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## 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 $\mathrm{z}=\mathrm{a}+\mathrm{ib}$ where $\mathrm{a},\mathrm{b}\in \mathrm{\mathbb{R}}$. 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.

## 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

${r}^{2}={\mathrm{a}}^{2}+{\mathrm{b}}^{2}$

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

$\left|\mathrm{z}\right|=\mathrm{r}=\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}$

where $\left|\mathrm{z}\right|$ 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. (${\left|\mathrm{z}\right|}^{2}={\mathrm{a}}^{2}+{\mathrm{b}}^{2}$)

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

**Solution: **

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

${\left|\mathrm{z}\right|}^{2}={(\mathrm{real}\mathrm{part})}^{2}+{(\mathrm{imaginary}\mathrm{part})}^{2}$

**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:

$\begin{array}{l}\left|\mathrm{z}\right|=\sqrt{{3}^{2}+{4}^{2}}\\ \Rightarrow \left|\mathrm{z}\right|=5\end{array}$

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

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

Let $\mathrm{z}=\mathrm{a}+\mathrm{ib}$ be a complex number, then the conjugate of it is $\overline{\mathrm{z}}=\mathrm{a}-\mathrm{ib}$ (recall the Complex Number article section about conjugates)!

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

$\mathrm{z}\stackrel{-}{\mathrm{z}}=\left(\mathrm{a}+\mathrm{ib}\right)\left(\mathrm{a}-\mathrm{ib}\right)$

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

$\mathrm{z}\stackrel{\mathrm{\_}}{\mathrm{z}}={\mathrm{a}}^{2}-\mathrm{abi}+\mathrm{abi}-{\mathrm{i}}^{2}{\mathrm{b}}^{2}\phantom{\rule{0ex}{0ex}}={\mathrm{a}}^{2}-{\mathrm{i}}^{2}{\mathrm{b}}^{2}$

And as $\mathrm{i}=\sqrt{-1}$, therefore

${\mathrm{i}}^{2}=-1$

**Step 3**: Substituting for ${\mathrm{i}}^{2}=-1$ into the equation in step 2, we get

$\mathrm{z}\stackrel{\mathrm{\_}}{\mathrm{z}}={\mathrm{a}}^{2}-\left(-1\right){\mathrm{b}}^{2}\phantom{\rule{0ex}{0ex}}\mathrm{z}\stackrel{\mathrm{\_}}{\mathrm{z}}={\mathrm{a}}^{2}+{\mathrm{b}}^{2}$

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

$\mathrm{z}\stackrel{\mathrm{\_}}{\mathrm{z}}={\left|\mathrm{z}\right|}^{2}$

In other words,

$(complexnumber)\times (it\text{'}sconjugate)=modulu{s}^{2}$

## Modulus and Phase Angle

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

Let $\mathrm{\theta}$ be the angle made by z with the positive x-axis. We call this angle $\mathrm{\theta}$ the **phase** angle of z, also known as the argument of z. In polar form, z is expressed as,

$\mathrm{z}=\mathrm{r}\left(\mathrm{cos\theta}+\mathrm{isin\theta}\right)$

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

$\mathrm{tan\theta}=\frac{\mathrm{b}}{\mathrm{a}}$

Solving for $\theta $ we get

$\mathrm{\theta}={\mathrm{tan}}^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$

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.

$\mathrm{sin}\theta =\frac{oppositeside}{hypotenuse}\phantom{\rule{0ex}{0ex}}$

Subbing in variables from figure 2 above, we get:

$\mathrm{sin}\theta =\frac{b}{r}$

We just learned that r is the modulus, $\sqrt{{a}^{2}+{b}^{2}}$. Substituting this expression for r in the equation above, we get

$\mathrm{sin\theta}=\frac{\mathrm{b}}{\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}}$

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

$r=\sqrt{{a}^{2}+{b}^{2}}$

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

$\mathrm{cos\theta}=\frac{\mathrm{a}}{\sqrt{{\mathrm{a}}^{2}+{\mathrm{b}}^{2}}}$

### Principal Arguments of Complex Numbers

It can be noticed that if the angle is changed by ${360}^{0},{720}^{0}$ and so on, the complex number remains the same. So there will not be a unique value of $\mathrm{\theta}$ but infinitely many. Hence, we call the first value of $\mathrm{\theta}$ the **principal**** argument of z, Arg(z). **

