4606,4612,4623,4629,4640,4646,4656,4662,4673,4678,4683,4691,4696,4707,4713,4723,4729,4740,4821,4855,4862,4879,4890,4896,4907,4913,4923,4928,4931,4941,4946,4957,4963,4973,4979,4990,4996,5009,5013,5023,5029,5040,5046,5056,5063,5073,5079,5090,5096,5107,5113],[6437,6444,6453,6463,6464,6475,6479,6480,6488,6488,6496,6497,6508,6514,6525,6531,6541,6547,6557,6564,6564,6574,6581,6581,6591,6597,6598,6608,6613,6614,6625,6630,6641,6647,6657,6664,6674,6698,6707,6714,6715,6725,6731,6731,6741,6747,6748,6758,6763,6764,6772,6780,6781,6788],[7020,7032,7038,7047,7059,7063,7074,7090,7093,7097,7108,7113,7124,7130,7141,7147,7158,7163,7164,7175,7181,7191,7197,7198,7208,7214,7214,7225,7231,7241,7247,7258,7264,7275,7283,7288],[7487,7499,7504,7514,7522,7530,7542,7547,7559,7564,7575,7583,7592],[7726,7756,7764,7775,7780,7781,7793,7797,7798,7813,7814,7814,7826,7830,7831,7843,7848,7859,7864],[8170,8183,8191,8197,8209,8231,8242,8247,8259,8264,8276,8280,8281,8293,8297,8309,8314,8339,8347,8358,8364,8365,8376,8380,8392,8397,8398,8406,8414,8424,8431,8440,8447,8458,8464,8475,8483,8492,8497,8508,8515,8515,8522],[8693,8704,8710,8723,8742,8756,8764,8776,8784,8793,8797,8809,8814,8821,8822,8831,8841,8848,8859,8864,8865,8876,8881,8892,8898,8909,8914,8925,8932]],"version":"2.0.0"}

    Principal Arguments of Complex Numbers

    It can be noticed that if the angle is changed by 3600,7200{"x":[[339,344,347,350,354,357,360,364,366,369,371,373,373,373,373,373,370,367,364,361,359,356,353,351,349,348,348,349,353,358,363,368,374,379,384,388,392,395,399,399,399,398,395,389,383,376,367,359,350,333,326,320,316,313,311,311],[483,476,472,465,452,446,442,438,435,432,431,430,430,430,431,436,440,445,449,454,458,462,466,469,472,474,476,476,476,476,473,468,462,455,449,444,439,435,431],[522,517,517,516,515,515,515,515,517,520,524,528,533,538,542,547,552,556,559,562,565,567,569,569,570,570,569,566,563,558,553,547,540,533,526,520,515,512,509,508,508,508,508,510],[595,595,595,599,602,606,610,615,619,623,626,629,631,632,632,632,629,625,621,616,611,606,601,596,593,591,591],[636,635,635,635,633,632,631,629],[726,733,739,749,760,772,782,789,794,797,799,800,799,798,797,796,795,794,793,793,793,792,791,789,787,785,783,781,778,776,774,772,770,769,769,768,768,768,768,768,768,769],[721,730,740,752,766,779,791,803,812,819,826,830,832,833,832,831,831,829],[859,857,857,857,858,861,866,872,878,884,890,894,898,900,902,902,903,903,903,903,901,898,894,890,884,878,873,869,865,862,860,860,863,868,873,879,885,892,898,904,910,916,921,924,927,928,929,929,928],[968,966,965,964,963,963,963,963,963,963,964,966,969,972,975,979,983,986,988,991,993,994,994,995,995,995,994,992,990,988,985,983,980,978,976,974,973,972],[1005,1004,1004,1004,1004,1005,1007,1008,1010,1012,1015,1016,1018,1019,1019,1020,1021,1021,1021,1021,1021,1021,1020,1018,1016,1014,1012,1010,1007,1005,1002,1001,1000,999]],"y":[[259,251,250,248,247,247,247,247,250,255,259,264,271,277,284,291,298,304,310,315,319,322,325,327,328,328,327,326,324,323,322,321,321,321,321,323,327,331,341,348,355,361,368,375,382,388,394,400,405,412,414,415,415,415,415,413],[265,273,279,286,299,305,311,318,324,329,334,340,344,348,351,355,355,355,355,355,354,351,348,345,342,339,336,335,333,331,331,331,331,331,332,334,338,343,348],[308,319,324,328,333,338,343,347,351,355,358,360,361,362,362,361,358,354,349,343,336,330,322,316,310,304,299,295,291,289,288,287,287,288,291,296,302,308,315,322,330,337,345,352],[242,246,248,254,256,257,258,258,258,255,251,247,243,238,231,228,227,226,226,226,230,234,240,246,253,267,272],[377,378,383,389,397,405,411,416],[265,266,268,271,273,274,274,274,274,273,272,271,271,271,271,271,271,271,272,274,275,277,281,285,291,297,306,315,325,334,343,351,358,364,368,372,375,377,379,380,379,379],[329,329,326,323,321,320,319,319,318,317,316,316,315,315,315,315,316,318],[311,307,305,301,297,293,289,286,283,282,282,282,282,283,286,288,292,296,302,307,314,320,327,334,341,348,353,359,363,366,368,369,369,369,367,366,365,363,363,362,361,361,361,361,361,361,361,362,362],[290,293,298,307,316,325,333,340,346,350,353,355,355,355,355,355,352,348,343,337,331,326,319,313,307,301,297,293,291,289,289,289,289,290,294,298,304,310],[256,256,260,265,270,275,280,282,283,283,282,278,274,271,267,264,261,259,256,255,253,252,250,250,250,250,250,250,252,255,258,262,267,271]],"t":[[0,41,53,60,70,76,87,101,101,110,120,126,136,143,151,160,168,176,188,198,205,210,219,226,236,243,298,310,319,326,336,343,353,361,369,376,389,397,411,419,426,436,443,453,460,470,476,489,493,511,522,526,538,543,552,560],[954,978,988,993,1013,1022,1027,1038,1043,1053,1060,1069,1078,1088,1093,1113,1122,1127,1138,1143,1153,1160,1169,1179,1188,1193,1203,1213,1222,1227,1238,1243,1253,1260,1269,1277,1288,1293,1302],[1738,1760,1772,1777,1789,1794,1805,1818,1819,1827,1838,1844,1853,1861,1870,1877,1890,1894,1905,1918,1919,1927,1938,1944,1953,1960,1970,1977,1987,1994,2002,2018,2023,2027,2038,2044,2053,2060,2070,2077,2087,2094,2103,2110],[2555,2601,2612,2631,2641,2644,2656,2661,2670,2677,2687,2694,2707,2713,2729,2739,2744,2755,2760,2770,2777,2787,2795,2806,2814,2830,2839],[7025,7070,7079,7091,7096,7106,7112,7122],[7945,8021,8031,8038,8046,8056,8063,8073,8079,8090,8096,8106,8154,8163,8175,8179,8188,8196,8206,8214,8227,8230,8241,8246,8257,8263,8274,8281,8290,8296,8307,8313,8323,8329,8340,8346,8357,8363,8373,8389,8438,8446],[8650,8696,8705,8713,8724,8730,8741,8746,8757,8763,8773,8780,8793,8796,8838,8856,8863,8878],[9135,9170,9184,9190,9197,9207,9213,9224,9230,9240,9247,9257,9263,9276,9283,9291,9296,9307,9313,9324,9330,9340,9347,9357,9363,9374,9380,9392,9397,9406,9413,9423,9462,9476,9483,9491,9497,9507,9513,9525,9530,9542,9547,9558,9563,9576,9592,9639,9647],[9876,9938,9947,9956,9964,9974,9980,9993,9997,10008,10013,10027,10030,10043,10047,10058,10063,10074,10080,10089,10097,10107,10113,10124,10130,10141,10147,10158,10163,10174,10180,10191,10197,10205,10213,10224,10230,10242],[10735,10788,10797,10807,10814,10826,10830,10841,10847,10861,10864,10875,10880,10894,10897,10908,10914,10927,10930,10943,10947,10959,10964,10975,10980,10995,10997,11008,11014,11025,11030,11042,11047,11058]],"version":"2.0.0"} and so on, the complex number remains the same. So there will not be a unique value of θ{"x":[[529,525,524,522,520,518,517,515,514,513,512,511,510,510,510,510,510,510,512,513,515,517,520,522,539,543,547,551,556,559,563,565,568,569,570,570,570,570,568,565,561,558,553,548,544,538,532,527,521,515,509,504,500,498,496,495,495,495,495,496,500,505,510,516,524,532,541,551,560,569,579,588,597,605,612,618,623]],"y":[[267,273,281,290,299,309,319,328,339,348,358,369,379,388,397,404,411,417,421,425,427,427,427,427,407,399,391,382,371,361,351,339,330,321,312,305,298,291,285,279,275,271,269,268,267,267,267,269,274,279,283,288,293,297,300,305,309,314,318,323,327,331,334,337,339,340,341,341,341,341,339,336,334,331,330,329,329]],"t":[[0,11,20,30,36,47,53,63,70,80,87,97,103,113,120,128,136,147,153,163,170,180,186,197,233,236,246,253,262,269,280,286,296,303,313,320,330,336,347,353,363,370,380,386,397,403,414,420,430,436,447,453,464,470,481,486,497,503,514,520,530,536,546,553,568,571,578,586,596,603,612,620,629,636,646,653,663]],"version":"2.0.0"} but infinitely many. Hence, we call the first value of θ{"x":[[507,500,498,495,492,489,487,484,482,480,479,477,476,476,476,476,476,476,476,477,480,483,486,490,505,511,517,524,530,535,540,544,546,549,550,550,550,547,544,540,531,527,522,517,512,507,501,496,490,486,482,478,476,474,473,473,473,476,480,487,495,505,515,528,539,549,559,568,575]],"y":[[200,207,214,222,230,239,250,261,272,284,295,305,315,324,333,341,349,356,363,368,371,374,375,375,371,365,359,351,343,334,324,315,305,294,285,275,267,240,234,228,221,218,216,215,215,215,218,223,228,234,241,248,255,262,269,274,277,281,283,285,286,286,286,286,286,285,284,284,284]],"t":[[0,26,35,44,54,61,71,77,88,94,104,110,121,127,138,144,156,160,173,177,187,194,204,210,238,244,254,261,273,277,287,294,304,310,321,327,337,363,369,377,401,402,411,419,432,439,444,455,460,473,477,487,494,504,511,520,527,536,544,555,560,570,577,587,594,603,611,620,627]],"version":"2.0.0"} the principal argument of z, Arg(z).

    Calculate the principal argument of the complex number z=1+i.

    Solution:

    Step 1: We know that tanθ = ba. From the question we have a=1 and b=1. Thus, we have the value of tangent as,

    tanθ=11

    Step 2: Solving for θ, we get

    θ=tan-1(1)θ=45°

    Thus, the principal argument, Arg(z), of z=1+i is 45°. This means that we can add or subtract multiples of 360 degrees to 45 and get the same complex number!

    De Moivre’s Theorem

    De Moivre’s Theorem is fundamental in complex numbers. The theorem is stated as follows:

    Theorem: Let z1,z2{"x":[[153,154,158,170,178,186,194,203,210,218,225,230,235,238,240,241,241,241,241,241,240,238,234,229,224,218,212,205,198,191,185,179,175,171,167,164,162,160,159,158,157,157,158,160,164,170,177,186,196,206,217,227,236,243,250,254,256,257,256,256,255],[128,133,142,155,170,184,197,210,220,229,236,241,245,246,245,243,240],[294,292,292,292,292,292,291,291,290,289,289,289,289,289,289],[396,397,397,397,397,397,397,397,397,397],[513,515,520,527,537,548,559,570,580,589,596,604,609,613,616,618,619,619,619,619,616,612,605,597,588,578,568,558,547,539,530,522,516,510,506,503,500,498,498,499,503,509,517,527,536,547,558,568,577,586,594,600,606,611,613,615,616,615],[481,488,499,515,533,549,565,580,592,602,611,616,620,621,622,620,617,613],[644,640,640,640,640,640,640,642,644,647,649,652,656,658,660,663,665,667,669,671,672,673,673,674,674,674,674,673,671,668,664,661,657,653,649,647,644,643,642,642,643,646,650,655,662,670,678,687,696,703,709,715,718,719,720,719,718,717,717]],"y":[[220,216,216,215,214,212,211,210,209,209,208,208,208,208,208,209,210,212,215,218,221,226,231,238,244,252,259,267,276,284,292,300,307,314,319,324,329,333,336,338,340,341,341,341,341,341,340,338,336,334,331,330,328,327,327,327,327,327,327,328,329],[291,288,287,285,283,281,279,277,275,273,271,269,268,267,268,269,272],[330,332,337,344,353,362,370,377,382,386,389,390,391,390,388],[329,339,341,343,346,349,353,357,362,366],[180,181,181,181,182,182,182,182,181,180,178,177,176,175,175,175,175,176,179,183,187,193,200,207,215,224,233,242,251,259,268,276,283,290,296,301,305,309,310,310,310,310,308,305,304,301,300,298,297,296,295,295,295,295,295,295,295,295],[252,248,247,246,245,244,243,242,240,239,237,236,235,235,235,235,238,241],[347,343,342,339,336,334,331,328,325,322,319,317,316,316,316,316,316,317,319,321,324,327,329,333,336,340,344,348,352,356,360,364,367,370,373,374,377,378,378,379,379,379,379,378,377,375,374,373,372,372,372,371,371,371,371,371,372,372,373]],"t":[[0,113,122,141,146,154,164,171,181,188,197,204,215,221,232,238,249,255,265,271,281,288,298,304,315,322,331,338,348,355,365,371,381,388,398,404,415,421,434,438,447,454,496,504,514,521,531,538,548,554,564,571,581,588,598,608,614,621,663,671,681],[913,938,947,955,964,971,981,988,998,1005,1015,1022,1030,1039,1079,1088,1098],[1530,1588,1597,1605,1615,1621,1632,1638,1647,1655,1665,1671,1682,1739,1746],[2462,2622,2634,2639,2647,2655,2665,2672,2682,2689],[3417,3480,3489,3498,3505,3516,3522,3532,3539,3549,3556,3566,3572,3582,3589,3599,3615,3622,3633,3639,3649,3656,3667,3672,3684,3689,3701,3706,3716,3722,3733,3739,3747,3756,3765,3772,3783,3789,3799,3829,3839,3849,3856,3866,3872,3883,3889,3899,3906,3916,3923,3932,3939,3949,3956,3966,3972,4005],[4202,4231,4239,4249,4256,4266,4272,4283,4289,4300,4306,4317,4323,4333,4339,4363,4372,4383],[4758,4805,4814,4823,4833,4840,4850,4856,4867,4873,4884,4890,4901,4906,4917,4923,4934,4940,4949,4956,4966,4973,4984,4989,5001,5006,5017,5022,5034,5039,5051,5056,5067,5073,5084,5089,5101,5106,5117,5123,5156,5165,5173,5184,5190,5201,5206,5217,5223,5234,5240,5248,5256,5266,5273,5314,5331,5339,5356]],"version":"2.0.0"} be two distinct complex numbers and their polar forms are

