Complex Numbers

Sometimes you try and graph a parabola, and discover it doesn't intersect the \(x\)-axis.  But what does that mean when you try and find the roots of the parabola by solving \(0=ax^2+bx+c\)?  It means the equation doesn't have any real roots. What is the opposite of real?  Imaginary, of course!  Here you will discover what an imaginary number is, and what the relationship is to a complex number.  Read on to take the complexity out of imaginary numbers! 

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Contents
Table of contents

    Complex Numbers Formulas

    A complex number is what you get when you try and take the square root of a negative number. You can't graph a complex number on the standard \((x,y)\)-coordinate plane because it isn't a "real" number. First let's look at what the square root of a negative number is defined as.

    The imaginary number \(i\) is defined as \(i=\sqrt{-1}\).

    From here you can talk about a more general complex number.

    A complex number is a number which is a combination of a real part and an imaginary part. It takes the form \(z=a+bi\) where \(a\) and \(b\) are real numbers, and \(i=\sqrt{-1}\).

    Remember that the set of all real numbers is written \(\mathbb{R}\). Similarly the set of all complex numbers is written \(\mathbb{C}\). Often you will see the notation:

    • \(\Re(z) = a\) for the real part of \(z=a+bi\); and

    • \(\Im(z) = b\) for the imaginary part of \(z=a+bi\).

    It is important to recognize that all real numbers are in fact complex numbers, since you can always write a number like \(5\) as \(5+0i\). Notice that \(\Im(5) = 0\), and that is true for any real number!

    Graphing Complex Numbers

    You can graph complex numbers on something called an argand diagram. When drawing an argand diagram there are two axes:

    • an axis representing the real part of the complex number, usually written \(\mathbb{R}\); and

    • an axis representing the imaginary part of the complex number, usually written \(i\mathbb{R}\).

    You will frequently see the phrases "complex plane" and "argand diagrams" used interchangeably. Below you can see how the complex plane is often graphed.

    Complex Numbers graph showing the real and complex axis StudySmarterFig. 1 - Graph of the complex plane.

    Notice that the real axis is the horizontal one, and the complex axis is the vertical one. Now let's graph a point.

    If you think about the complex number \(z=-6+4i\), the real part is \(\Re(z)=-6\) and the imaginary part is \(\Im(z) = 4\). That means to graph the point \(z=-6+4i\) on the complex plane, you would go left \(6\) units on the horizontal axis, and up \(4\) units on the imaginary axis. You can see this in the picture below.

    Complex Numbers plot of the complex point z=-6+4i StudySmarterFig. 2 - Plot of the complex point \(z=-6+4i\).

    So this is a lot like graphing a point in the regular \((x,y)\) Cartesian Coordinate System!

    Of course, there are other ways to represent complex numbers than the way you have seen so far. For more information on that, see the articles Representation of Complex Numbers and Polar Coordinates.

    Absolute Value of Complex Numbers

    You can take the absolute value of a real number (for example \(|-10|=10\), but can you do the same with complex numbers? Sort of. In the case of complex numbers, it is called a modulus instead.

    The modulus of the complex number \(z=a+bi\) is

    \[ |z| = \sqrt{a^2+b^2}.\]

    This should look very familiar to you! In fact it is the length of the hypotenuse of a right triangle found using the Pythagoras Theorem! You can see it graphically in the picture below.

    Complex Numbers the modulus of a complex number as the length of the hypotenuse of a right triangle StudySmarterFig. 3 - The modulus of a complex number is the length of the hypotenuse of the right triangle formed by plotting the point in the complex plane.

    Let's finish off this example.

    Find the modulus of the complex number \(z=-6+4i\).

    Answer:

    Here \(a=-6\) and \(b=4\), so

    \[ \begin{align} |z| &= \sqrt{a^2+b^2} \\ &= \sqrt{6^2 + (-4)^2} \\ &= \sqrt{36+16} \\ &= \sqrt{52} \\ &= 2\sqrt{13}.\end{align}\]

    Now that you know that, how about adding and subtracting complex numbers?