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

**Solution: **

**Step 1:** We know that $\mathrm{tan}\theta =\frac{b}{a}$. From the question we have $a=1andb=1$. Thus, we have the value of tangent as,

$\begin{array}{l}\mathrm{tan\theta}=\frac{1}{1}\\ \\ \end{array}$

**Step 2: Solving for $\theta $, we get**

**$\begin{array}{l}\mathrm{\theta}={\mathrm{tan}}^{-1}\left(1\right)\\ \theta =45\xb0\end{array}$**

Thus, the principal argument, Arg(z), of $z=1+i$ is $45\xb0$. 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 ${\mathrm{z}}_{1},{\mathrm{z}}_{2}$ be two distinct complex numbers and their polar forms are

${\mathrm{z}}_{1}={\mathrm{r}}_{1}\left({\mathrm{cos\theta}}_{1}+{\mathrm{isin\theta}}_{1}\right)$ and ${\mathrm{z}}_{2}={\mathrm{r}}_{2}\left({\mathrm{cos\theta}}_{2}+{\mathrm{isin\theta}}_{2}\right)$

Then the product ${\mathrm{z}}_{1}{\mathrm{z}}_{2}$ will be a complex number with the phase as ${\mathrm{\theta}}_{1}+{\mathrm{\theta}}_{2}$.

**A significant consequence of De Moivre’s Theorem: **We have defined $\mathrm{\theta}$** **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, ${\mathrm{z}}^{\mathrm{n}}$, with a different phase angle. A consequence of the above theorem is that

${\mathrm{z}}^{\mathrm{n}}={\mathrm{r}}^{\mathrm{n}}{\left(\mathrm{cos\theta}+\mathrm{isin\theta}\right)}^{\mathrm{n}}$

simplifies to

${\mathrm{z}}^{\mathrm{n}}={\mathrm{r}}^{\mathrm{n}}\left(\mathrm{cosn\theta}+\mathrm{isinn\theta}\right)$

which has a phase angle of $\mathrm{n\theta}$. This is true for all integer values of n i.e. $\mathrm{n}\in \mathrm{\mathbb{Z}}$

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 $z={\left(1+i\right)}^{6}$, find its argument.

**Solution:**

**Step1**: Convert the complex number into its polar form:

$z={r}^{6}{\left(\mathrm{cos}\theta +is\mathrm{in}\theta \right)}^{6}$$={r}^{6}\left(\mathrm{cos}n\theta +i\mathrm{sin}n\theta \right)$

$z={\left(\sqrt{2}\right)}^{6}{\left(\frac{1+i}{\sqrt{2}}\right)}^{6}$

**Step2**: Comparing with the general form, we deduce that

Giving us the principle argument as follows:

$\mathrm{tan}n\theta =\frac{1}{1}=1$

$n\theta =\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

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

$6\theta =\frac{\pi}{4}$

$\theta =\frac{\pi}{24}$

Hence the principle argument is $\pi /24$ 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 ${x}^{2}-x+4=0$.

**Solution:**

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

$x=\frac{1\pm \sqrt{1-16}}{2}$

**Step 2: **Upon simplification, we get:

$x=\frac{1\pm \sqrt{-15}}{2}$

But how do we simplify the part $\sqrt{-15}$ because we do not know the square root of a negative number?

This is where we employ our knowledge of using $i=\sqrt{-1}$ which simplifies the given form into:

$x=\frac{1\pm 15i}{2}$

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:**

**$\stackrel{\_}{h}\frac{\partial \psi}{\partial t}=\stackrel{\wedge}{H}\psi $**

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 equation$\mathrm{z}\stackrel{\mathrm{\_}}{\mathrm{z}}={\left|\mathrm{z}\right|}^{2}$
- 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 $\mathrm{\theta}={\mathrm{tan}}^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$.
- 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 ${\mathrm{z}}^{\mathrm{n}}={\mathrm{r}}^{\mathrm{n}}\left(\mathrm{cosn}+\mathrm{isinn\theta}\right)$ (the argument of a complex number raised to power n is n times the argument of the original complex

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##### 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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