Complex Numbers

Introduction

Imaginary Numbers

Definition In Mathematics, the imaginary number i is defined such that $\mathrm{i}^2 = -1$. So, $\mathrm{i} = \sqrt{-1}$.

Cartesian form of a complex number

$z = a + b\mathrm{i}$ , where $a,b \in \mathbb{R}$ is called the Cartesian form of a complex number.

$a$ is called the real part of $z$, written as Re(z) = a

and $b$ is called the imaginary part of $z$, written as Im(z) = b.

Geometric representation

Argand diagram

The real number line is extended to a plane of points called an Argand diagram. The plane is denoted by $\mathbb{C}$.

Modulus

Given the complex number $z = x + y\mathrm{i}$, $x,y \in \mathbb{R}$, the modulus of $z$ is given by $|z| = |x + y\mathrm{i}| = \sqrt{ x^2 + y^2} = \sqrt{\left(\mathrm{Re}(z)\right)^2 + \left(\mathrm{Im}(z)\right)^2}$

The modulus is the distance r from the origin.

Operations

Two complex numbers are equal …

addition

$z_1 + z _2 = (a + b\mathrm{i}) + (c + d\mathrm{i}) = (a + c) + (b + d)\mathrm{i}$

scalar multiplication

$\lambda z = \lambda (a + b\mathrm{i}) = (\lambda a) + (\lambda b)\mathrm{i}$

multiplication

$z_1 \cdot z _2 = (a + b\mathrm{i}) \cdot (c + d\mathrm{i}) = (ac - bd) + (ad + bc)\mathrm{i}$

conjugates

For every complex number $z = a + b\mathrm{i}$ there is a conjugate complex number of the form $z^* = a - b\mathrm{i}$.

Their real parts are equal and their imaginary parts are opposite.

The conjugates are useful when dividing two complex numbers.

example

Find $\frac{1 + 3\mathrm{i}}{2 - \mathrm{i}}$

$\frac{1 + 3\mathrm{i}}{2 - \mathrm{i}} \cdot \frac{2 + \mathrm{i}}{2 + \mathrm{i}} = \frac{2 - 3 + 6\mathrm{i} + \mathrm{i}}{4 + 1} = \frac{1}{5}(-1 +7\mathrm{i})$

Powers and Roots

argument

The argument $\theta = \arg(z)$ is the angle formed by the complex number $z=a+ib$ in the Complex plane in anti-clockwise direction.

$$\theta = \arg(z) = \arctan{\left(\frac{b}{a}\right)}$$

Polar form

In the the Argand plane

$$x = r\cos{\theta}$$ $$y = r\sin{\theta}$$

where $r = |z|$

So the complex number $z = x+iy$ in Cartesian form can be written as

$$z= r(\cos{\theta} + i\sin{\theta})$$

short form

Some literature uses a short form:

$$z= r \cdot \mathrm{cis~} {\theta}$$

Euler’s or exponential form of a complex number

A complex number can written in Euler’s form.

$$z = r\mathrm{e}^{i\theta}$$

where $r = |z|$ and $\theta = \arg(z) = \arctan{\left(\frac{b}{a}\right)}$

opposite

conjugate

opposite-conjugate

operations with complex numbers in polar form

addition

multiplication by a scalar

multiplication of complex numbers in polar form

multiplying the moduli and adding the arguments

$$z_1 \cdot z_2$$

reciprocal complex number

$$\frac{1}{z}$$

division of complex numbers in polar form

$$\frac{z_1}{z_2}$$

Powers and Roots of Complex Numbers in Polar Form

De Moivre’s theorem

$$z^n = r^n(\cos{n\theta} + i \sin{n\theta})$$

Also, it can be shown that $$z^{\frac{1}{n}} = r^{\frac{1}{n}}(\cos{\frac{\theta}{n}} + i \sin{\frac{\theta}{n}})$$

The periodic nature of the trigonometric functions allows for multiple roots.

Given

$$z^n = r(\cos{\theta} + i \sin{\theta})$$

Say, w is the nth root of z, where $z = r\mathrm{cis~}\theta$.

Taking the nth root allows for multiples of $2\pi$ as $z = r\mathrm{cis~}(\theta + 2\pi k)$ all represent the same complex number.

So, $$w = z^{\frac{1}{n}} = r\left(\cos{\left(\frac{\theta+ 2\pi k}{n}\right)} + i \sin{\left(\frac{\theta+ 2\pi k}{n}\right)}\right)$$


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