    z1=r1cosθ1+isinθ1 and z2=r2cosθ2+isinθ2{"x":[[44,45,45,46,47,48,49,51,53,55,58,61,65,68,71,74,76,77,78,78,78,78,77,75,73,70,67,63,60,56,52,47,42,38,34,30,26,23,20,19,18,18,19,21,24,27,31,36,41,46,51,56,61,64,67,70,71,73],[24,25,28,33,40,48,55,62,67,70,73,74,74],[104,103,102,102,102,103,104,106,109,111,114,117,118,119,118,116,113,110,106,102,99,97,96,98,100,103,107,112,117,122,127],[151,150,148,147,147,148,150,153,157,161,166,171,176,180,182,183],[137,136,135,136,137,138,142,147,154,162,170,175,179,183,185,187,188],[236,236,235,236,238,241,245,249,253,257,260,263,265,266,266,266,266,265,263,262,260,259,258,257,257,257,257,257,258,260,263,265,267,269,270,271,270,268,266,265,263,261,261,261,262,264,267,269,271,275,279,283,288],[294,293,294,295,296,297,298,300,303,305,307,308,309,309,309,308,306,303,300,298,296,296,298,301,304,308,312,315,318,320,322,323,325,326],[372,369,368,367,366,364,362,359,356,352,348,344,341,338,337,337,339,343,349,353],[426,427,426,425,424,422,418,414,411,408,406,406,407,409,412,414,418,421,424,427,429,430,431,433,433,434,434,434,434,433,433,433,434,434,436,437,439,441,443,445,446,447,447,447,447,446,444,443,442,441,440,439,439,439,439,440,441,444,447,450,455,459,463,467,470,472,474,475,476,476,476,475,473,472,470,469,468,468,469,471,474,477,480,481,482,483,482,481,479,477,475,473,472],[514,514,516,516,516,516,516,516,515,514,513,512,511,511,510,510,511,514,517,520,525,528,532,535,536,536,536,533,531,527,524,520,517,514,512,510,510,511,514,519,525,531,537],[549,548,548,547,547,548,549,551,553,556,558,560,560,560,558,557,555,552,551,550,552,556,562,569,575,581,587,588],[600,599,598,597,598,600,602,609,611,620,624,626],[610,610,611,611,612,612,612,612,611,610,610,610],[667,667,667,668,669,669,670,671,671,671,672,672,673,675,677],[673,672,671,672,673,673],[730,731,732,732,732,732,732,731,729,726,724,723,722,720,719,719,719,719,719,721,722,725,728,731,733,734,734,732,729,726,723,720,718,716,716,716],[750,750,750,749,748,748,748,748,749,749,749,750,750,750],[754,754,754,754,755],[761,762,763,763,764,765,765,765,765,765,765,765,765,765,766,767,769,771,774,776,779,781,783,784,785,786,786,787,787,788,789,791,792],[815,815,816,816,817,817,816,815,814,813,813,813,814,817,820,824,828,832,836,839,842,844,844,843,840,836,831,824,818,816,812,808,807,807,808,810,814,817,823,830,833],[849,849,849,849,850,850,853,855,857,857,857,857,855,852,851,849,848,849,851,854,858,861,864,870,875,881],[889,888,888,888,889,891,893,896,900,902,904,907,909,909,909,907,905,900,897,894,887,883,873,871,867]],"y":[[184,184,183,183,183,183,183,183,182,182,182,181,181,181,181,181,181,181,181,182,183,184,185,187,189,192,194,197,200,204,208,212,216,220,223,226,229,232,234,236,237,238,239,240,240,241,241,241,240,240,239,239,238,238,238,238,238,238],[213,213,213,213,213,213,212,211,211,210,210,210,211],[278,278,278,277,275,274,272,270,269,269,269,270,272,275,278,282,287,291,295,299,301,303,304,304,303,303,302,301,300,299,299],[216,216,216,216,215,215,215,215,215,214,213,212,211,211,211,211],[240,240,240,240,240,239,239,237,236,235,234,233,232,232,231,231,231],[237,238,238,237,236,234,231,229,225,222,217,213,208,205,203,202,201,201,201,202,203,205,207,209,212,214,216,219,220,222,222,222,222,222,221,221,222,224,227,230,233,236,240,242,244,245,246,246,246,245,244,242,240],[264,264,263,262,260,259,258,257,255,255,255,255,256,258,260,262,265,268,271,273,275,276,276,276,275,274,274,274,273,273,273,274,274,274],[180,177,175,175,175,175,176,179,184,191,199,208,218,227,236,243,250,254,257,258],[207,207,207,207,207,207,209,211,215,219,223,227,230,232,233,233,232,231,228,226,223,220,219,217,216,215,214,215,216,218,220,223,226,229,231,232,233,233,231,229,226,223,220,218,216,213,212,211,211,211,212,213,215,218,220,222,224,226,227,227,226,225,222,220,217,215,212,209,207,205,204,203,203,203,205,207,208,210,213,215,217,219,221,224,226,227,229,230,230,230,230,230,229],[206,205,203,202,200,199,198,197,197,198,202,207,213,219,227,235,240,244,245,244,241,237,232,227,221,215,208,202,196,192,190,189,190,194,198,203,208,213,216,218,219,219,218],[251,251,250,249,248,248,248,248,249,250,252,256,257,259,262,263,265,267,269,269,269,269,268,266,265,264,263,263],[220,220,220,220,220,221,221,221,221,221,221,220],[215,216,217,220,223,227,230,235,241,243,245,246],[209,207,206,207,210,212,215,219,223,227,230,233,234,234,232],[185,185,183,184,184,185],[211,210,209,208,206,205,204,202,200,200,199,199,200,201,202,203,205,206,208,211,213,216,219,222,225,226,228,230,231,231,232,232,230,229,227,225],[207,206,205,203,203,204,206,209,214,218,221,226,229,230],[185,184,183,184,184],[214,213,212,211,211,211,212,214,215,217,219,220,221,222,220,218,215,212,209,207,206,206,207,210,212,217,219,222,225,227,229,229,229],[201,199,198,197,198,200,204,208,213,218,223,228,231,233,234,233,231,228,224,219,212,204,196,189,184,181,179,179,182,184,188,196,198,201,205,209,213,214,216,217,217],[245,244,243,242,242,241,241,242,243,246,248,251,254,258,259,261,262,262,261,261,259,259,258,256,255,254],[187,186,184,183,183,184,186,189,194,197,201,210,215,220,231,236,241,250,254,258,266,270,278,280,282]],"t":[[0,20,37,46,72,79,89,96,106,112,122,129,139,146,155,162,172,179,189,196,206,212,222,229,240,246,258,262,272,279,289,296,306,314,322,329,339,346,356,362,371,379,389,396,406,413,422,429,439,446,455,462,472,480,491,496,506,512],[768,804,813,822,829,839,846,855,863,875,879,892,896],[5487,5497,5505,5531,5539,5548,5557,5565,5574,5581,5590,5598,5607,5614,5627,5631,5642,5648,5658,5664,5677,5681,5694,5714,5726,5731,5740,5748,5760,5765,5777],[6316,6323,6332,6335,6356,6365,6374,6381,6391,6398,6407,6415,6424,6431,6444,6447],[6636,6649,6655,6658,6662,6665,6677,6681,6691,6698,6708,6715,6724,6731,6741,6748,6757],[7159,7169,7175,7198,7208,7215,7225,7232,7243,7248,7258,7265,7275,7282,7292,7298,7312,7315,7324,7332,7344,7348,7358,7365,7375,7382,7393,7398,7410,7415,7426,7432,7444,7448,7460,7465,7492,7498,7511,7515,7524,7532,7543,7548,7558,7565,7577,7582,7583,7593,7598,7613,7615],[7925,7960,7991,7999,8008,8015,8025,8032,8042,8049,8059,8065,8078,8082,8094,8100,8110,8115,8128,8132,8142,8149,8165,8175,8183,8192,8199,8210,8215,8225,8232,8242,8249,8262],[8699,8721,8727,8729,8732,8742,8749,8759,8766,8776,8782,8792,8799,8809,8816,8826,8832,8842,8849,8857],[12632,12685,12727,12735,12743,12751,12761,12767,12778,12784,12794,12800,12813,12818,12829,12835,12845,12852,12863,12868,12879,12885,12896,12900,12914,12917,12927,12948,12950,12960,12967,12977,12984,12996,13001,13011,13017,13028,13034,13044,13051,13061,13067,13077,13084,13094,13101,13111,13117,13127,13134,13144,13151,13161,13167,13180,13184,13196,13201,13211,13217,13231,13234,13244,13251,13261,13267,13277,13284,13294,13301,13311,13317,13327,13334,13344,13351,13361,13367,13380,13385,13394,13401,13411,13417,13427,13434,13444,13451,13461,13467,13477,13480],[13938,13947,13957,13960,13968,13977,13984,14001,14011,14018,14028,14034,14045,14051,14061,14068,14078,14085,14095,14101,14111,14118,14128,14135,14143,14151,14161,14168,14178,14184,14195,14201,14212,14218,14231,14234,14245,14251,14262,14268,14278,14284,14294],[14593,14606,14613,14636,14660,14677,14685,14693,14701,14711,14718,14730,14736,14736,14748,14751,14763,14768,14778,14785,14802,14812,14819,14831,14836,14845,14852,14862],[15141,15147,15153,15168,15186,15195,15201,15212,15218,15228,15236,15245],[15355,15402,15411,15419,15431,15436,15446,15451,15462,15469,15470,15480],[15808,15820,15831,15861,15869,15878,15885,15896,15902,15913,15918,15929,15935,15946,15952],[16044,16049,16055,16087,16094,16102],[16416,16424,16431,16433,16447,16453,16463,16468,16479,16486,16496,16503,16504,16514,16519,16520,16530,16536,16537,16547,16552,16563,16570,16580,16585,16596,16602,16613,16619,16630,16636,16646,16652,16663,16669,16672],[16792,16802,16809,16812,16827,16852,16862,16871,16880,16886,16896,16903,16913,16919],[17047,17060,17065,17077,17080],[17223,17235,17240,17253,17262,17279,17289,17297,17302,17313,17319,17330,17336,17347,17369,17379,17386,17397,17402,17413,17419,17430,17435,17447,17452,17465,17471,17480,17487,17497,17503,17513,17520],[17707,17722,17730,17737,17772,17777,17786,17797,17803,17814,17819,17830,17836,17847,17852,17864,17869,17880,17886,17897,17902,17911,17919,17929,17936,17947,17953,17964,17970,17980,17986,17997,18004,18004,18015,18020,18031,18036,18047,18053,18056],[18345,18387,18403,18420,18429,18436,18447,18461,18469,18480,18486,18497,18503,18514,18519,18530,18537,18553,18562,18570,18580,18587,18587,18598,18603,18614],[18804,18810,18817,18824,18829,18837,18849,18854,18864,18870,18871,18882,18886,18887,18899,18904,18905,18915,18920,18921,18931,18936,18947,18953,18964]],"version":"2.0.0"}

    Then the product z1z2{"x":[[174,176,182,189,197,207,217,224,231,238,243,247,251,253,255,256,257,257,255,254,251,247,242,236,229,222,215,208,201,195,189,183,179,174,171,168,166,165,165,167,171,177,185,193,203,214,225,235,243,251,256,260,263,262,261,258],[183,187,193,202,215,227,239,250,257,262,265,267,268,269,270,270,270,270],[301,301,301,301,301,301,301,300,299,299,299],[368,371,376,383,392,401,410,419,426,432,438,442,445,448,449,450,450,449,448,446,443,440,436,430,424,417,410,403,397,390,385,380,375,372,369,367,365,364,363,363,363,363,365,369,373,379,385,391,399,406,414,422,429,435,440,444,447,450,451,452,452],[361,367,374,383,392,403,412,419,425,429,432,433,434,435,435,434,433,433],[469,472,472,474,477,480,484,487,490,492,493,494,495,496,496,496,496,496,496,496,496,494,492,490,488,486,484,482,481,480,479,480,483,486,490,494,499,505,510,516,521,525,528,530,532]],"y":[[165,164,164,164,164,164,164,164,164,164,164,164,163,163,163,163,163,166,168,172,176,182,187,194,201,208,214,222,228,235,243,249,255,261,266,270,274,276,277,277,277,277,277,276,275,274,274,273,272,272,271,271,271,271,272,272],[219,213,213,213,213,213,213,213,213,213,213,213,213,213,214,216,219,222],[286,290,296,303,311,319,326,330,333,335,334],[168,164,164,164,164,164,164,163,162,162,161,161,161,161,161,161,162,163,164,166,170,174,179,185,191,197,204,211,217,223,228,233,238,242,245,249,251,254,256,257,259,260,261,261,261,261,260,258,257,255,255,254,254,254,254,254,254,254,254,254,255],[204,201,201,201,202,203,204,204,204,204,204,204,204,204,203,203,203,204],[293,286,283,281,278,275,272,270,268,267,267,267,267,269,271,274,277,281,284,287,291,293,297,299,301,303,305,306,306,307,307,307,307,307,307,307,307,307,307,306,306,305,305,305,305]],"t":[[0,58,68,75,85,92,102,108,118,125,135,142,152,159,169,175,185,218,225,235,244,252,258,269,275,286,292,302,308,319,330,335,342,352,358,369,375,385,402,442,452,458,469,475,485,492,505,509,519,525,535,545,552,609,617,631],[853,892,901,909,919,933,936,942,956,959,968,975,986,993,1002,1009,1019,1026],[1520,1575,1584,1592,1602,1609,1619,1626,1636,1642,1717],[2232,2309,2318,2326,2336,2343,2351,2359,2369,2376,2386,2393,2403,2409,2419,2426,2458,2477,2485,2492,2502,2509,2519,2526,2537,2543,2553,2560,2570,2576,2587,2593,2603,2610,2620,2626,2637,2643,2653,2660,2670,2686,2703,2709,2720,2726,2736,2743,2752,2759,2769,2776,2786,2793,2803,2810,2819,2826,2836,2843,2853],[3140,3197,3201,3210,3219,3226,3237,3243,3253,3259,3269,3276,3287,3293,3303,3350,3359,3369],[3864,3943,3946,3952,3961,3970,3976,3987,3993,4003,4010,4020,4026,4043,4052,4060,4070,4077,4088,4093,4104,4110,4121,4127,4138,4143,4151,4160,4170,4176,4188,4242,4253,4262,4270,4277,4287,4293,4304,4310,4320,4327,4337,4343,4354]],"version":"2.0.0"} will be a complex number with the phase as θ1+θ2{"x":[[190,186,184,181,179,176,174,172,170,168,167,166,165,165,165,165,167,169,172,176,181,186,192,198,203,208,214,218,222,227,229,231,233,234,235,236,237,237,237,237,237,237,235,233,230,226,222,218,212,205,198,192,186,180,175,171,167,166,165,165,165,166,168,172,177,183,190,198,206,212,219,224,229,231,233,234,234,233,233],[281,279,279,279,279,279,279,278,277,277,277,277,277],[340,344,350,358,369,380,392,402,410,417,422,426,428,429,429,428],[396,397,397,396,395,393,392,390,389,389,388,387,387,387,387,387,387,387,387],[541,536,532,528,524,521,518,516,515,514,514,514,514,515,517,519,522,526,529,533,537,541,544,548,550,552,553,555,557,559,561,563,565,567,568,568,568,568,566,565,564,563,560,556,550,545,541,536,533,530,527,526,525,525,525,525,525,526,529,531,534,537,541,545,549,553,556,560,563,565,568,571,573,575,576,577,578,578],[572,574,576,580,584,588,592,596,599,601,602,603,603,603,603,603,601,599,597,595,594,593,596,601,610,620,630,640,649,657,664,669,671,673,673]],"y":[[235,249,255,262,269,278,286,295,304,312,320,329,335,341,346,350,353,355,356,356,356,356,354,349,344,338,331,324,317,309,302,294,285,277,269,260,253,247,240,235,231,228,225,224,222,221,221,221,223,228,234,241,250,259,268,277,286,292,298,305,309,313,317,319,322,323,324,324,324,324,323,321,320,319,319,318,319,320,321],[355,362,369,378,387,396,405,411,415,419,420,419,418],[305,309,309,309,308,307,306,305,304,303,303,303,303,303,304,305],[260,261,266,274,284,294,304,314,322,329,335,340,345,350,353,356,358,359,360],[233,242,247,255,262,270,279,288,295,303,309,315,320,324,327,329,329,330,330,330,327,324,319,314,309,303,297,291,283,274,266,257,248,241,235,232,231,230,230,230,230,230,230,232,235,238,241,245,249,253,259,264,269,275,280,284,288,292,295,299,302,304,306,308,309,309,309,309,309,308,306,304,303,302,301,300,301,304],[371,362,360,357,355,353,352,351,351,351,351,353,356,363,367,372,376,380,383,386,387,389,389,389,389,389,388,387,387,386,386,385,385,385,386]],"t":[[0,10,16,25,32,39,49,56,65,72,82,89,99,106,116,122,132,139,149,156,166,172,182,189,199,206,216,222,233,239,250,256,266,273,283,289,300,309,316,327,332,339,349,356,366,372,383,389,400,406,416,422,433,439,450,456,466,473,483,489,499,506,516,523,532,539,549,556,565,573,582,589,599,610,615,632,671,681,689],[977,1039,1048,1056,1068,1073,1082,1090,1100,1106,1116,1172,1182],[1498,1538,1540,1548,1557,1566,1573,1583,1590,1600,1606,1617,1623,1634,1673,1682],[1853,1897,1907,1916,1923,1933,1940,1950,1957,1967,1973,1983,1990,2001,2007,2017,2023,2034,2040],[2442,2457,2465,2473,2483,2490,2500,2507,2517,2524,2534,2541,2551,2557,2567,2574,2584,2590,2601,2607,2617,2624,2634,2640,2651,2657,2668,2674,2684,2690,2701,2714,2717,2724,2738,2740,2750,2767,2790,2800,2807,2817,2823,2834,2840,2851,2857,2867,2873,2884,2890,2900,2907,2917,2924,2935,2941,2951,2957,2968,2974,2985,2990,3001,3007,3017,3024,3038,3043,3053,3057,3067,3073,3084,3091,3101,3124,3134],[3453,3532,3541,3549,3558,3568,3574,3585,3591,3602,3607,3619,3624,3646,3649,3657,3667,3674,3684,3691,3701,3707,3757,3766,3774,3784,3791,3802,3807,3819,3824,3835,3841,3852,3898]],"version":"2.0.0"}.