    Examples with Complex Numbers

    One of the first things you learn to do with numbers is to add and subtract them, so let's use those as the basic examples. Remember that if you had \(2x+3y\), and \(-4x+7y\), to add them together you would gather up the like terms and add those getting

    \[ \begin{align} (2x+3y) + (-4x+7y) &= (2x-4x) + (3y+7y) \\ &= -2x+10y,\end{align}\]

    and you could perform a similar action with subtraction. Working with complex numbers is very similar!

    The idea is to gather up the like terms and add or subtract them. The difference here is that you are gathering together the real parts and the complex parts. So if you have two complex numbers, \(z = a+bi\) and \(t = c+di\), you could add them like so:

    \[ \begin{align} z+t &= (a+bi) + (c+di) \\ &= (a+c) + (bi+di) \\ &= (a+c)+(b+d)i.\end{align}\]

    Subtracting them would work similarly:

    \[ \begin{align} z-t &= (a+bi) - (c+di) \\ &= a+bi-c-di\\&= (a-c) + (bi-di) \\ &= (a-c)+(b-d)i.\end{align}\]

    From those two equations you can see that complex numbers have the following properties:

    • \(\Re(z) + \Re(t) = \Re(z+t)\);

    • \(\Re(z) - \Re(t) = \Re(z-t)\);

    • \(\Im(z) + \Im(t) = \Im(z+t)\); and

    • \(\Im(z) - \Im(t) = \Im(z-t)\).

    Let's take a look at a quick example.

    For \(z=2-3i\) and \(t=-5+6i\), find \(z+t\) and \(z-t\).

    Answer:

    Remember to gather up like terms! So

    \[ \begin{align} z+t &= (2-3i) + (-5+6i) \\ &= (2-5) + (-3i+6i) \\ &= -3+3i,\end{align}\]

    and

    \[ \begin{align} z-t &=(2-3i) - (-5+6i) \\ &= 2-3i-(-5)-(6i)\\ &= (2+5)+(-3i-6i)\\ &= 7-9i.\end{align}\]

    What about multiplying complex numbers?

    Multiplying Complex Numbers

    Remember when you learned to multiply and factor to solve quadratic functions and equations? Multiplying together to complex numbers is uses exactly the same FOIL method you saw there. Remember that FOIL stands for

    first - multiply the first two terms in the parenthesis together;

    inner - multiply the inner two terms of the parenthesis together;

    outer - multiply the outer two terms of the parenthesis together; and

    last - multiply the last two terms in the parenthesis together.

    So in terms of complex numbers, if you have \(z = a+bi\) and \(t = c+di\), then

    \[ \begin{align} zt &= (a+bi)(c+di) \\ &= ac + bci + adi + bdi^2 \\ &= ac + (bc+ad)i + bd(-1) \\ &= ac-bd + (bc+ad)i ,\end{align}\]

    where you have used the fact that \(i^2 = -1\).

    Let's take a look at a quick example.

    For \(z=2-3i\) and \(t=-5+6i\), find \(zt\).

    Answer:

    Don't forget to use the fact that \(i^2 = -1\) when you calculate:

    \[ \begin{align} zt &= (2-3i)(-5+6i) \\ &= (2)(-5) + (-3i)(-5) + (2)(6i)+(-3i)(6i) \\ &= -10 + 15i+12i-18i^2 \\ &= -10 +27i -18(-1) \\ &= -10+18+27i \\ &= 8+27i. \end{align}\]

    Notice that you can't just multiply the real parts together and the imaginary parts together!

    Dividing Complex Numbers

    Dividing complex numbers isn't quite as nice as multiplying them. To make life easier, you will need to know what a complex conjugate is.

    The complex conjugate of the complex number \(z=a+bi\) is the complex number \(\bar{z} = a-bi\).