    A significant consequence of De Moivre’s Theorem: We have defined θ{"x":[[472,469,464,460,456,455,454,454,454,455,457,461,466,472,478,485,492,499,505,512,517,522,536,538,539,540,540,540,539,536,532,526,520,513,505,497,489,481,473,465,460,455,452,452,452,452,454,459,464,472,480,489,500,510,522,533,543,553,561,568,575,581,585]],"y":[[288,299,313,324,336,347,357,367,375,383,390,395,399,401,402,402,400,395,389,381,373,364,319,308,298,288,279,271,265,260,256,253,252,252,252,254,259,265,272,281,289,297,306,313,321,328,334,339,343,346,347,347,347,347,344,341,337,335,334,333,332,332,332]],"t":[[0,38,61,66,73,81,87,97,104,114,120,131,137,148,154,164,171,181,187,198,204,214,246,254,264,271,281,287,297,304,314,321,332,337,348,359,364,373,380,387,397,404,414,421,431,437,447,454,464,471,481,487,497,504,512,521,530,537,546,554,563,571,580]],"version":"2.0.0"} as the phase angle of a complex number z. We shall now see how the phase angle changes according to the power raised to z. Let us multiply z with itself n times. We have a new complex number, zn{"x":[[380,380,383,387,392,403,409,416,423,429,435,441,445,449,452,454,457,458,460,460,460,458,454,449,443,436,428,420,412,404,396,389,382,376,372,369,367,368,369,373,379,387,395,404,414,423,433,442,450,456,463,467,471,475,477,479,481,482,483,484,484,485],[371,373,379,389,401,414,425,435,444,450,455,464,467,470,471,472,471],[493,493,493,493,493,493,493,493,493,493,493,494,495,495,496,496,496,496,496,496,498,500,503,507,510,514,517,520,522,524,525,525,526,526,526,526,526,526,526,526,526,527,527,527,529,530,531,533,534,535,536,537,537,538,538,539]],"y":[[268,267,267,267,267,267,267,267,266,266,265,265,265,265,265,265,267,268,258,262,265,270,276,282,288,295,304,312,322,331,340,348,356,363,368,372,374,374,374,374,373,371,370,368,366,364,362,361,360,360,360,360,360,360,360,360,360,361,361,361,362,362],[319,317,317,317,316,316,316,316,316,316,316,315,314,314,313,313,314],[197,200,204,208,213,218,222,226,230,233,234,235,235,233,232,231,229,226,223,219,214,208,202,196,191,187,185,184,183,183,184,186,188,190,193,196,200,203,207,211,215,219,223,226,229,232,234,236,238,240,240,241,242,243,244,244]],"t":[[0,79,87,100,104,122,130,137,148,154,164,171,180,187,197,204,214,220,252,254,263,270,280,287,297,304,312,320,330,337,347,354,364,370,380,387,397,436,447,454,463,470,480,487,497,504,514,520,530,537,546,554,563,570,580,587,596,604,613,621,630,637],[900,928,938,947,954,964,971,981,987,997,1004,1025,1029,1038,1047,1065,1111],[1466,1537,1546,1555,1565,1571,1584,1592,1598,1604,1615,1623,1654,1679,1688,1697,1704,1714,1721,1731,1738,1748,1754,1764,1771,1781,1788,1798,1804,1815,1831,1838,1849,1855,1865,1871,1884,1891,1897,1904,1915,1923,1931,1938,1948,1954,1964,1972,1980,1988,1997,2004,2014,2021,2046,2078]],"version":"2.0.0"}, with a different phase angle. A consequence of the above theorem is that

    zn=rncosθ+isinθn{"x":[[65,65,65,66,68,70,71,76,77,79,80,82,84,86,87,90,92,93,96,99,102,105,106,107,108,108,107,107,106,104,102,98,95,90,86,79,74,70,69,65,63,61,61,61,64,67,69,71,73,75,77,82,88,94,103,108,112,114,115,117,118,119],[66,67,68,69,72,78,84,91,99,104,108,112,115,117,118],[126,126,127,127,128,129,129,129,129,130,130,130,130,130,129,129,128,128,128,129,130,132,133,136,138,140,141,142,142,142,142,143,144,145,147,148,149,149,150,150,150,150],[176,175,174,174,176,179,183,189,195,201,207,213,216,219,221,222,223,223],[172,172,171,170,171,174,179,186,194,203,211,216,220,222],[298,297,298,298,300,302,305,309,313,316,319,320,321,321,320,319,318,317,316,316,316,317,318,319,321,322,324,326,327,330,332,333,335,336,337,337,337,335,334,332,331,330,329,329,330,332,335,339,342,344,348,349],[409,408,406,405,402,398,393,391,385,381,377,374,373,375,377,380,386,395,398],[437,437,437,437,437,437,436,435,433,431,429,427,425,424,424,425,427,431,435,442,446,449,454,457,460,464,464,464,464,464,463,463,462,462,461,461,461,461,461,462,463,465,466,468,470,472,473,473,474,475,475,476,476,476,475,474,472,471,470,468,468,468,468,470,472,473,478,480,483,490,496,502,504,510,515,518,522,524,525,525,525,523,521,517,516,515,514,514,514,514,514,516,518,520,521,524,524,525,525,525,525,525,524,521,517,516],[562,562,562,561,561,560,560,560,560,560,560,559,559,559,559,560,561,563,565,570,573,576,579,581,583,584,584,584,582,579,576,572,567,565,563,560,558,557,556,556,558,560,564,569,575,582,588,591],[618,617,616,615,616,618,623,629,633,637,645,649,652,658,662],[642,642,644,644,644,644,644,643,643,643,643,643,643,643,643],[706,707,707,708,708,709,709,709,709,710,710,711,712,714,716,717],[715,714,713,712,713,714,715],[765,764,764,764,764,764,764,763,761,758,755,751,749,747,746,747,750,753,758,762,765,766,767,767,766,765,763,761,759,756,754,752,749,748],[783,783,783,783,783,783,783,783,783,784,784,785],[785,784,783,783,784,785],[808,809,809,810,810,811,811,812,812,812,812,812,812,811,811,811,812,812,814,817,822,826,829,831,834,835,836,837,837,837,837,837,837,837,837,837,837],[867,868,868,868,869,869,870,870,870,870,870,870,870,870,872,873,875,878,881,884,886,889,891,892,892,889,885,880,873,871,867,865,864,864,867,871,876,881,886,889,891,896],[917,916,916,917,919,921,924,928,933,936,937,936,932,927,921,918,915,909,906],[924,925,926,926,927,928,928,928,928,928,928,927,926,925,925,925,925,925,926,927,928,933,936,939,941,944,945,946,948,949,949,950,950,950,950,950,949,949,949,950],[353,352,351,351,351,351,351,351,351,351,350,350,350,351,352,354,357,359,362,365,367,369,370,371,372,373,374,374,375,376,376,377,379,380,381]],"y":[[215,214,213,212,211,211,211,210,209,209,209,209,209,209,208,208,208,208,208,208,208,208,208,210,211,213,214,215,216,218,221,224,227,230,234,239,242,245,247,250,252,254,256,257,258,258,258,258,258,258,257,256,255,254,253,252,252,252,252,252,252,252],[235,235,234,234,233,232,231,230,229,228,228,228,228,228,228],[182,181,181,180,180,181,182,184,186,188,190,192,193,194,194,193,192,190,188,186,183,181,178,175,173,171,170,170,172,174,177,180,184,187,191,194,197,200,202,204,206,207],[225,225,225,224,224,224,223,223,222,222,221,221,221,221,221,221,221,222],[239,240,240,240,240,240,239,238,236,235,234,233,232,232],[254,254,254,253,251,250,245,239,231,224,218,214,212,211,210,210,210,210,210,211,213,214,216,218,220,222,223,223,223,223,223,222,221,220,220,222,224,227,232,237,239,244,248,253,254,255,255,253,250,249,246,244],[195,193,192,191,190,190,192,194,200,209,220,238,250,260,264,272,277,280,280],[227,226,225,224,223,222,221,221,221,221,224,228,234,240,247,252,257,260,261,261,259,257,254,250,246,241,238,235,234,233,233,232,233,234,236,239,242,246,249,252,254,256,257,257,257,254,253,251,248,246,244,240,238,235,231,229,228,227,228,230,232,238,241,245,246,247,249,250,250,250,248,246,245,242,240,237,234,231,228,226,224,222,220,220,220,220,220,221,222,224,225,228,231,235,238,244,246,247,249,250,251,252,252,252,252,252],[217,216,215,214,213,212,211,215,217,220,227,236,244,248,251,257,259,261,263,262,259,254,249,242,234,225,221,216,208,202,199,197,197,198,199,202,204,210,216,221,226,228,231,233,235,235,235,234],[230,230,230,230,230,230,230,229,229,228,227,227,226,226,226],[213,212,211,212,215,220,225,232,238,244,249,251,255,258,260],[217,217,218,221,225,231,238,243,247,249,250,251,251,251,249,247],[198,198,198,197,197,198,198],[226,225,224,223,222,221,219,216,214,213,212,212,213,215,218,222,226,229,233,236,238,240,242,243,244,245,245,245,245,245,245,245,243,242],[216,215,216,219,224,229,235,240,245,247,248,250],[201,200,200,199,200,201],[220,219,218,219,220,222,225,228,231,234,236,237,238,236,235,234,230,228,224,219,213,211,211,212,214,218,223,225,228,233,238,243,244,246,248,249,250],[212,212,211,212,213,216,218,221,226,229,238,240,244,246,247,247,245,242,237,232,230,223,216,209,202,197,194,193,193,195,198,203,208,212,218,221,223,224,225,225,225,223],[191,191,190,191,193,195,199,206,215,223,231,239,245,251,258,260,263,267,269],[153,153,154,155,156,158,161,164,166,169,171,172,172,171,170,168,165,164,160,158,155,149,145,143,142,142,144,146,149,155,157,159,165,167,169,172,173,175,176,177],[179,179,179,178,179,181,183,186,189,192,194,195,194,191,188,184,180,176,172,170,169,170,171,174,178,181,185,188,192,194,195,196,196,194,193]],"t":[[0,14,34,52,58,64,73,81,88,89,101,102,102,105,114,122,125,131,140,147,155,164,171,181,188,199,205,206,214,222,231,238,247,257,264,271,280,291,297,305,314,321,330,338,354,363,372,373,380,381,390,397,404,414,421,430,440,447,456,457,464,471],[835,846,851,855,864,871,880,891,897,906,914,921,931,938,947],[1347,1359,1364,1380,1388,1421,1431,1438,1449,1455,1464,1472,1482,1488,1507,1522,1531,1538,1548,1555,1565,1572,1581,1589,1601,1605,1617,1621,1638,1648,1655,1665,1672,1681,1688,1698,1705,1714,1722,1732,1739,1751],[2299,2312,2322,2339,2348,2355,2365,2372,2382,2388,2400,2405,2415,2422,2431,2439,2448,2455],[2597,2605,2612,2614,2630,2639,2648,2655,2665,2672,2682,2689,2700,2704],[4160,4179,4240,4242,4249,4256,4265,4273,4282,4289,4299,4306,4316,4323,4332,4339,4349,4356,4366,4373,4390,4399,4406,4415,4423,4432,4439,4449,4456,4465,4473,4482,4490,4503,4508,4515,4524,4532,4540,4549,4556,4566,4573,4582,4590,4599,4607,4620,4624,4632,4639,4649],[9964,9972,9980,9984,9992,10002,10009,10018,10025,10035,10042,10052,10059,10068,10075,10085,10092,10102,10109],[10575,10583,10591,10593,10602,10609,10625,10635,10642,10652,10659,10669,10675,10685,10692,10702,10709,10719,10725,10735,10743,10753,10759,10771,10775,10787,10793,10802,10809,10822,10826,10835,10842,10851,10858,10869,10875,10885,10892,10902,10909,10918,10925,10935,10942,10954,10959,10960,10969,10976,10977,10985,10992,11002,11008,11020,11025,11037,11042,11054,11059,11071,11076,11087,11093,11094,11103,11109,11111,11123,11127,11136,11142,11152,11159,11169,11175,11186,11192,11202,11209,11219,11225,11235,11243,11245,11252,11260,11261,11269,11275,11285,11292,11302,11308,11319,11326,11328,11336,11343,11345,11352,11358,11369,11376,11385],[11754,11763,11769,11771,11777,11785,11793,11818,11826,11827,11837,11843,11853,11860,11860,11870,11875,11886,11892,11902,11910,11919,11926,11936,11943,11952,11959,11960,11969,11976,11986,11994,12002,12010,12010,12023,12026,12035,12042,12052,12059,12060,12070,12076,12086,12093,12102,12110],[12362,12369,12376,12385,12393,12403,12410,12419,12426,12427,12436,12443,12443,12453,12460],[12615,12641,12650,12655,12660,12670,12677,12686,12693,12703,12710,12720,12726,12736,12743],[13078,13118,13120,13127,13137,13144,13153,13160,13170,13177,13177,13188,13194,13203,13210,13220],[13305,13313,13320,13322,13361,13369,13376],[13582,13591,13597,13602,13610,13620,13626,13636,13644,13653,13662,13670,13676,13687,13693,13703,13710,13720,13727,13737,13744,13753,13760,13770,13776,13777,13787,13794,13794,13804,13811,13811,13821,13827],[13957,13977,14020,14027,14037,14044,14053,14060,14070,14076,14077,14088],[14212,14220,14235,14244,14263,14270],[14532,14552,14587,14610,14610,14621,14628,14638,14644,14654,14661,14668,14669,14703,14711,14711,14721,14727,14738,14743,14754,14761,14770,14778,14788,14794,14804,14811,14811,14822,14828,14838,14844,14845,14856,14860,14871],[15056,15069,15102,15119,15127,15138,15144,15145,15156,15160,15171,15177,15188,15193,15205,15211,15221,15228,15238,15245,15254,15260,15272,15277,15288,15294,15302,15310,15320,15327,15338,15343,15353,15360,15370,15377,15387,15395,15404,15411,15412,15422],[15573,15580,15587,15619,15627,15627,15639,15645,15657,15661,15671,15677,15688,15694,15704,15710,15711,15722,15728],[16200,16219,16227,16236,16245,16260,16266,16271,16278,16288,16295,16310,16328,16337,16345,16345,16356,16361,16369,16373,16377,16388,16395,16405,16411,16421,16428,16438,16444,16455,16461,16462,16472,16478,16479,16490,16495,16496,16506,16521],[19690,19700,19709,19722,19754,19762,19772,19779,19790,19795,19807,19812,19837,19845,19856,19862,19872,19878,19894,19898,19905,19912,19924,19929,19940,19945,19957,19962,19974,19979,19990,19995,20005,20012,20015]],"version":"2.0.0"}

    simplifies to

    zn=rncosnθ+isinnθ

    which has a phase angle of {"x":[[356,352,352,352,352,352,352,352,352,352,352,352,353,353,354,354,354,354,355,355,355,355,355,355,355,355,355,355,355,355,355,355,356,357,359,361,364,366,369,372,375,378,379,381,383,384,386,388,390,392,395,397,399,400,401,402,402,402,403,403,404,404,404,405,406,407,408,409],[464,461,461,461,460,459,458,457,456,455,455,455,455,455,455,455,457,459,462,465,468,471,474,477,480,483,485,487,489,490,491,491,492,493,493,493,493,493,493,493,492,490,487,484,481,477,474,470,465,461,456,452,449,447,446,446,446,446,450,456,463,471,480,489,498,506,515,521,527,532,534,536,537]],"y":[[227,235,239,243,248,252,256,258,260,262,263,264,265,266,267,269,270,272,273,274,275,276,277,278,279,280,279,278,277,276,275,274,271,266,259,252,245,238,232,228,225,223,223,222,222,222,222,223,225,228,231,235,240,245,250,257,262,268,274,279,285,290,293,296,299,301,302,303],[206,210,215,221,229,237,245,253,259,265,270,274,277,280,282,285,288,290,292,294,295,296,296,296,295,291,285,278,271,264,256,248,241,233,227,221,214,209,204,199,195,192,189,187,186,185,185,185,189,195,200,206,211,216,221,223,226,228,231,232,234,235,235,235,235,235,235,235,234,234,233,233,233]],"t":[[0,96,105,115,125,131,138,148,155,165,171,181,198,205,215,222,232,238,249,255,266,272,288,299,316,339,481,499,505,515,522,531,539,550,555,565,572,582,589,598,605,615,622,632,639,648,655,664,672,680,694,698,705,715,725,731,738,748,755,765,772,781,789,798,805,815,822,832],[1198,1230,1239,1249,1256,1268,1272,1282,1295,1298,1305,1315,1323,1332,1339,1348,1355,1365,1372,1382,1389,1399,1406,1415,1422,1432,1439,1449,1455,1465,1472,1482,1489,1499,1505,1515,1522,1532,1539,1549,1556,1568,1572,1582,1594,1598,1605,1615,1622,1631,1639,1647,1655,1665,1672,1682,1689,1698,1705,1715,1722,1732,1739,1749,1756,1765,1772,1782,1789,1799,1805,1815,1822]],"version":"2.0.0"}. This is true for all integer values of n i.e. n{"x":[[268,267,267,267,267,266,265,264,263,261,260,259,258,257,256,255,254,254,254,254,254,255,256,257,258,259,260,262,265,268,272,275,279,282,287,290,294,298,301,303,305,306,307,307,307,307,307,307,307,306,306,306,306,307,308,309],[461,460,454,451,449,444,438,432,425,418,411,405,399,393,388,384,381,379,378,377,378,381,384,388,394,400,407,414,421,428,434,439],[385,384,383,383,382,384,387,390,395,401,408,414,421,426,431,435,438,441,444,445,446],[578,577,577,575,575,575,576,581,586,591,597,604,610,616,622,628,633,639,644,650,654,657,660,660,660,659,656,653,649,644,640,635,629,624,618,614,609,604,600,597,595,595,596,598,600,603,607,612,617,623,630,639,646,654,661,666,671,676,678,680],[645,644,642,641,640,638,635,632,628,624,620,617,613,610,607,605,603,601,600,599,597,596,595,594,593,593,593],[602,601,601,602,605,609,615,621,627,634,640,646,652,657,661,664,665,666,667,666]],"y":[[243,242,244,247,251,256,261,268,274,281,287,293,298,304,308,312,316,318,319,317,314,310,306,301,296,290,284,278,271,265,260,256,253,251,249,249,249,249,251,254,258,263,268,275,281,288,295,302,308,315,320,325,329,333,335,336],[225,225,222,222,222,223,225,227,231,235,240,245,252,258,265,272,279,286,293,299,305,310,315,318,321,323,324,324,324,323,322,321],[286,286,286,285,284,284,284,284,284,283,283,283,282,281,280,280,280,279,279,280,281],[220,220,219,218,217,216,216,214,214,213,213,213,213,213,213,213,213,213,213,214,215,216,218,221,224,229,233,238,244,248,254,259,264,270,276,281,287,293,297,300,303,304,304,304,304,303,302,301,300,298,297,294,292,290,287,286,284,283,282,281],[208,208,208,209,210,214,220,227,235,243,251,258,265,272,279,285,291,297,302,305,309,313,316,318,319,320,319],[262,262,261,261,261,260,260,259,259,259,258,257,256,255,254,254,254,254,255,256]],"t":[[0,18,44,54,61,71,78,88,94,104,114,121,128,138,144,154,161,171,179,214,227,230,240,244,254,261,271,278,288,295,304,311,321,328,338,344,355,361,371,378,388,394,404,411,419,428,436,444,454,461,473,478,488,495,505,510],[986,993,999,1001,1005,1011,1023,1028,1037,1045,1054,1061,1071,1081,1087,1095,1104,1111,1121,1131,1137,1145,1154,1162,1170,1178,1187,1195,1204,1211,1228,1229],[1492,1499,1504,1507,1512,1537,1545,1554,1562,1571,1578,1588,1595,1605,1612,1621,1628,1638,1645,1655,1662],[7082,7089,7094,7097,7106,7114,7123,7139,7147,7156,7164,7173,7180,7190,7197,7207,7214,7223,7230,7240,7247,7257,7264,7274,7280,7290,7297,7307,7314,7324,7330,7340,7347,7357,7364,7374,7381,7390,7397,7407,7414,7424,7443,7447,7457,7464,7473,7480,7490,7497,7507,7514,7523,7531,7540,7547,7557,7564,7573,7581],[7947,7960,7966,7968,7973,7981,7991,8000,8007,8014,8024,8031,8043,8047,8058,8064,8074,8081,8091,8097,8107,8114,8124,8131,8141,8147,8164],[8566,8578,8583,8606,8614,8624,8631,8643,8648,8658,8664,8674,8681,8691,8698,8707,8714,8724,8731,8741]],"version":"2.0.0"}

    This consequence helps us find the argument of a complex number raised to a power much more efficiently rather than multiplying all the terms.

    Modulus and Phase Examples

    Without expanding the complex number z=1+i6{"x":[[62,63,125,146,153,156,158,160,167,169,171,173,174,175,176,173,172,168,160,157,146,142,138,129,119,114,100,96,82,78,66,60,56,51,48,48,50,52,56,61,68,76,84,93,98,111,116,130,137,144,152,157,160,165,172,176],[66,64,60,58,57,56,56,57,59,61,68,71,80,84,98,109,121,127,137,143,148,153,157,166,170,177,183,188,192,194],[260,261,262,263,268,271,279,283,293,301,304,315,318,328,334,338,342,345],[286,289,291,296,303,310,318,327,331,344,348,361,369,377,383,388,392,394,395],[577,577,577,577,578,579,580,581,581,582,582,582,581,580,578,577,576,574,574,573,573,572,571,571,571,572,573],[642,642,643,644,648,654,662,671,677,691,696,711,715,728,737,744,748,752,758,763,764,768,769,770],[730,728,727,727,727,726,725,724,723,721,720,719,718,718,718,718,718,718,718,718,718,719],[851,851,851,852,852,853,854,854,854,854,853,852,851,851,851,854,856,861,863,870,873,881,886,891,897,901],[876,874,873,871,871,870,870,870],[526,524,524,523,521,520,518,515,513,511,508,500,497,491,489,486,486,486,488,490,497,500,510,514,518,529,534,540],[948,949,951,957,960,963,966,972,980,993,996,1004,1005,1007,1008,1007,1006,1004,998,995,987,981,975,973,970,966],[1068,1069,1069,1069,1068,1067,1064,1063,1062,1058,1055,1052,1050,1049,1048,1047,1047,1047,1048,1051,1054,1057,1058,1062,1067,1069,1073,1074,1076,1077,1077,1078,1078,1077,1077,1073,1072,1065,1063,1054,1048,1041,1037,1036]],"y":[[242,242,242,239,238,238,238,238,239,239,240,242,243,244,250,254,256,261,271,274,287,291,295,305,317,323,341,347,365,371,385,393,399,408,414,415,419,421,422,423,423,423,421,419,418,415,414,412,412,412,414,415,416,418,421,422],[321,322,322,322,322,322,321,321,321,321,321,321,321,321,318,315,312,311,308,307,306,305,304,303,302,301,301,300,300,300],[333,333,332,332,330,329,325,324,320,318,318,316,315,315,315,315,315,316],[371,371,370,369,367,364,362,359,358,355,354,352,351,350,349,349,349,349,349],[247,246,244,242,241,241,246,254,264,270,277,285,309,317,338,345,358,375,379,392,397,405,408,411,412,412,411],[334,335,335,335,334,332,331,329,328,327,326,324,324,322,321,320,319,319,319,318,318,318,318,319],[267,265,264,265,268,278,288,301,309,323,341,346,358,361,371,377,381,382,384,386,387,387],[270,269,265,263,262,263,267,269,282,288,307,314,335,344,352,358,359,364,365,366,366,364,360,355,347,338],[220,218,217,216,217,219,220,221],[229,228,227,225,227,228,230,236,240,245,251,273,281,306,314,340,359,379,400,411,437,444,465,470,475,484,488,490],[221,221,218,214,214,213,213,215,218,232,237,254,260,281,287,307,321,328,350,358,378,391,401,405,409,415],[118,116,114,112,111,111,114,115,118,125,134,144,149,155,161,175,179,188,191,195,197,198,198,198,194,193,187,185,183,179,178,176,172,168,167,163,162,162,164,169,177,185,192,196]],"t":[[0,13,30,52,59,68,68,76,84,93,96,101,109,112,126,134,143,151,159,168,176,184,185,193,201,209,219,227,234,251,252,265,268,287,293,301,310,320,328,334,343,351,363,368,376,384,393,403,410,420,428,435,443,451,460,468],[875,879,890,895,901,910,940,947,951,960,968,976,985,993,1001,1010,1022,1029,1035,1035,1043,1043,1051,1060,1060,1068,1076,1085,1093,1101],[1554,1562,1566,1572,1585,1593,1603,1610,1620,1629,1641,1643,1654,1660,1671,1677,1687,1693],[1929,1935,1943,1952,1960,1970,1977,1987,1993,2004,2010,2021,2027,2037,2043,2054,2060,2071,2075],[2923,2928,2931,2936,2944,2960,2971,2977,2988,2994,2994,3005,3010,3021,3027,3038,3044,3054,3060,3071,3077,3088,3094,3105,3121,3127,3138],[3424,3428,3444,3454,3461,3471,3477,3488,3494,3505,3511,3521,3527,3538,3544,3555,3561,3561,3571,3577,3588,3594,3602,3611],[3842,3848,3861,3877,3888,3894,3905,3911,3921,3927,3938,3944,3955,3961,3972,3977,3988,3994,3994,4002,4011,4021],[4424,4429,4432,4436,4444,4456,4461,4472,4478,4489,4494,4505,4511,4521,4528,4538,4544,4555,4561,4571,4578,4588,4594,4606,4611,4622],[4776,4787,4795,4811,4822,4828,4838,4843],[5429,5438,5443,5453,5470,5478,5481,5489,5495,5495,5506,5511,5522,5528,5539,5545,5555,5561,5572,5578,5589,5595,5605,5611,5612,5622,5628,5628],[6200,6204,6214,6215,6220,6228,6229,6239,6245,6265,6270,6278,6287,6295,6303,6312,6322,6329,6339,6345,6356,6362,6373,6378,6379,6387],[6904,6909,6915,6921,6939,6945,6956,6962,6962,6973,6979,6990,6995,6996,7007,7012,7023,7029,7039,7045,7056,7062,7073,7079,7090,7095,7106,7112,7112,7124,7129,7129,7137,7145,7156,7162,7170,7179,7189,7195,7206,7212,7223,7229]],"version":"2.0.0"}, find its argument.

    Solution:

    Step1: Convert the complex number into its polar form:

    z=r6cosθ+isinθ6=r6cosnθ+isinnθ{"x":[[77,76,77,82,86,92,95,106,110,118,123,132,140,149,157,164,168,176,178,182],[89,91,94,95,101,104,106,112,115,123,135,139,152,156,167,174,180,184,187,188],[249,250,251,251,252,254,257,260,262,264,269,270,274,275,278,278,279,280,280,281,281,280,280,279,278,277,275,274,273,271,271,271,272,274,276,276,278,279,280,282,283,285,288,290,294,296,302,304,313,316,326,329,336,345,349,358,360,366,368,370,373,378,381,382,383,384],[420,421,422,423,423,423,423,422,420,419,417,414,412,410,407,399,396,390,387,383,380,379,379,379,380,383,385,387,390,393,399,402,404,406,410,411,413,414,415,415,414,411,408,403,400,393,391,388],[489,487,480,475,469,462,459,455,448,445,436,433,425,423,421,418,417,416,418,419,423,425,428,436,444,451,455],[522,521,519,517,517,516,515,514,513,512,509,506,501,499,497,495,491,489,488,485,485,484,484,486,488,491,493,498,500,507,509,516,518,520,524,527,529,531,533,533,533,533,533,534,535,535,538,539,544,545,550,553,556,557,558,560,562,562,562,562,560,556,553,550,549,544,543,537,536,535,534,534,534,535,536,540,544,546,549,557,563,570,577,584,590,593,600,602,605,606,606,606,605,604,602,599,597,596,596,594,594,595,596,597,599,601,604,605,606,606,606,606,605,604,603,601,599,598,596,593,591,590,585,583],[647,647,647,647,647,646,646,645,645,645,645,645,647,647,649,650,653,656,660,664,668,672,674,680,681,684,685,686,686,686,686,686,686,687,687,687,687,687,686,686],[736,736,735,734,733,731,729,728,726,725,724,721,721,720,720,720,721,722,724,730,732,738,741,748,752,757,759,760,764,765,766,767,767,767,764,762,758,754,752,750,746,744,737,734,728,725,724,724,725,726,732,738,741,747,751],[798,799,800,804,806,813,815,823,826,833,837,841,842,844,845,847,848],[826,826,826,826,825,824,823,822,822,822,822,822,823,824,825,826,827,828],[883,883,883,883,883,883,881,880,879,878,878,878,878,878,879,880,881,885,887,890,893,894],[891,890,889,889,888,888],[958,957,956,956,955,953,950,948,946,942,939,937,930,927,925,924,924,925,926,931,935,939,941,943,947,949,951,952,956,957,957,958,958,957,956,952,951,946,944,937,936,931],[975,976,979,980,982,982,981,980,979,978,978,978,978,978,979,979,980,981],[979,979,979,979,979,980,980],[1006,1006,1006,1006,1006,1006,1006,1006,1006,1006,1005,1004,1004,1003,1003,1002,1001,1001,1001,1001,1002,1003,1005,1007,1008,1011,1016,1018,1024,1027,1029,1031,1033,1034,1034,1035,1035,1036,1036,1036,1036],[1058,1058,1059,1059,1059,1059,1059,1058,1058,1058,1057,1056,1056,1056,1056,1056,1056,1056,1057,1059,1061,1063,1066,1068,1071,1073,1075,1075,1076,1076,1076,1076,1076,1076,1076,1077,1077,1078],[1099,1098,1098,1098,1097,1097,1097,1097,1098,1099,1100,1102,1104,1106,1107,1108,1110,1111,1113,1114,1115,1115,1115,1115,1115,1114,1114,1113,1110,1109,1107,1105,1103,1100,1099,1096,1096,1095,1095,1098,1099,1101,1105,1109,1111],[1111,1111,1112,1113,1116,1117,1118,1121,1123,1123,1125,1126,1127,1128,1129,1129,1128,1126,1125,1119,1117,1109,1106,1103]],"y":[[254,253,252,250,249,247,247,245,244,243,242,241,239,238,236,236,235,235,235,237],[288,288,288,287,286,286,285,284,283,282,279,278,276,276,274,274,274,274,274,274],[211,211,209,208,208,207,207,207,210,212,222,225,239,244,257,262,271,286,291,306,311,321,326,334,343,351,358,364,367,373,374,375,371,367,362,359,353,349,341,333,325,315,305,295,286,280,266,262,248,245,234,231,226,220,218,215,215,215,215,215,216,218,220,222,223,224],[144,142,133,130,127,125,124,123,122,121,121,121,122,123,125,132,135,142,147,155,164,172,175,182,184,188,188,189,190,190,189,188,187,185,183,181,179,177,176,175,175,176,179,184,187,196,199,206],[156,156,157,160,165,172,176,181,192,199,222,230,255,264,273,289,297,321,333,339,351,357,362,375,382,386,387],[275,275,273,271,270,269,269,267,266,266,266,266,268,269,271,273,279,282,285,293,298,302,310,318,322,324,325,326,326,325,324,319,317,315,311,308,306,302,301,300,301,303,304,309,313,314,318,319,321,321,320,318,315,313,311,310,303,297,290,286,279,270,267,266,266,269,271,280,284,287,294,297,300,305,306,308,309,309,309,306,302,298,292,287,281,278,270,268,262,261,260,258,258,258,259,260,262,263,264,267,270,272,275,277,280,282,287,288,290,293,295,297,300,302,303,306,307,309,310,312,313,314,315,315],[262,263,265,269,272,283,287,298,301,308,309,312,309,307,300,296,286,279,273,268,265,263,262,262,263,268,272,277,280,283,288,291,294,296,303,306,309,311,315,317],[248,247,245,244,243,243,244,246,250,253,256,270,277,288,293,304,314,321,324,331,332,334,334,333,329,324,321,317,308,303,298,292,273,262,253,245,240,236,235,234,233,233,236,238,246,253,259,263,266,266,266,265,264,262,261],[292,292,292,291,291,291,291,291,291,291,291,291,291,291,291,291,291],[263,262,263,268,271,284,294,303,310,315,317,319,322,325,326,328,328,329],[262,264,266,268,271,278,287,296,304,312,314,317,320,322,323,324,325,325,325,323,320,317],[224,222,220,219,219,220],[274,267,265,264,263,262,261,261,261,263,264,266,271,275,278,282,283,286,287,289,291,293,293,294,296,297,298,299,301,303,304,307,310,312,313,315,316,319,320,323,324,325],[275,274,272,272,275,277,286,294,302,307,314,317,323,328,331,332,334,334],[243,241,234,233,232,231,233],[275,276,277,278,279,283,285,290,295,298,304,308,310,313,314,314,314,313,312,310,307,302,297,295,292,289,286,285,285,287,290,294,298,303,305,312,314,315,318,319,321],[271,272,274,278,283,285,294,297,303,305,308,309,310,309,308,307,304,302,298,293,288,283,280,280,280,281,286,288,294,296,297,301,304,306,308,309,311,311],[260,261,262,263,268,276,280,292,296,305,307,311,313,314,314,314,312,311,305,303,293,286,278,269,266,260,258,256,252,250,249,249,250,252,254,262,264,271,272,276,277,277,278,278,278],[193,192,192,191,193,194,196,202,212,218,237,244,262,275,287,300,314,328,334,352,358,373,378,382]],"t":[[0,11,25,48,50,61,69,75,86,91,92,102,108,119,125,135,141,152,158,169],[440,444,452,458,469,475,475,485,491,500,508,519,525,535,542,552,558,568,575,585],[10827,10838,10841,10845,10855,10862,10872,10879,10889,10895,10907,10912,10925,10929,10939,10945,10955,10962,10975,10979,10988,10995,10996,11007,11012,11025,11029,11041,11045,11058,11062,11075,11100,11104,11112,11123,11129,11129,11139,11145,11156,11162,11171,11179,11189,11195,11206,11212,11223,11229,11239,11245,11254,11262,11272,11279,11289,11295,11296,11306,11312,11323,11329,11339,11346,11354],[11650,11663,11666,11671,11679,11690,11696,11696,11706,11712,11713,11723,11729,11729,11740,11746,11756,11763,11763,11773,11779,11790,11796,11808,11812,11823,11829,11829,11840,11846,11860,11863,11863,11873,11879,11880,11890,11896,11896,11910,11913,11923,11930,11942,11946,11956,11963,11973],[12337,12350,12353,12354,12363,12372,12379,12380,12390,12396,12407,12413,12423,12429,12429,12440,12446,12458,12463,12473,12479,12480,12490,12496,12506,12513,12520],[12837,12847,12850,12851,12855,12863,12866,12873,12879,12880,12890,12896,12907,12913,12913,12922,12930,12930,12940,12946,12947,12957,12963,12973,12984,12990,12996,13007,13013,13024,13029,13040,13046,13047,13057,13063,13073,13080,13090,13102,13105,13113,13121,13130,13138,13146,13156,13164,13174,13180,13190,13196,13207,13213,13214,13224,13230,13240,13246,13257,13263,13274,13280,13290,13297,13307,13313,13324,13330,13330,13341,13346,13347,13357,13363,13374,13380,13381,13391,13396,13407,13414,13424,13430,13440,13446,13457,13464,13474,13480,13481,13491,13496,13507,13513,13524,13530,13531,13541,13546,13557,13566,13574,13580,13590,13596,13607,13613,13614,13624,13630,13631,13641,13646,13651,13658,13663,13665,13675,13680,13681,13691,13696,13707],[13971,13982,13997,14007,14014,14024,14030,14041,14047,14058,14063,14074,14113,14124,14130,14141,14147,14158,14163,14174,14180,14191,14197,14207,14214,14224,14230,14241,14247,14247,14258,14263,14264,14275,14280,14291,14297,14297,14308,14314],[14504,14514,14517,14522,14530,14541,14547,14558,14563,14564,14575,14580,14591,14597,14597,14608,14613,14625,14630,14641,14647,14658,14663,14675,14680,14689,14697,14697,14708,14714,14714,14725,14730,14741,14747,14758,14763,14775,14780,14782,14792,14797,14808,14814,14825,14830,14839,14847,14857,14863,14875,14880,14891,14897,14904],[15196,15222,15230,15241,15247,15258,15264,15275,15280,15292,15297,15308,15314,15314,15325,15330,15347],[15500,15514,15531,15541,15547,15558,15564,15575,15580,15589,15597,15598,15608,15614,15625,15631,15631,15642],[15911,15922,15926,15930,15935,15942,15947,15959,15964,15975,15981,15983,15993,15997,16000,16009,16014,16025,16031,16042,16047,16055],[16195,16199,16214,16225,16231,16239],[16620,16633,16640,16647,16648,16656,16664,16665,16672,16681,16681,16692,16698,16709,16714,16726,16731,16742,16748,16759,16764,16776,16781,16781,16793,16798,16798,16810,16814,16815,16826,16831,16842,16849,16859,16864,16865,16876,16881,16892,16898,16909],[17062,17074,17077,17081,17103,17106,17118,17125,17131,17139,17148,17148,17159,17164,17176,17181,17192,17198],[17364,17374,17377,17382,17392,17398,17414],[17658,17670,17674,17675,17681,17690,17698,17709,17715,17715,17727,17731,17743,17748,17760,17765,17776,17792,17798,17801,17810,17815,17826,17831,17832,17843,17848,17856,17865,17876,17881,17893,17898,17910,17915,17926,17931,17934,17943,17948,17956],[18138,18165,18174,18182,18193,18198,18210,18215,18226,18232,18243,18248,18256,18274,18282,18292,18298,18299,18310,18315,18327,18332,18343,18348,18360,18365,18376,18382,18393,18398,18399,18410,18415,18426,18432,18443,18448,18460],[18671,18691,18698,18701,18710,18715,18727,18732,18743,18748,18760,18765,18777,18784,18793,18798,18810,18815,18827,18832,18844,18848,18860,18865,18873,18882,18882,18894,18898,18910,18915,18927,18932,18943,18948,18960,18965,18977,18982,18994,18998,18999,19011,19015,19027],[19264,19269,19275,19282,19293,19298,19299,19311,19315,19327,19332,19344,19349,19360,19365,19377,19382,19394,19399,19410,19415,19427,19432,19436]],"version":"2.0.0"}