    There is something very nice that happens when you multiply a complex number and its conjugate together.

    Multiply together \(z=a+bi\) and its complex conjugate (\bar{z} =a-bi\).

    Answer:

    You would multiply them together exactly like in the previous section:

    \[ \begin{align} z\bar{z} &= (a+bi)(a-bi) \\ &= a^2 +abi-abi-b^2i^2 \\ &= a^2 -b^2(-1) \\ &= a^2+b^2. \end{align}\]

    So when you multiply a complex number and its conjugate together, you get a real number! You can see from the formula in the example that it is also true that

    \[z\bar{z} = |z|^2.\]

    What does that have to do with dividing two complex numbers? Well, the way to divide complex numbers is to multiply the numerator and denominator of the fraction by the complex conjugate of the denominator. In other words, if you divide \(z=a+bi\) by \(t = c+di\), you get

    \[\begin{align} \frac{z}{t} &= \frac{a+bi}{c+di} \\ &= \frac{(a+bi)(c-di)}{(c+di)(c-di)} \\ &= \frac{(a+bi)(c-di)}{c^2+d^2} .\end{align}\]

    So really what you have done is multiply by a very fancy \(1\) instead of doing actual division!

    Let's take a look at an example applying this.

    For \(z=2-3i\) and \(t=-5+6i\), divide \(z\) by \(t\).

    Answer:

    Don't forget to use the fancy \(1\)! Then you get

    \[ \begin{align} \frac{z}{t} &= \frac{ 2-3i}{-5+6i} \\ &= \frac{ (2-3i)(-5-6i)}{(-5+6i)(-5-6i)} \\ &= \frac{(2)(-5)+ (-5)(-3i)+(2)(-6i)+(-3i)(-6i) }{5^2 + 6^2} \\ &= \frac{-10-15i-12i+18i^2}{25+36} \\ &= \frac{-10-27i +18(-1)}{61} \\ &= \frac{-28 -27i}{61} \\ &= -\frac{28}{61} -\frac{27}{61}i.\end{align}\]

    There are plenty more things you can do with complex numbers. Check out our related articles Imaginary Unit and Polar Bijection, Operations with Complex Numbers, Modulus and Phase, Representation of Complex Numbers, and Roots of Complex numbers.

    Complex Numbers - Key takeaways

    • A complex number is a number which is a combination of a real part and an imaginary part. It takes the form \(z=a+bi\) where \(a\) and \(b\) are real numbers,
    • The imaginary number \(i\) is defined as \(i=\sqrt{-1}\).
    • All real numbers are actually complex numbers.
    • The notation for the real and imaginary parts of complex numbers are:
      • \(\Re(z) = a\) for the real part of \(z=a+bi\); and

      • \(\Im(z) = b\) for the imaginary part of \(z=a+bi\).

    • The modulus (or absolute value) of the complex number \(z=a+bi\) is

      \[ |z| = \sqrt{a^2+b^2}.\]

    • The complex conjugate of \(z=a+bi\) is \(\bar{z} = a-bi\).

      • To divide two complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.

    Frequently Asked Questions about Complex Numbers

    What is a complex number?

    A complex number is a type of number which has the form of a+bi, where a is a real part and b is an imaginary part.

    How to divide complex numbers?    

     Complex numbers are divided with the help of conjugate on the denominator.

    How to add complex numbers? 

    Complex numbers are added by adding the real parts together and the imaginary parts together respectively.  

    How to graph complex numbers?     

     Complex numbers are graphed with an argand diagram using real and imaginary parts as the axes.

    What is a complex number example?

     Examples of complex numbers are 5+8i, 3-2i, and so on.

    Test your knowledge with multiple choice flashcards

    What is the value of \(i^2\)?

    What is the imaginary part of the complex number \(a+bi\)?

    What is the value of \((x+iy)^2\)?

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