    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    Step2: Comparing with the general form, we deduce that

    sinnθ=cosnθ=12

    Giving us the principle argument as follows:

    tannθ=11=1

    nθ=π4{"x":[[405,398,397,394,391,389,384,383,379,378,377,376,376,376,378,379,384,386,391,393,399,405,411,418,425,428,430,436,439,446,450,452,454,454,454,452,448,446,438,432,426,419,412,405,401,391,388,385,380,378,378,378,382,386,389,392,404,414,425,437,450,456],[546,545,545,548,552,556,559,566,577,581,592,595,601],[539,540,541,544,550,557,561,573,582,585,592,596,599,605,608],[720,719,719,718,716,714,712,710,709,707,706,704,704,703,703,703],[756,759,760,760,760,759,758,756,755,754,753,751,750,749,749,749,749,749,749,749],[665,666,669,675,684,694,700,716,721,738,744,758,763,767,775,782,788,791,793,794],[800,799,793,791,788,786,783,778,775,770,764,761,758,751,745,736,733,730,726,722,719,717,716,716,716],[802,802,799,798,797,796,793,792,790,787,784,780,776,775,770,768,768,768,768,769,771,774,777,779,782,789,792,796,806,814,818,829,835,841,844,850,852,855],[823,823,823,823,823,823,822,821,820,820,819,819,819,819,820],[265.99999999999994,265.99999999999994,263.99999999999994,262.99999999999994,262.99999999999994,262.99999999999994,261.99999999999994,261.99999999999994,261.99999999999994,261.99999999999994,260.99999999999994,260.99999999999994,260.99999999999994,259.99999999999994,259.99999999999994,258.99999999999994,259.99999999999994,260.99999999999994,261.99999999999994,261.99999999999994,262.99999999999994,263.99999999999994,265.99999999999994,268.99999999999994,272.99999999999994,277.99999999999994,283.99999999999994,286.99999999999994,295.99999999999994,298.99999999999994,306.99999999999994,309.99999999999994,314.99999999999994,316.99999999999994,322.99999999999994,323.99999999999994,327.99999999999994,328.99999999999994,329.99999999999994,330.99999999999994,330.99999999999994,330.99999999999994,330.99999999999994,329.99999999999994,328.99999999999994,327.99999999999994,327.99999999999994,327.99999999999994,330.99999999999994,331.99999999999994,333.99999999999994]],"y":[[220,219,220,224,231,235,250,255,274,280,287,302,310,328,338,343,352,354,358,359,360,359,356,351,343,338,333,322,316,298,285,271,260,249,243,230,222,218,210,206,204,204,206,211,214,228,233,239,248,257,264,266,270,272,273,274,274,274,272,270,267,266],[265,266,267,268,269,269,269,269,268,267,266,266,265],[319,320,320,320,319,318,317,314,312,311,310,310,309,309,309],[189,188,190,195,202,212,222,232,237,248,251,261,267,272,277,280],[188,180,179,182,186,189,193,200,203,208,212,226,230,239,242,249,255,260,265,267],[204,202,201,199,196,194,193,190,189,186,185,183,182,181,179,179,178,178,179,180],[233,235,244,248,257,262,267,277,282,293,305,312,318,331,344,361,366,371,380,388,394,400,402,406,407],[319,321,328,331,334,338,345,350,354,363,371,379,388,392,403,410,415,419,422,423,424,425,425,425,425,425,425,425,425,424,424,423,422,422,422,422,422,422],[386,389,401,410,418,423,432,437,445,449,462,469,475,481,484],[259,262,277,281,288,297,304,311,315,325,329,339,341,347,349,350,349,342,336,332,324,319,310,300,290,279,270,266,255,253,248,247,248,250,256,259,269,274,283,293,298,305,316,322,333,344,354,358,370,372,377]],"t":[[0,13,16,23,33,39,50,56,66,73,73,83,90,99,106,119,123,134,140,140,152,156,168,173,185,190,193,200,206,218,223,233,240,250,256,269,273,282,290,300,306,320,323,334,340,352,356,359,366,373,383,390,399,406,407,417,423,433,440,449,456,464],[795,799,815,832,840,850,856,867,873,883,890,900,907],[1110,1123,1126,1132,1140,1150,1157,1167,1173,1184,1190,1190,1200,1207,1207],[1523,1538,1548,1557,1567,1573,1584,1590,1603,1607,1619,1624,1634,1640,1650,1657],[1818,1829,1832,1857,1865,1874,1874,1884,1890,1890,1901,1907,1917,1924,1924,1932,1940,1951,1957,1965],[2206,2219,2224,2234,2240,2251,2257,2267,2274,2284,2290,2303,2307,2307,2318,2324,2338,2340,2351,2357],[2597,2607,2610,2611,2616,2624,2624,2635,2641,2651,2657,2658,2668,2674,2682,2694,2699,2699,2709,2718,2724,2734,2741,2751,2765],[2957,2970,2973,2975,2982,2984,2991,2999,3002,3007,3018,3024,3034,3041,3051,3057,3069,3074,3087,3091,3091,3101,3107,3108,3118,3124,3124,3133,3141,3151,3157,3168,3175,3187,3191,3201,3207,3218],[3433,3445,3448,3449,3458,3467,3474,3475,3485,3491,3502,3507,3518,3524,3535],[1653247102346,1653247102356,1653247102360,1653247102362,1653247102367,1653247102377,1653247102383,1653247102394,1653247102400,1653247102411,1653247102417,1653247102427,1653247102433,1653247102444,1653247102450,1653247102461,1653247102483,1653247102494,1653247102500,1653247102511,1653247102517,1653247102517,1653247102528,1653247102533,1653247102544,1653247102550,1653247102561,1653247102567,1653247102581,1653247102584,1653247102594,1653247102600,1653247102611,1653247102617,1653247102627,1653247102633,1653247102644,1653247102650,1653247102661,1653247102667,1653247102667,1653247102678,1653247102683,1653247102684,1653247102694,1653247102703,1653247102710,1653247102717,1653247102725,1653247102734,1653247102742]],"version":"2.0.0"}

    Step3: Plug in the value of n , which is 6:

    6θ=π4{"x":[[238,238,235,234,232,214,211,199,195,183,179,171,169,167,165,163,162,162,162,163,166,170,175,181,188,191,200,203,212,215,218,221,222,221,218,214,207,204,198,195,189,187,180,178,176,175,176,177,182,184,187],[315,313,307,303,301,294,289,286,283,281,280,280,281,282,288,290,298,304,311,314,317,323,326,330,332,339,341,345,347,348,348,346,343,341,334,331,323,317,312,307,303,302,300,299,299,299,302,305,307,318,326,330,335,346,351,368],[413,412,413,415,419,424,427,437,440,447,450,453,456,461,466,468,473,474,475],[415,416,420,424,427,430,433,440,443,447,453,459,467,472,475,478,480],[548,549,551,551,552,552,552,552,550,548,546,541,540,535,535,534,533,533,534,534,535],[577,577,578,578,579,579,578,577,576,573,572,569,567,563,562,560,560,559,559,560,560,561,561,562],[518,516,512,511,511,511,512,516,518,527,531,544,549,565,571,576,586,595,598,612,614,615],[528,528,531,533,541,549,559,568,573,588,593,607,613,617,625,633,640,643,647,649,652],[577,577,577,575,574,572,571,569,568,568,567,567,567,568,570,573,577,582,588,591,601,604,614,617,621,627,633,635,641,643,645],[614,614,613,612,611,610,610,610,610,610,611,612,613,614,615,617]],"y":[[177,176,173,173,172,186,190,206,213,235,244,268,277,286,295,312,320,328,341,345,353,359,363,364,363,361,353,350,336,326,315,305,290,287,284,284,287,289,294,297,306,311,326,332,345,349,358,359,363,363,363],[210,210,210,212,214,224,233,244,257,272,285,297,310,314,326,329,334,335,335,334,332,327,323,319,314,297,289,270,256,243,233,224,217,215,210,209,211,214,220,228,238,244,254,264,269,273,278,280,281,282,280,278,277,272,269,260],[244,245,245,245,245,244,243,242,241,241,241,240,240,239,239,238,238,238,238],[284,284,284,283,282,281,281,279,278,277,275,273,271,270,269,268,268],[173,172,167,166,166,167,170,176,183,192,197,215,221,237,240,249,254,258,261,262,264],[190,189,187,185,183,182,184,188,191,203,208,226,232,253,258,274,278,290,295,298,302,303,304,305],[174,174,174,174,175,176,176,178,178,178,177,175,174,169,168,166,165,163,163,163,164,164],[347,346,344,343,341,339,337,335,335,333,333,331,330,330,329,328,327,327,327,327,328],[361,363,365,376,386,396,400,413,416,419,422,430,433,435,437,438,438,438,438,437,434,433,431,430,429,428,427,426,426,426,426],[405,404,404,409,412,431,446,451,455,461,464,469,472,473,474,475]],"t":[[0,9,13,15,20,52,59,70,75,86,92,103,109,109,120,125,126,136,142,142,153,159,167,176,186,192,203,209,220,225,237,245,263,267,277,286,292,303,309,309,320,325,336,342,353,359,370,375,386,392,393],[694,705,708,710,717,726,736,742,752,759,769,776,786,792,803,809,822,826,835,842,843,855,859,859,870,876,890,892,905,909,919,926,936,942,953,959,969,976,986,992,1003,1009,1021,1026,1026,1036,1042,1043,1053,1059,1068,1076,1078,1086,1092,1103],[1412,1419,1434,1443,1453,1459,1470,1476,1486,1492,1493,1501,1504,1510,1520,1526,1537,1543,1551],[1716,1727,1730,1735,1743,1743,1753,1759,1759,1770,1776,1786,1793,1803,1809,1820,1827],[2071,2085,2088,2093,2095,2103,2110,2120,2126,2136,2143,2153,2159,2170,2176,2187,2193,2202,2209,2220,2226],[2363,2367,2370,2372,2376,2393,2410,2420,2426,2437,2443,2453,2460,2470,2476,2487,2493,2505,2509,2520,2526,2527,2537,2543],[2838,2847,2853,2854,2860,2870,2876,2887,2893,2904,2910,2920,2926,2937,2943,2943,2952,2960,2970,2989,2993,3004],[3392,3410,3427,3437,3443,3454,3460,3470,3477,3487,3493,3504,3510,3510,3521,3527,3537,3543,3552,3552,3560],[3775,3802,3810,3820,3827,3837,3843,3854,3860,3860,3871,3877,3887,3893,3904,3910,3923,3927,3938,3944,3954,3961,3971,3977,3977,3988,3993,4002,4010,4021,4027],[4200,4208,4214,4235,4244,4266,4269,4277,4277,4285,4293,4304,4310,4311,4322,4324]],"version":"2.0.0"}

    θ=π24{"x":[[271,266,265,262,259,255,251,247,243,242,239,238,238,238,238,240,243,248,253,255,261,264,271,277,283,289,295,298,305,308,310,311,311,311,308,305,301,299,297,292,285,282,275,272,268,256,250,245,240,239,239,239,244,246,254,261,269,278,283,300,306,318],[400,400,402,406,408,417,424,430,438,446,453,456,467,470,477,479,482,484,486,486],[410,412,413,416,419,425,432,440,447,450,462,465,476,483,489,492,497],[597,597,598,599,599,599,598,597,595,593,591,590,587,586,584,584,583,583,583,584,585,586,589,591],[638,638,639,639,639,639,638,637,635,633,632,630,629,626,625,623,623,622,622,622,623,624,625,626,627,629],[562,561,557,556,559,560,568,579,588,593,600,617,622,640,651,663,668,681,689,691,696,698,702],[546,549,566,571,583,595,607,613,620,633,640,646,653,673,680,692,698,709,719,727,734,739,741,744],[552,554,560,568,576,584,586,587,590,592,592,591,588,584,582,572,569,559,556,553,548,543,540,539,538,538,539,541,543,548,553,560,563,574,578,590,597,604,607,616,619,623,625],[668,668,667,666,664,662,658,654,650,648,647,643,642,641,640,640,641,644,648,654,661,669,677,682,695,699,703,711,719,725,731,733,738,739],[699,698,696,696,696,696,696,696,696,696,696,697,698,699,701,703]],"y":[[207,208,208,209,211,215,220,228,238,243,261,268,287,293,299,312,321,330,337,339,342,342,342,340,335,328,319,314,295,283,269,262,243,237,221,212,204,201,197,193,191,191,193,195,198,212,222,234,246,250,261,263,269,270,274,275,275,274,273,267,264,257],[232,233,234,233,233,231,230,228,226,224,223,222,219,218,216,216,216,216,216,217],[270,269,268,267,266,263,261,258,256,256,253,253,251,250,248,248,247],[147,146,145,145,146,147,155,158,167,178,188,193,204,213,221,224,233,236,242,243,244,246,247,247],[173,171,166,163,162,163,167,172,177,184,191,199,204,217,222,233,236,246,249,252,256,259,260,260,260,260],[154,154,154,154,155,156,158,158,158,158,158,156,155,154,153,153,153,154,156,157,159,161,164],[309,309,304,303,301,299,296,295,294,291,290,289,288,286,285,284,283,282,281,281,282,283,284,286],[383,382,378,375,374,375,376,377,381,386,392,398,406,414,418,432,437,451,455,460,467,473,479,480,483,487,487,488,489,489,488,487,487,484,483,481,479,477,477,475,475,474,474],[379,380,387,390,397,401,413,422,430,434,438,447,450,456,461,462,464,466,467,466,465,463,460,458,454,452,451,449,447,445,444,444,444,443],[417,418,425,427,430,442,451,470,475,484,492,499,503,509,513,517]],"t":[[0,13,22,30,40,46,56,63,73,80,92,96,107,113,113,123,130,140,146,157,163,163,176,181,192,197,207,213,226,230,242,246,260,263,277,280,290,296,298,308,313,322,330,330,340,346,358,363,376,380,389,397,407,413,423,430,439,454,456,463,471,480],[892,907,910,914,923,930,940,947,957,963,974,980,991,997,1007,1013,1014,1024,1030,1041],[1219,1230,1233,1238,1241,1247,1257,1264,1274,1280,1291,1297,1307,1313,1324,1330,1341],[1624,1634,1637,1638,1647,1656,1664,1674,1680,1691,1697,1697,1707,1714,1724,1730,1740,1747,1757,1764,1764,1774,1780,1791],[1937,1944,1947,1949,1964,1973,1980,1991,1997,2008,2014,2024,2030,2041,2047,2058,2064,2074,2080,2081,2093,2097,2108,2114,2114,2125],[2384,2397,2400,2401,2430,2440,2449,2467,2472,2473,2482,2491,2498,2508,2514,2522,2531,2541,2547,2558,2564,2564,2575],[2938,2945,2948,2952,2956,2964,2975,2981,2981,2991,2997,2998,3006,3014,3024,3031,3031,3041,3048,3060,3064,3075,3082,3090],[3426,3429,3434,3435,3450,3467,3473,3474,3482,3489,3498,3508,3515,3525,3531,3541,3548,3558,3564,3565,3575,3581,3592,3598,3608,3614,3625,3631,3631,3642,3648,3658,3664,3675,3681,3692,3698,3708,3714,3725,3731,3742,3744],[3967,3978,3982,3986,3990,3998,4009,4014,4025,4031,4031,4042,4048,4058,4064,4065,4073,4081,4092,4098,4112,4115,4125,4131,4144,4148,4151,4159,4164,4174,4181,4192,4198,4198],[4410,4420,4423,4424,4431,4441,4450,4468,4473,4481,4492,4498,4498,4509,4515,4526]],"version":"2.0.0"}

    Hence the principle argument is π/24{"x":[[287,287,287,287,287,287,286,286,285,284,283,282,281,279,277,276,276,274,274,273,273,274],[354,355,356,356,356,356,356,355,354,353,350,349,348,347,346,345,345,345,345,345,345,345,346,347,348,348],[240,240,241,242,244,254,258,262,267,285,290,303,309,323,335,348,354,371,376,389,392,395,400,403],[447,444,443,439,438,436,432,430,425,423,416,413,405,400,397,392,389,387,382,379,376,374,373,371,371,370,371],[445,445,448,449,454,457,461,464,466,472,475,478,479,480,480,479,476,474,467,462,457,454,450,445,445,444,444,445,446,449,454,460,464,475,479,491,496,506,512,516,518,520],[546,546,546,545,543,541,538,534,532,528,527,525,525,526,528,530,536,539,547,551,561,564,567,573,576,581,584,586],[566,566,566,566,566,566,565,562,560,557,555,555,554,554,554,554]],"y":[[136,138,149,153,157,166,171,175,185,190,194,204,209,219,228,232,236,243,246,250,252,252],[128,127,127,128,131,133,136,142,151,156,173,179,189,194,205,215,224,234,242,245,254,257,261,262,262,261],[155,156,156,157,157,157,156,154,152,145,143,139,137,133,130,126,125,122,121,119,119,119,119,119],[129,135,138,149,154,159,169,174,192,199,219,227,251,266,274,289,295,301,320,331,340,349,352,360,362,366,365],[302,301,292,290,285,284,281,280,279,279,281,285,289,295,302,306,318,323,336,345,354,358,365,373,375,378,379,380,380,380,379,377,376,371,370,365,364,361,360,360,360,360],[281,280,281,285,292,296,306,319,323,333,339,344,346,348,350,350,350,350,347,346,343,342,340,338,337,335,334,334],[311,313,314,315,320,331,335,349,353,366,373,377,382,385,387,391]],"t":[[0,7,12,13,20,21,29,30,40,46,46,57,63,74,79,80,91,96,97,107,113,124],[299,307,324,329,341,346,346,357,363,374,380,390,396,397,407,413,424,430,441,446,457,463,474,479,490,496],[765,773,780,788,796,807,813,813,823,830,840,846,847,856,863,873,880,890,896,907,913,913,923,930],[1166,1172,1180,1189,1196,1198,1207,1213,1223,1230,1240,1246,1257,1264,1274,1280,1280,1290,1297,1307,1313,1323,1330,1340,1347,1357,1374],[5139,5149,5152,5157,5165,5175,5181,5182,5192,5198,5208,5215,5225,5232,5242,5248,5262,5265,5277,5281,5292,5298,5310,5315,5327,5331,5332,5348,5348,5358,5365,5377,5381,5392,5398,5408,5415,5425,5431,5444,5448,5448],[5676,5682,5707,5715,5725,5731,5742,5748,5759,5765,5775,5782,5792,5798,5809,5815,5825,5832,5842,5849,5859,5865,5865,5876,5881,5892,5898,5899],[6100,6114,6117,6124,6132,6142,6148,6159,6165,6176,6182,6192,6198,6199,6209,6215]],"version":"2.0.0"} radians.

    Applications of Complex numbers

    The concept of complex numbers is not entirely pure and abstract but has applications everywhere.

    One of the most straightforward applications of complex numbers is finding the roots of a quadratic equation. It is also one of the places where the concept of a complex number arose.

    Find the root of the quadratic equation x2-x+4=0{"x":[[233,237,242,249,257,259,263,265,265,265,263,258,256,248,246,238,236,229,225,224,221],[302,299,297,294,292,289,288,287,282,281,276,274,271,270,269,269,270,272,276,280,286,290,300,304,317,321,334,339],[361,361,361,361,361,361,361,363,364,369,371,380,383,393,398,403,404,407,408,408,406,404,400,398,395,394,393,392,391,391,391,392,397,402,408,417,426,430,439],[424,424,428,431,441,446,451,461,466,470,474,486,490,497],[563,563,563,563,564,568,574,578,586,597,600,608,610,611,614,614,615,615,613,607,606,597,594,587,582,580],[635,633,629,627,625,621,620,616,615,614,615,616,620,623,625,630,633,643,647,651,660,670,681],[751,752,753,754,757,761,768,771,781,784,788,791,799,801],[773,773,773,774,774,774,774,774,774,772,771,770,769,768,768,768],[883,881,878,874,873,870,868,866,865,865,865,865,868,870,878,881,885,893,897,909,917,921,927],[903,904,904,906,906,906,906,905,903,902],[1008,1011,1015,1017,1020,1023,1027,1038,1042,1053,1056,1064],[982,981,983,986,997,1003,1019,1024,1034,1043,1047,1051,1061],[1117,1118,1118,1118,1119,1119,1118,1117,1117,1116,1116,1116,1117,1118,1124,1127,1134,1137,1144,1149,1154,1156,1159,1163,1164,1166,1166,1166,1166,1166,1163,1159,1157,1146,1141,1127,1123,1108,1099,1091]],"y":[[377,377,378,380,385,387,395,401,407,411,419,430,434,447,450,461,464,470,472,472,472],[386,386,386,386,387,388,389,390,398,402,416,421,429,434,440,443,445,449,452,454,455,455,455,454,450,448,440,437],[308,303,301,298,296,295,292,288,285,280,279,274,272,269,269,270,271,277,279,285,294,297,307,310,318,321,323,325,330,331,333,333,332,330,327,322,318,315,311],[384,385,384,384,381,380,379,378,377,376,375,374,374,374],[361,360,359,357,354,351,347,346,344,344,344,348,349,351,355,357,359,365,374,384,387,397,399,408,412,414],[349,350,351,351,352,355,358,367,371,381,388,391,398,401,402,405,406,407,407,407,404,401,397],[366,366,366,365,364,364,363,362,361,361,361,361,360,360],[345,344,342,342,344,349,355,359,363,372,377,381,385,395,398,400],[343,347,351,358,361,369,374,379,382,386,388,389,391,391,392,392,391,390,389,387,385,385,384],[359,362,367,376,381,397,402,411,424,432],[375,375,375,374,374,374,373,372,371,371,370,370],[411,412,412,411,408,407,402,402,400,399,399,398,397],[343,345,348,351,364,370,389,396,410,431,444,449,455,458,465,467,468,468,464,459,451,446,435,415,407,385,378,358,352,347,337,329,326,321,320,322,324,334,341,349]],"t":[[0,8,17,34,39,50,56,66,73,81,89,98,109,114,123,131,139,148,156,164,171],[487,496,501,506,514,525,526,533,540,548,557,566,573,573,581,590,590,598,606,614,626,636,643,648,656,665,673,690],[879,886,892,899,906,907,915,928,936,940,950,956,965,977,982,990,998,1007,1015,1028,1037,1040,1048,1056,1067,1073,1074,1081,1090,1098,1109,1115,1126,1137,1146,1149,1156,1165,1177],[1440,1449,1482,1490,1498,1507,1507,1515,1523,1524,1532,1544,1553,1557],[1916,1924,1932,1940,1951,1959,1965,1975,1983,1991,1998,2007,2015,2015,2023,2032,2032,2040,2059,2070,2074,2085,2090,2100,2107,2116],[2333,2341,2353,2360,2365,2366,2376,2382,2392,2399,2407,2415,2424,2432,2435,2441,2449,2461,2465,2466,2476,2484,2493],[2933,2941,2949,2963,2966,2974,2982,2990,3001,3007,3007,3015,3024,3032],[3242,3249,3262,3274,3284,3291,3299,3307,3307,3315,3324,3324,3332,3343,3349,3353],[3763,3771,3781,3791,3799,3810,3816,3827,3833,3841,3849,3860,3866,3874,3883,3891,3894,3900,3910,3916,3927,3933,3944],[4180,4209,4216,4227,4232,4244,4250,4260,4266,4277],[4720,4721,4724,4725,4733,4738,4741,4749,4760,4766,4778,4784],[4987,5005,5012,5018,5025,5033,5041,5050,5058,5066,5074,5075,5084],[5389,5397,5409,5416,5428,5434,5444,5450,5461,5467,5478,5483,5484,5494,5501,5511,5517,5528,5534,5542,5550,5561,5567,5578,5584,5594,5600,5611,5617,5617,5628,5634,5644,5650,5661,5667,5678,5684,5694,5700]],"version":"2.0.0"}.

    Solution:

    Step 1: Using the quadratic formula to find the roots of this equation:

    x=1±1-162{"x":[[80,79,77,77,77,77,77,84,90,98,103,108,113,122,131,143,146,146,146,144,142,134,127,123,115,111,107,97,94,85],[181,182,182,181,178,177,173,171,168,157,153,143,137,136,135,136,138,140,143,148,152,155,167,177,181],[256,257,258,260,261,262,266,268,278,281,293,296,302],[261,262,263,264,268,270,277,286,294,298,303,315,318,324],[467,468,469,470,470,470,470,470,470,470,468,467,467,464,464,463,462,461,460,458,458,458],[581,582,583,584,584,586,586,586,586,586,584,584,583,581,581,581,580,580,580],[551,549,546,544,543,541,541,543,547,549,551,558,565,568,576,580,584,594,597,603,605,608,610],[556,556,555,554,556,561,564,574,589,597,603,608,610,613,614,615,616],[663,665,666,666,668,670,673,674,678,683,687,690,691,694,696,697,698,700,700,701,701,703,704,706,707,709,711,713,713,715,716,717,718,720,720,721,721,721,721,720,720,718,717,716,715,715,715,715,716,717,719,720,722,729,734,741,746,750,760,770,777,797,805,831,840,870,880,903,939,951,976,1000,1025,1038,1060,1092,1111,1119,1127,1134,1140,1156,1160,1164],[798,798,798,798,797,795,794,793,793,793,793,793,793,791,791,789,788,788,789],[861,865,866,868,870,871,878,884,890,896,899,907,910,916,918,921,923,925,926,928],[998,999,1000,1000,1000,1000,1000,999,997,996,996,994,993,993,993,993,992,992],[1060,1058,1057,1055,1054,1053,1051,1050,1047,1043,1039,1036,1034,1031,1031,1033,1034,1038,1040,1044,1046,1050,1053,1057,1061,1066,1067,1068,1070,1070,1070,1069,1068,1063,1060,1053,1050,1043,1039],[506,510,514,518,522,528,534,540,556,573,601,611,633,643,667,678,717,730,771,786,798,826,840,854,881,907,920,956,967,988,998,1007,1023,1030,1043,1053,1057,1065,1071,1075,1077],[720,720,719,718,718,718,718,718,720,722,726,729,732,738,743,746,753,756,758,764,764,765,764,762,760,758,753,747,740,734,728,722,718,716,716,715,716,717,723,725,728,734,743,747,762,767,783,787,801]],"y":[[332,332,331,330,329,326,324,318,314,311,309,309,309,309,312,322,328,331,339,347,351,364,372,376,383,387,389,395,395,394],[309,308,307,306,306,306,307,309,311,323,328,348,361,370,378,381,386,388,389,390,391,391,388,384,382],[318,318,318,318,318,318,318,318,318,318,317,317,317],[347,347,348,348,348,348,347,345,344,343,342,340,339,339],[290,290,290,290,292,296,302,307,312,317,327,332,337,348,352,358,361,366,370,374,375,378],[288,288,288,290,291,300,305,309,320,325,331,335,342,345,348,352,354,355,357],[312,312,312,312,311,311,310,310,309,309,309,309,308,308,307,307,306,305,304,304,304,304,304],[387,388,388,388,388,385,385,382,379,378,377,377,377,376,375,375,375],[371,371,371,370,368,368,367,367,367,368,371,375,377,382,388,392,399,402,404,407,409,411,411,411,411,408,404,400,398,388,381,377,367,359,351,339,332,325,321,311,308,297,293,281,275,268,262,256,253,248,242,240,238,231,228,225,224,223,222,221,220,219,219,219,219,218,218,218,217,217,217,217,217,218,218,219,221,221,221,222,222,225,225,226],[288,289,293,301,307,328,335,350,354,357,361,365,366,371,373,378,379,381,381],[335,335,335,335,334,334,333,332,332,331,331,330,329,329,329,329,329,329,329,329],[290,289,289,291,297,300,312,317,326,331,335,346,352,354,358,361,363,364],[287,288,288,288,288,290,294,297,304,313,322,331,335,345,348,355,357,361,362,363,363,363,363,361,358,355,354,349,345,341,338,335,334,332,331,332,332,335,337],[459,459,458,458,457,457,456,455,454,452,449,448,446,445,444,443,441,441,440,440,440,439,439,439,438,438,437,437,436,436,436,436,436,436,437,437,437,437,437,437,437],[542,541,540,538,537,535,534,532,529,528,526,525,524,521,521,521,521,521,522,527,529,537,539,546,549,552,559,566,572,577,582,587,588,589,590,590,590,590,591,591,591,591,591,591,589,589,587,586,583]],"t":[[0,8,15,24,32,35,42,49,58,65,74,82,82,90,99,118,124,124,133,141,156,159,165,174,182,183,190,199,208,215],[475,491,507,519,524,535,541,541,553,561,566,575,583,592,599,607,616,616,624,632,633,642,652,660,666],[938,943,955,963,966,966,974,982,992,999,1008,1016,1024],[1200,1208,1212,1224,1232,1243,1249,1257,1266,1266,1274,1283,1291,1299],[1809,1817,1824,1841,1849,1858,1866,1874,1875,1883,1891,1891,1899,1910,1916,1924,1931,1933,1941,1950,1959,1966],[2267,2272,2284,2303,2308,2320,2325,2325,2336,2341,2341,2353,2358,2358,2370,2375,2375,2387,2391],[2555,2560,2567,2575,2586,2591,2603,2608,2620,2624,2625,2637,2641,2653,2658,2658,2670,2675,2687,2691,2692,2703,2708],[3035,3043,3050,3058,3084,3091,3103,3111,3129,3133,3142,3150,3158,3166,3167,3175,3182],[3527,3534,3542,3550,3558,3570,3575,3575,3587,3592,3604,3609,3620,3625,3637,3642,3654,3658,3658,3671,3675,3687,3692,3708,3720,3725,3737,3742,3754,3759,3770,3775,3787,3792,3804,3809,3821,3825,3837,3842,3854,3859,3871,3876,3887,3892,3904,3909,3920,3925,3934,3939,3943,3954,3959,3970,3975,3976,3988,3992,4004,4009,4021,4026,4037,4043,4054,4059,4071,4076,4087,4093,4100,4106,4111,4131,4134,4135,4142,4143,4150,4159,4168,4174],[8332,8368,8377,8389,8394,8405,8410,8422,8427,8427,8439,8444,8456,8460,8472,8477,8489,8493,8510],[8986,9003,9012,9019,9023,9028,9040,9044,9056,9060,9073,9077,9089,9094,9106,9111,9111,9123,9129,9148],[9758,9774,9794,9806,9811,9823,9828,9840,9844,9844,9856,9861,9873,9877,9889,9894,9906,9911],[10179,10189,10194,10206,10211,10223,10228,10228,10240,10245,10256,10261,10273,10278,10290,10295,10307,10311,10323,10327,10328,10340,10345,10356,10361,10369,10374,10379,10389,10394,10408,10412,10423,10428,10440,10445,10457,10461,10473],[11144,11145,11148,11155,11161,11161,11163,11173,11178,11187,11194,11203,11211,11211,11220,11228,11237,11244,11253,11261,11262,11269,11278,11278,11286,11294,11303,11311,11320,11328,11328,11338,11345,11345,11355,11362,11371,11378,11387,11395,11399],[12104,12118,12126,12139,12153,12155,12161,12170,12178,12187,12195,12195,12203,12212,12212,12220,12228,12229,12238,12245,12253,12262,12270,12278,12279,12288,12295,12304,12311,12320,12328,12337,12345,12353,12361,12362,12378,12388,12395,12403,12407,12414,12420,12428,12437,12446,12457,12463,12470]],"version":"2.0.0"}

    Step 2: Upon simplification, we get:

    x=1±-152{"x":[[104,96,94,92,88,88,86,86,87,90,91,96,98,103,108,117,124,136,140,148,150,154,156,157,158,158,156,152,150,143,140,131,129,122,117,113,110,108],[198,198,198,196,193,192,187,185,183,177,167,164,157,155,154,153,153,153,154,157,159,165,168,172,181,186,191,201,217,228,233,243,256],[346,353,354,359,361,364,374,381,387,390,394,399,402,404],[364,369,372,375,381,386,399,403,415,419],[605,604,604,603,603,602,601,601,600,599,598,598,598,598,598,597,597,597],[727,728,729,729,729,729,728,726,724,723,721,720],[684,684,685,690,699,704,716,726,734,742,744,747,750,754,756],[687,686,686,686,686,687,689,697,700,707,711,715,718,726,733,735,740,742,744,748,749,751,752],[817,818,820,821,823,824,826,831,837,840,843,844,846,846,848,848,850,850,851,852,853,854,855,856,857,858,860,861,862,864,866,868,868,870,870,870,871,871,871,871,871,871,873,875,881,887,894,899,917,923,946,954,983,993,1026,1046,1056,1077,1096,1120,1134,1143,1150,1152],[942,941,940,939,938,940,943,948,952,964,968,978,987,995,1003,1008],[1042,1043,1043,1044,1044,1045,1046,1046,1045,1043,1041,1040,1039],[1116,1116,1115,1114,1109,1107,1102,1097,1094,1090,1087,1086,1083,1082,1081,1081,1080,1079,1078,1078,1078,1078,1078,1079,1079,1080,1081,1082,1083,1084,1086,1087,1088,1090,1093,1095,1097,1101,1104,1106,1108,1111,1114,1116,1116,1116,1115,1114,1113,1107,1104,1098,1094,1087,1083,1078],[614,631,637,648,661,669,685,711,721,740,762,774,797,822,834,861,872,884,908,921,933,955,966,987,1006,1014,1023,1038,1051,1062,1070,1075,1077],[826,826,826,826,826,826,826,827,829,830,833,836,841,844,851,854,857,865,866,869,869,869,868,867,863,855,852,846,839,833,828,826,823,822,824,826,834,838,847,858,866,878,891,897]],"y":[[369,368,367,367,365,364,359,356,353,350,348,343,341,338,334,331,329,330,331,335,337,340,342,344,349,358,364,371,375,385,388,397,399,405,407,408,408,407],[326,327,328,328,328,328,330,331,333,338,349,353,367,372,377,385,388,392,397,401,403,406,407,407,408,408,408,407,403,401,399,397,392],[336,337,337,337,337,337,337,336,336,336,336,336,337,337],[369,369,369,369,368,368,367,367,365,365],[271,285,291,297,304,311,318,325,344,348,361,365,370,372,377,379,380,381],[287,289,290,291,296,305,312,325,338,344,354,357],[319,321,321,321,319,319,318,317,316,315,315,315,315,315,315],[389,391,392,393,394,394,394,392,391,388,387,386,385,384,382,382,381,381,380,379,379,378,378],[364,364,364,364,364,364,364,365,371,377,385,389,398,401,408,409,412,413,414,414,414,412,408,403,396,384,373,360,354,335,328,313,308,295,291,287,279,272,268,258,256,251,245,241,235,231,229,228,225,225,224,224,224,224,224,225,226,228,230,234,237,239,241,242],[319,320,320,320,320,320,321,321,321,320,319,319,318,317,315,315],[284,284,285,286,287,289,296,303,308,317,327,336,344],[287,288,289,289,291,292,294,295,297,298,298,299,300,300,302,304,308,310,315,317,320,321,323,325,327,328,329,329,329,329,329,329,329,329,329,328,328,328,328,329,329,332,335,338,341,343,346,347,348,351,351,352,353,354,354,354],[463,460,459,459,458,458,458,458,458,458,457,457,457,457,457,457,457,457,457,457,458,459,459,461,461,462,462,463,464,464,464,464,464],[544,543,542,541,540,538,534,532,527,526,524,522,521,520,520,521,522,528,530,540,544,552,556,561,569,582,586,594,601,607,611,613,615,617,618,618,618,618,617,615,614,611,608,607]],"t":[[0,10,16,23,28,32,40,51,58,68,74,85,91,102,107,119,125,144,149,157,165,173,181,182,190,198,207,215,224,232,240,252,257,269,273,285,290,302],[471,475,490,502,507,518,524,524,535,540,552,557,569,574,574,582,590,593,602,607,619,624,624,636,640,641,652,657,669,674,685,690,702],[984,999,1008,1016,1019,1028,1032,1042,1049,1057,1065,1074,1082,1087],[1275,1283,1291,1295,1299,1308,1316,1324,1335,1340],[3923,3931,3937,3940,3942,3950,3953,3960,3967,3976,3983,3992,4000,4000,4008,4017,4025,4032],[4350,4354,4361,4367,4376,4383,4384,4392,4400,4409,4417,4417],[4655,4664,4675,4686,4692,4702,4708,4717,4725,4734,4740,4742,4742,4751,4759],[4968,4975,4978,4985,5000,5003,5009,5017,5025,5034,5034,5042,5049,5050,5059,5067,5075,5082,5084,5092,5100,5109,5117],[5472,5478,5485,5501,5501,5505,5509,5521,5525,5538,5542,5554,5559,5571,5580,5587,5592,5593,5604,5609,5621,5626,5638,5642,5654,5659,5671,5675,5688,5692,5705,5709,5721,5726,5734,5739,5743,5754,5759,5771,5776,5788,5793,5804,5810,5821,5826,5838,5843,5855,5860,5871,5876,5888,5893,5904,5909,5921,5926,5938,5943,5955,5960,5971],[6276,6281,6287,6293,6309,6317,6326,6337,6343,6355,6359,6371,6376,6388,6393,6405],[6589,6594,6602,6609,6621,6626,6638,6643,6655,6659,6671,6676,6688],[6917,6922,6953,6960,6971,6976,6988,6993,6993,7006,7010,7022,7026,7038,7043,7055,7061,7072,7076,7088,7093,7093,7106,7110,7121,7126,7149,7152,7159,7160,7168,7176,7177,7184,7193,7193,7201,7209,7210,7218,7221,7229,7234,7243,7251,7259,7268,7268,7277,7284,7293,7301,7302,7310,7318,7322],[7731,7748,7755,7761,7768,7777,7785,7794,7802,7810,7818,7818,7827,7835,7843,7851,7852,7860,7868,7869,7877,7885,7885,7894,7902,7910,7917,7918,7928,7935,7944,7951,7960],[8270,8277,8286,8294,8302,8310,8319,8327,8336,8343,8343,8352,8360,8368,8377,8377,8386,8393,8403,8410,8418,8427,8427,8436,8443,8452,8460,8469,8477,8485,8493,8494,8502,8510,8527,8535,8543,8552,8561,8568,8569,8577,8585,8594]],"version":"2.0.0"}

    But how do we simplify the part -15{"x":[[287,288,289,290,294,298,303,309,316,318,326,328,331,332,333,334,335,336,336,337,338,338,340,341,342,343,344,345,348,350,354,355,357,358,360,360,361,363,368,370,371,372,373,374,374,374,374,374,374,374,374,374,374,374,375,376,378,381,386,391,402,411,423,437,445,461,480,501,521,556,579,602,614,626,638,671,682,711,720,736,755,764,771,774],[453,453,455,456,458,460,467,471,475,489,494,507,512,520,523,530,533,538,540,542],[583,584,585,585,586,586,587,588,588,590,590,590,590,588,588,587,587,586,586,586,587],[683,684,684,685,685,683,682,678,672,661,656,650,646,645,643,642,642,642,642,643,643,644,644,644,644,644,645,646,646,647,648,650,652,664,670,677,680,689,691,694,696,696,697,697,696,696,692,688,686,679,676,672,664,659,654,647,643,640]],"y":[[496,496,496,496,495,494,492,491,491,493,502,507,518,524,529,545,554,558,568,571,576,577,578,577,575,574,571,568,556,552,537,531,514,508,494,488,482,469,441,427,419,412,392,378,364,357,344,337,325,319,308,303,297,289,277,272,266,259,252,248,241,238,234,230,228,225,221,219,217,214,213,212,212,212,212,212,212,214,214,216,219,221,224,225],[428,429,429,429,429,429,429,429,429,427,427,425,424,422,421,419,418,417,416,415],[356,354,352,351,350,351,354,357,361,375,381,402,409,424,436,442,448,462,465,469,469],[341,339,338,337,336,336,337,339,342,349,354,358,362,364,367,367,369,370,373,378,384,390,393,403,407,414,418,421,422,424,424,424,422,417,414,412,412,415,417,423,427,430,437,441,444,447,452,457,459,463,465,467,469,471,472,474,474,474]],"t":[[0,7,24,40,44,56,61,73,77,90,94,106,111,111,122,128,139,144,156,162,173,178,189,194,207,211,211,223,227,240,244,256,261,273,277,278,290,294,317,319,320,328,336,347,353,361,369,370,378,386,394,397,406,411,423,428,440,444,456,461,473,478,492,496,503,513,519,529,536,545,553,562,569,570,578,586,602,608,612,619,629,636,644,653],[991,1000,1028,1036,1038,1045,1053,1053,1061,1069,1078,1086,1094,1103,1103,1111,1120,1128,1135,1136],[1316,1320,1328,1338,1345,1361,1370,1372,1378,1386,1395,1403,1411,1420,1428,1435,1436,1445,1453,1461,1469],[1677,1682,1688,1695,1703,1728,1737,1745,1753,1762,1770,1778,1786,1795,1806,1812,1823,1829,1840,1845,1857,1861,1874,1878,1890,1895,1907,1912,1923,1929,1945,1956,1961,1982,1987,1995,2003,2012,2020,2028,2036,2036,2045,2052,2053,2054,2061,2070,2078,2087,2088,2095,2103,2104,2112,2120,2121,2128]],"version":"2.0.0"} because we do not know the square root of a negative number?

    This is where we employ our knowledge of using i=-1{"x":[[339,336,336,333,331,328,326,324,323,323,323,324,328,333,338,345,352,358,364,368,374,380,387,394,401],[314],[451,457,465,474,488,502,514,527,539,550,560,570,577,583,588,591],[489,494,503,515,531,547,563,576,587,596,601,607,610,611,613],[698,703,706,710,715,721,726,731,736,739,742,744,746,748,751,753,755,757,760,763,765,767,768,770,773,776,779,782,786,789,792,794,795,796,796,796,794,793,791,786,784,782,782,782,782,782,787,794,802,813,824,837,853,867,883,900,915,930,946,961,977,992,1007,1021,1036,1047,1058,1067,1073,1078,1080],[855,860,870,883,898,913,926,936,945,951,956,958,960],[1017,1020,1020,1020,1020,1020,1020,1020,1020,1020,1020,1020,1021,1023,1024,1026,1028]],"y":[[239,250,259,272,287,304,322,340,376,391,405,417,425,431,435,435,435,431,426,421,414,405,397,388,378],[165],[271,267,267,264,262,260,258,258,257,256,255,255,255,255,255,256],[321,321,321,318,315,312,309,308,307,305,304,304,304,304,304],[394,388,385,383,381,379,379,379,379,380,384,387,391,396,401,405,411,416,420,424,427,428,429,429,428,422,415,407,397,386,374,361,347,333,320,306,293,281,268,245,233,222,211,201,192,184,175,168,162,157,154,150,147,145,144,144,144,144,144,144,144,145,145,147,148,149,151,154,157,159,161],[330,330,329,326,323,320,318,317,315,314,313,312,311],[259,260,265,273,283,295,309,323,337,350,361,371,381,389,397,404,409]],"t":[[0,29,42,46,58,63,75,80,102,104,113,121,130,138,148,154,163,175,179,189,196,208,213,225,230],[433],[863,881,890,896,905,913,922,930,939,947,955,963,972,980,988,996],[1199,1263,1272,1288,1297,1298,1308,1314,1325,1330,1342,1347,1358,1364,1375],[1736,1780,1788,1797,1805,1814,1822,1830,1839,1847,1855,1863,1872,1880,1888,1897,1908,1913,1925,1930,1942,1947,1958,1975,1980,1992,1997,2008,2013,2026,2031,2042,2048,2058,2064,2075,2080,2092,2099,2117,2122,2133,2140,2147,2160,2164,2172,2180,2189,2198,2205,2215,2222,2231,2239,2247,2256,2265,2272,2281,2289,2298,2306,2314,2322,2330,2339,2348,2356,2364,2372],[2719,2755,2764,2775,2781,2792,2797,2809,2814,2825,2831,2842,2847],[3032,3072,3081,3089,3100,3113,3114,3122,3132,3139,3149,3158,3164,3176,3181,3192,3197]],"version":"2.0.0"} which simplifies the given form into:

    x=1±15i2{"x":[[188,188,187,186,186,186,190,193,198,201,205,211,214,220,224,227,233,234,237,239,240,240,239,236,231,229,226,223,220,215,210,207,207,206],[267,267,267],[263,263,261,258,257,256,252,250,247,246,241,240,238,236,236,236,237,240,242,258,261,268],[375,376,378,380,389,393,398,406,413,418,424,426],[396,395,394,393,394,396,398,400,406,410,414,427,436,444],[666,666,666,667,667,667,667,667,667,667,667,666,666,666,665,664,664,664,664,664,664,664,664,666],[776,777,777,777,777,777,777,777,776,776,775,774,774,774,774,775,776],[747,749,751,754,761,770,778,782,785,788,795,797,799,800,801,802,802],[752,751,751,753,754,758,760,763,769,776,780,787,790,794,800,804,807,811,812,813],[900,901,901,901,902,903,904,904,906,907,907,909,909,910,910,911,911,911,913,913,914],[979,978,977,972,968,966,963,959,955,954,951,951,950,949,949,949,949,949,949,950,950,950,950,950,951,953,956,960,968,971,975,976,977,978,980,980,980,979,976,973,970,968,965,961,960,959],[1015,1016,1017,1017,1018,1018,1018,1020,1020,1021,1023,1024,1027,1029,1033,1034,1037,1040],[1033,1032],[703,705,707,711,719,731,737,744,751,773,781,797,816,833,851,880,890,899,908,928,944,969,977,996,1006,1013,1019,1021,1027,1028],[864,866,866,866,866,866,867,869,871,880,883,891,894,904,908,913,914,915,916,916,916,914,913,910,906,904,900,894,892,887,886,887,889,891,900,903,911,918,927,939,945,947]],"y":[[332,331,330,328,327,326,321,319,315,314,312,311,311,311,314,316,322,324,328,334,337,339,345,352,359,363,366,369,374,379,383,385,386,386],[314,313,312],[310,311,312,313,314,315,318,320,323,326,332,336,339,348,353,357,358,360,361,361,361,359],[314,314,314,314,313,313,312,312,312,311,311,311],[345,345,346,346,346,346,346,345,344,344,342,339,338,336],[266,265,264,262,261,260,262,265,276,286,298,304,311,316,330,334,341,343,345,348,349,352,353,354],[262,262,266,269,281,291,301,308,311,319,321,324,325,326,327,327,325],[285,285,285,284,284,284,284,284,284,284,284,284,285,285,285,286,287],[354,355,356,356,356,355,355,354,353,352,351,351,350,349,349,348,348,347,347,347],[272,271,270,269,269,269,270,272,280,284,291,304,308,312,318,321,324,325,327,328,328],[261,261,261,264,267,268,270,273,276,277,278,279,279,281,283,285,286,289,290,294,297,298,299,300,300,300,299,298,295,295,295,295,295,297,300,301,307,308,312,315,318,319,321,323,324,324],[265,265,267,268,277,281,289,299,302,308,312,313,314,313,311,309,305,299],[224,224],[424,424,422,421,419,418,418,418,417,415,415,414,413,412,411,409,408,407,405,403,401,399,398,397,397,396,395,395,395,395],[487,487,486,485,484,482,481,478,475,469,468,466,466,467,469,473,477,480,486,489,492,498,501,507,511,514,518,523,524,528,529,529,530,531,531,531,531,531,529,528,526,525]],"t":[[0,7,15,23,31,41,48,56,65,65,74,81,81,90,98,106,115,123,131,140,147,148,156,165,173,181,181,182,191,198,206,215,215,223],[387,400,409],[433,444,448,460,465,465,477,481,482,493,498,498,510,515,526,532,543,548,548,574,581,590],[936,957,965,973,982,990,990,998,1007,1015,1023,1032],[1178,1183,1190,1198,1224,1232,1234,1243,1248,1249,1260,1265,1277,1282],[2027,2031,2035,2040,2041,2060,2077,2082,2090,2099,2107,2115,2118,2127,2132,2144,2149,2149,2161,2166,2166,2177,2182,2194],[2630,2657,2674,2682,2694,2699,2711,2716,2729,2733,2744,2749,2749,2761,2766,2777,2783],[3013,3049,3057,3060,3067,3074,3084,3091,3091,3099,3107,3116,3124,3125,3133,3141,3153],[3422,3426,3441,3466,3474,3483,3490,3491,3499,3508,3508,3516,3524,3525,3534,3541,3550,3558,3568,3574],[4015,4024,4028,4033,4050,4056,4058,4066,4074,4083,4091,4100,4108,4111,4116,4124,4133,4135,4145,4150,4162],[4363,4399,4416,4428,4433,4441,4442,4450,4461,4467,4475,4479,4488,4494,4512,4516,4529,4533,4545,4550,4566,4578,4583,4600,4616,4628,4633,4645,4662,4666,4679,4683,4683,4696,4700,4712,4717,4729,4733,4745,4750,4762,4767,4779,4784,4795],[4959,4967,5000,5009,5016,5025,5033,5041,5042,5050,5058,5066,5075,5083,5092,5092,5100,5108],[5243,5259],[5749,5754,5761,5768,5775,5783,5792,5792,5800,5812,5817,5829,5834,5846,5850,5859,5867,5870,5879,5884,5896,5901,5912,5917,5929,5934,5946,5950,5958,5970],[6285,6300,6306,6308,6309,6317,6322,6330,6334,6356,6359,6367,6375,6386,6392,6400,6409,6418,6425,6426,6434,6442,6442,6451,6459,6468,6475,6484,6492,6501,6517,6534,6542,6550,6559,6567,6579,6584,6596,6601,6613,6618]],"version":"2.0.0"}

    Which can now be treated according to the laws of complex numbers.

    A whole branch of mathematics known as Complex Analysis is based on studying complex numbers and applying their properties to other fields of mathematics.

    A very popular equation in physics also employs the use of complex numbers, the Schrodinger Equation:

    h_ψt=Hψ

    Which is a wave equation used to model the orbits of electrons around the nucleus of an atom.

    The applications are plenty, from the differential equation of an inductor-capacitor (LC) circuit to modeling the signals received.

    Modulus and Phase - Key takeaways

    • The modulus of a complex number is its distance from the origin on a complex plane.
    • A complex number is related to its modulus and conjugate by the equationzz_=z2
    • The phase angle or argument of a complex number is the angle r(line segment connecting the origin to the complex number) makes with the positive x-axis.
    • The phase angle of a complex number z=a+ib is θ=tan-1ba{"x":[[124,123,123,122,122,122,122,122,121,120,119,118,117,117,117,119,122,125,128,132,137,142,147,151,154,157,158,159,158,156,153,149,145,141,137,133,129,125,122,120,118,117,118,121,126,133,141,150,160,169,175],[213,211,210,211,213,218,224,231,239,246,252,257,259,260],[210,209,210,212,216,221,228,235,242,249,254,258,261,262,262],[366,366,366,366,366,366,366,365,364,362,360,359,359,359,359,359],[340,339,336,337,342,348,357,366,376,384,389,393,395,395,394],[412,412,412,412,412,411,409,407,404,401,397,393,390,387,385,384,384,385,387,389,393,397,401,405,408,411,412,413,413,413,413,413,413,414,415,416,418,421,423,425,427,429,429,430,430,430,430,430,429,429,428,428,429,429,430,430,430,430,430,430,431,432,434,437,440,443,447,450,453,455,457,458,459,459,459,459,460,461,462,463,464,465,466],[463,462,460,458,457,458,460,464,468,472,477,481,484,487],[507,507,507,507,507,507,508,509,510,510,511,511,511,511,511,512],[588,588,589,590,590,590,590,590,589,589,588,588,588,588,588,588,588,588,589,589,590,591,592,593,595,597,599,602,604,607,609,610,611,611,609,607,605,604,599,596,593,591,589,589,589,592],[578,576,577,581,588,596,607,616,624,630,633,635,635,634,632,631],[601,601,602,602,602,602,601,599,596,592,588,584,580,578,577,577,578,581,583,586,590,592,595,598,599,601,602,602,602,602,602,602,603,604,605,607,608,609,610],[650,650,650,650,651,652,654,657,661,665,668,671,672,672,671,669,667,665,663,661,659,657,656,655,655,654],[557,557,557,556,556,555,554,553,551,550,547,544,541,538,536,534,534,534,535,536,538,539,541,544,546,549]],"y":[[157,156,155,154,153,154,156,160,168,177,190,204,218,230,238,244,248,248,247,242,236,229,220,211,201,190,179,169,159,150,142,135,131,129,129,131,135,141,149,157,165,173,179,184,187,187,187,185,182,180,178],[173,173,173,173,174,175,177,178,179,179,179,179,179,178],[208,208,208,208,207,206,204,203,202,201,201,200,200,199,198],[145,143,142,144,147,153,163,176,190,202,212,219,224,228,231,232],[188,188,188,188,187,186,184,182,179,178,177,177,177,178,179],[205,203,202,200,198,196,194,193,192,192,193,196,200,205,210,214,217,220,221,221,221,219,216,214,211,208,207,206,207,208,211,213,215,217,219,219,219,219,217,215,212,210,208,205,203,201,199,198,197,196,196,197,199,205,208,211,214,216,218,219,218,215,211,207,203,199,196,194,193,193,194,197,199,203,206,210,214,217,220,221,222,222,222],[153,153,153,152,152,152,152,151,150,149,149,148,148,148],[134,132,131,130,129,130,132,135,139,144,149,154,159,163,167,169],[152,150,147,144,141,139,136,135,134,136,140,145,151,157,163,170,177,182,186,188,189,188,187,185,182,180,178,177,177,177,179,181,184,187,190,193,196,197,198,199,198,197,195,192,188,185],[231,230,230,229,229,229,229,229,230,231,232,232,233,233,233,233],[275,274,272,271,270,269,268,267,267,268,269,272,276,279,283,286,288,290,290,290,288,286,283,280,277,274,272,273,276,279,282,285,288,291,295,298,300,301,301],[170,169,168,166,166,166,169,174,181,191,205,220,233,245,256,268,279,289,298,305,311,315,317,318,316,315],[176,175,174,173,172,171,170,170,171,175,182,192,205,218,231,243,254,263,273,281,288,295,300,303,305,305]],"t":[[0,9,16,21,39,46,56,63,73,79,88,96,105,113,122,129,138,146,155,163,172,179,189,196,210,213,222,230,241,246,255,263,272,286,288,296,305,313,322,330,340,347,358,363,373,379,390,397,408,413,425],[851,859,871,889,896,905,913,922,930,939,946,955,963,972],[1106,1116,1130,1138,1146,1156,1165,1172,1180,1189,1196,1206,1213,1223,1230],[1680,1689,1694,1719,1721,1730,1739,1747,1756,1763,1773,1780,1789,1797,1806,1814],[1947,1964,1970,1972,1980,1989,1997,2006,2013,2023,2030,2039,2047,2056,2063],[2425,2436,2442,2447,2456,2463,2473,2480,2489,2497,2506,2514,2523,2530,2540,2547,2561,2563,2573,2580,2592,2597,2606,2615,2625,2630,2640,2647,2664,2673,2680,2691,2697,2706,2714,2724,2730,2741,2747,2756,2763,2773,2780,2790,2797,2806,2814,2823,2831,2843,2862,2864,2873,2889,2897,2906,2914,2923,2930,2940,2964,2973,2980,2990,2997,3006,3014,3023,3030,3039,3047,3056,3064,3073,3080,3090,3097,3107,3114,3123,3130,3140,3146],[3379,3387,3393,3395,3406,3414,3423,3431,3440,3447,3457,3464,3474,3481],[3623,3635,3640,3656,3672,3681,3690,3697,3707,3714,3724,3731,3741,3747,3759,3764],[4241,4250,4255,4257,4264,4274,4281,4291,4298,4323,4331,4340,4348,4358,4364,4374,4381,4391,4398,4408,4414,4431,4441,4448,4458,4464,4474,4481,4494,4498,4508,4514,4524,4531,4541,4548,4558,4564,4577,4581,4593,4598,4609,4614,4627,4631],[4936,4948,4956,4964,4975,4981,4991,4998,5008,5015,5025,5031,5041,5058,5065,5077],[5375,5385,5390,5395,5398,5411,5415,5425,5431,5442,5448,5458,5465,5475,5481,5492,5498,5508,5515,5525,5531,5542,5548,5558,5566,5575,5581,5616,5623,5632,5642,5648,5658,5665,5677,5682,5692,5698,5703],[6017,6025,6030,6032,6048,6057,6065,6075,6082,6092,6098,6109,6115,6125,6132,6142,6148,6159,6165,6175,6182,6192,6198,6209,6215,6222],[6793,6801,6807,6815,6825,6832,6849,6859,6865,6876,6884,6892,6899,6909,6915,6926,6932,6943,6949,6959,6965,6976,6982,6992,6999,7009]],"version":"2.0.0"}.
    • De Moivre’s theorem describes how a complex number raised to an integral power can be calculated easily without using the binomial theorem.
    • A significant consequence of De Moivre’s theorem states that zn=rncosn+isinnθ{"x":[[107,106,105,106,108,112,116,122,127,133,137,140,143,145,146,147,147,147,146,145,142,138,134,130,125,120,116,113,111,109,109,109,110,112,116,122,128,134,141,148,154,159,165,168,170,171,170,168,166,164,161,160],[105,104,101,102,106,111,119,128,137,146,154,160,164,167,167,166,163,159],[183,183,183,182,182,183,183,183,183,184,184,184,184,184,184,184,184,184,185,187,189,191,193,195,197,199,200,201,202,203,203,204,204,204,205,205,205,206,207,207,208],[260,259,257,256,257,259,263,268,273,278,283,287,290,292,293],[252,251,250,252,255,260,266,273,280,286,292,296,298,300,300,298,297],[345,344,344,343,344,345,347,350,351,353,358,363,367,369,373,374,375,374,373,371,369,367,366,364,363,363,363,365,367,371,376,381,386,391,394,396,397,396,394,392,389,386,383,381,380,380,380,381,384,387,392,396,401,405,407],[404,403,403,403,403,403,403,403,403,403,403,404,405,407,409,411,413,414,415,416,418,419,420,421,422,423,424,424,424,424,424,425],[532,532,532,532,532,531,530,529,527,524,521,516,511,506,501,496,492,490,490,490,493,497,501,506,511,516],[562,561,561,560,559,558,556,554,552,550,548,547,546,546,546,548,552,555,559,564,569,574,578,581,584,585,586,586,586,586,585,585,585,585,585,584,584,585,586,587,589,591,593,596,598,600,601,601,601,601,599,598,596,595,593,592,591,591,591,591,592,593,596,599,603,608,614,619,624,628,632,635,638,640,640,641,641,640,638,636,633,630,628,626,625,626,628,632,635,639,643,645,646,647,647,646,644,642,640,637,635,633,630],[666,665,664,664,664,664,664,664,664,664,664,664,664,664,665,665,666,668,670,672,675,678,681,683,685,685,686,686,686,686,686,686,686,687,687,688,689,690],[710,710,710,710,710,711,711,711,711,711,711,711,713,714,716,718,721,724,727,730,732,734,735,735,734,732,729,726,724,719,716,713,712,711,710,710,711,712,713,717,721,726,728,732,736,737,738],[767,766,765,764,766,768,772,777,782,785,792,797,799,803,804,805,803,801],[785,785,785,785,785,785,785,785,785,784,784,784,784,784,784,785],[840,840,841,841,841,842,842,842,843,843,844,844,845,846,847,849,851,853,854,856],[847,846,847],[899,899,899,899,900,900,900,900,899,897,895,892,890,888,887,887,887,889,892,896,899,903,905,907,907,906,904,902,899,896,894,891,889,887],[918,918,918,919,919,919,920,921,921,922,924,925,927,929],[923,922,921,921,922,923,924,926],[937,937,937,937,937,937,937,938,938,939,939,939,940,940,940,941,941,942,943,944,945,947,949,951,952,954,955,955,955,956,956,957,957,958,959,959,960,962,963,964,965,966],[985,985,985,985,985,985,985,985,985,985,985,985,985,985,986,986,987,988,990,992,994,996,997,999,1000,1002,1003,1004,1004,1005,1005,1005,1006,1006,1006,1006,1007,1007,1008,1009],[1027,1027,1027,1027,1028,1028,1028,1029,1029,1030,1031,1032,1033,1035,1037,1039,1041,1043,1045,1046,1047,1047,1046,1044,1041,1037,1033,1030,1027,1025,1025,1025,1026,1028,1030,1033,1036,1040,1044,1047,1049,1051,1051],[1065,1065,1065,1065,1067,1071,1075,1080,1084,1085,1084,1083,1081,1078,1076,1073,1071,1070,1068,1067,1066,1065,1063]],"y":[[152,152,152,152,152,152,152,151,151,151,151,152,153,154,155,156,157,158,159,162,165,170,174,179,184,188,193,197,200,203,206,208,209,210,211,211,210,209,208,207,206,205,204,204,204,204,204,204,204,204,204,204],[184,184,184,184,183,182,181,179,178,177,176,176,175,175,174,174,175,175],[117,116,115,114,113,114,116,118,121,123,124,125,126,127,128,127,125,122,119,116,112,109,107,105,105,105,107,109,112,115,119,122,126,129,132,134,135,136,136,135,134],[162,162,161,161,161,161,161,161,161,161,162,162,162,162,163],[190,189,189,189,189,188,188,187,186,186,185,185,185,185,184,184,184],[201,201,202,202,202,202,200,198,197,195,190,184,178,174,165,162,159,158,158,158,158,159,160,163,165,167,170,172,174,176,177,177,176,175,175,175,175,177,179,181,185,189,192,196,200,202,205,207,208,208,208,207,205,202,200],[129,129,128,129,131,132,134,135,137,138,137,135,132,129,125,123,121,120,120,121,123,126,129,132,136,139,142,145,147,148,149,149],[133,132,131,130,128,127,125,123,122,121,121,124,129,137,147,159,173,185,195,203,208,212,214,215,215,212],[167,166,165,165,164,163,163,162,163,164,167,170,174,179,183,186,189,190,190,190,188,186,183,181,179,177,175,174,172,171,171,172,174,177,180,183,186,188,190,191,192,191,190,188,185,181,178,175,172,169,167,165,164,164,165,166,168,170,173,176,178,181,183,184,185,185,184,182,180,178,176,174,172,169,167,164,162,160,158,157,157,157,157,159,160,162,165,168,170,173,176,179,181,184,185,186,187,188,188,188,188,187,185],[169,167,166,165,166,167,170,173,177,180,184,186,188,189,189,187,185,182,178,173,169,166,163,161,161,162,164,166,169,173,176,179,183,186,188,189,190,189],[163,162,161,162,163,167,171,176,181,186,190,192,194,194,194,193,191,188,184,179,174,169,162,157,152,147,145,143,143,142,145,148,150,152,156,161,165,167,168,171,172,173,173,173,171,170,169],[171,171,171,171,171,171,171,171,171,170,170,170,169,169,169,169,169,169],[162,161,160,159,160,162,165,169,173,177,182,185,189,191,192,193],[166,165,165,164,165,166,169,170,173,176,180,183,186,188,190,190,190,189,187,183],[145,145,145],[171,170,169,168,167,165,163,161,160,159,159,159,160,162,164,166,169,172,174,177,179,181,183,184,185,186,187,187,188,188,188,188,188,187],[166,165,166,168,171,175,179,182,185,186,187,187,186,183],[148,147,146,145,145,145,146,149],[170,169,168,167,166,165,164,164,166,167,170,172,175,178,180,182,183,184,183,181,178,175,172,168,166,164,163,164,165,167,169,172,174,177,180,182,184,186,187,187,187,186],[161,160,159,158,159,161,163,167,170,174,177,179,181,182,181,179,176,173,168,164,161,157,155,153,153,153,154,156,158,161,165,168,171,175,177,179,181,182,182,182],[157,156,155,156,157,160,163,167,172,176,180,184,185,186,186,184,181,177,173,169,163,158,153,148,145,142,141,141,143,145,149,152,156,160,163,166,168,169,170,170,169,168,167],[126,125,124,123,125,128,134,143,153,164,174,181,188,195,201,206,212,215,218,219,220,219,218]],"t":[[0,35,72,89,99,106,115,122,132,139,149,155,165,172,184,199,205,215,222,231,239,248,255,265,272,281,289,298,306,318,322,334,341,348,356,369,372,381,389,398,406,415,422,431,440,449,467,473,482,489,498,505],[792,798,802,806,815,822,832,839,848,856,865,872,882,889,899,915,922,933],[1575,1584,1589,1598,1606,1656,1664,1673,1682,1689,1700,1706,1707,1718,1736,1758,1768,1773,1783,1790,1801,1806,1817,1823,1832,1840,1849,1856,1867,1873,1882,1889,1899,1906,1916,1923,1932,1939,1957,1968,1973],[2583,2596,2602,2604,2640,2648,2656,2666,2673,2683,2690,2699,2706,2716,2723],[2904,2910,2917,2940,2949,2957,2966,2973,2983,2990,3000,3007,3020,3023,3033,3050,3055],[3477,3489,3495,3507,3523,3533,3540,3550,3557,3557,3567,3573,3583,3591,3600,3607,3617,3633,3640,3651,3657,3666,3678,3683,3692,3700,3707,3716,3723,3733,3740,3753,3757,3768,3773,3785,3790,3807,3816,3824,3833,3840,3850,3857,3867,3874,3883,3890,3902,3907,3918,3923,3936,3940,3953],[4288,4293,4307,4374,4382,4391,4400,4407,4417,4433,4457,4469,4475,4486,4491,4502,4507,4520,4524,4534,4540,4552,4557,4567,4574,4584,4593,4601,4607,4617,4624,4634],[5176,5184,5190,5193,5199,5207,5219,5224,5234,5241,5249,5257,5267,5274,5284,5292,5301,5307,5318,5324,5334,5341,5351,5357,5368,5374],[5911,5922,5928,5933,5941,5952,5958,5968,5974,5985,5991,6001,6008,6018,6024,6035,6041,6051,6060,6068,6074,6084,6091,6101,6108,6118,6124,6135,6141,6151,6174,6194,6201,6208,6218,6224,6235,6241,6251,6258,6268,6274,6285,6291,6303,6308,6321,6325,6336,6341,6352,6360,6368,6375,6385,6391,6401,6408,6418,6424,6435,6441,6452,6458,6468,6475,6485,6494,6502,6508,6518,6525,6535,6542,6552,6558,6568,6575,6585,6591,6602,6608,6618,6625,6635,6641,6652,6660,6668,6675,6685,6691,6700,6708,6718,6725,6735,6741,6751,6758,6768,6775,6785],[7416,7427,7438,7451,7475,7485,7492,7502,7508,7519,7525,7535,7542,7552,7568,7575,7583,7592,7602,7608,7618,7625,7635,7642,7652,7658,7669,7675,7686,7692,7702,7708,7719,7725,7736,7744,7752,7775],[8034,8047,8068,8094,8102,8109,8119,8125,8136,8142,8153,8158,8169,8175,8176,8187,8192,8203,8210,8219,8225,8236,8242,8253,8259,8269,8275,8286,8292,8303,8309,8319,8325,8326,8337,8344,8352,8358,8359,8369,8376,8386,8393,8402,8409,8410,8420],[8880,8889,8895,8897,8917,8925,8936,8945,8953,8959,8969,8975,8976,8986,8992,9003,9021,9030],[9163,9172,9179,9193,9218,9226,9236,9245,9253,9259,9270,9276,9286,9292,9303,9309],[9720,9729,9734,9740,9773,9781,9789,9792,9803,9809,9820,9826,9837,9843,9854,9859,9870,9876,9887,9892],[10001,10018,10068],[10468,10475,10479,10485,10493,10504,10509,10520,10526,10537,10543,10554,10559,10571,10576,10584,10593,10603,10609,10620,10626,10637,10643,10654,10660,10671,10676,10684,10693,10703,10710,10721,10726,10737],[10909,10915,10976,10985,10993,11003,11010,11021,11026,11037,11043,11054,11060,11071],[11165,11176,11184,11193,11218,11226,11238,11243],[11371,11379,11385,11393,11403,11410,11421,11443,11454,11460,11471,11476,11488,11493,11503,11510,11520,11526,11543,11553,11560,11570,11576,11587,11593,11604,11610,11626,11636,11643,11652,11660,11670,11676,11687,11693,11704,11710,11721,11726,11737,11742],[12160,12171,12177,12193,12219,12227,12238,12243,12253,12260,12270,12277,12288,12293,12319,12327,12338,12343,12354,12360,12371,12378,12388,12394,12406,12411,12420,12427,12437,12444,12455,12460,12472,12477,12488,12494,12505,12511,12527,12537],[12752,12773,12794,12812,12821,12827,12839,12844,12854,12860,12871,12877,12889,12894,12905,12911,12922,12927,12939,12944,12955,12960,12972,12980,12989,12994,13005,13012,13021,13027,13038,13044,13055,13060,13072,13077,13089,13094,13106,13112,13122,13127,13134],[13339,13350,13360,13373,13394,13404,13412,13422,13427,13439,13444,13458,13460,13473,13477,13489,13494,13506,13511,13522,13527,13539,13544]],"version":"2.0.0"} (the argument of a complex number raised to power n is n times the argument of the original complex
    Frequently Asked Questions about Modulus and Phase

    What is a phase of a complex number?

    The phase of a complex number is the angle it makes with the positive x axis in a complex plane.

    What is meant by "modulus"?

    Modulus essentially means the magnitude of a complex number i.e. the distance of a complex point from the origin.

    How do you find the phase of a complex function?

    The phase angle of a complex number z=a+ib can be found using the formula "inverse tan of (y/x) = phase"

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