Complex Numbers is an important topic. Here are list of Complex Numbers Formulas

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## Complex Numbers

Complex numbers are defined as an ordered pair of real numbers like (x,y) where

$z=(x,y)=x+iy$ and $i = \sqrt {-1}$

and both x and y are real numbers and x is known as real part of complex number and y is known as imaginary part of the complex number.

## Addition of complex numbers

Let z_{1}=x_{1}+iy_{1} and z_{2}=x_{2}+iy_{2} then

z_{1}+z_{2}=(x_{1}+x_{2})+i(y_{1}+y_{2})

## Subtraction

z_{1}-z_{2}=(x_{1}-x_{2})+i(y_{1}-y_{2})

## Multiplication

(z_{1}.z_{2})=(x_{1}+iy_{1}).(x_{2}+iy_{2})

## Multiplicative Inverse

for $z=a+ib$

z^{-1} is given by

=$(\frac{a}{a^{2}+b^{2}})+i(\frac{-b}{a^{2}+b^{2}})$

## Complex conjugate

$z=x+iy$ Then complex conjugate is given by

$\bar{z}=x-iy$

Properties of Complex Conjugate

- $\bar(\bar{z})=z$
- $z+\bar{z}=2x$
- $z-\bar{z}=2iy$
- $z\bar{z}=(x^{2}+y^{2})$
- $\overline {z_1 -z_2}=\bar {z_1} -\bar {z_2}$
- $\overline {z_1 +z_2}=\bar {z_1} +\bar {z_2}$
- $\overline {z_1 z_2}=\bar {z_1} \bar {z_2}$
- $\overline {\frac{z_1}{z_2}}=\frac {\bar {z_1}}{\bar {z_2}}$

## Modulus of Complex Number

Modulus of the absolute value of z is denoted by |z| and is defined by

|z|=$\sqrt{(x^2+y^2)}$

Properties of modulus

- $z \bar{z}= |z|^2$
- $|z_1z_2| =|z_1 z_2|$
- $|\frac {z_1}{z_2}|=\frac {|z_1|}{|z_2|}$
- $|z_1+z_2|\leq|z_1|+|z_2|$
- $|z_1-z_2|\geq |z_1>|-|z_2>|$

## Polar Form

- Let r be any non negative number and $\theta$ any real number. If we take $x=rcos\theta$ and $y=rsin\theta$ then, $r=\sqrt{x^2+y^2}$which is the modulus of z and $\theta=tan^{-1}\frac{y}{x}$ which is the argument or amplitude of z and is denoted by arg.z
- we also have $x+iy=r(cos\theta+isin\theta)=r[cos(2n\pi+\theta)+isin(2n\pi+\theta)]$ , where n=0, ±1, ±2, ….
- Argument of a complex number is not unique since if $\theta$ is the value of argument then $2n\pi + \theta$ (n=0, ±1, ±2, ….) are also values of the argument. Thus, argument of complex number can have infinite number of values which differ from each other by any multiple of $2 \pi$
- The unique value of $\theta$ such that $\pi < \theta <= \pi $ is called the principal value of the amplitude or principal argument
- The principal argument of the complex number is find using the below steps

Step 1) for z=a+ib , find the acute angle value of $\theta=tan^{-1}|\frac{y}{x}|$

Step 2) Look for the values of a ,b

if (a,b) lies in First quadrant then Argument=$\theta$

if (a,b) lies in second quadrant then Argument =$\pi-\theta$

if (a,b) lies in third quadrant then Argument =$-\pi+\theta$

if (a,b) lies in Fourth quadrant then Argument =$-\theta$

- Arg(0) is not defined.
- argument of positive real number is zero.
- argument of negative real number is $\pm \pi$

## Identities of Complex Number

For all complex numbers z_{1} and z_{2}

- (z
_{1}+ z_{2})^{2}= z_{1}^{2}+ z_{2}^{2}+ 2z_{1}z_{2} - (z
_{1}– z_{2})^{2}= z_{1}^{2}+ z_{2}^{2}– 2z_{1}z_{2} - (z
_{1}+ z_{2})^{3}= z_{1}^{3}+ z_{2}^{3}+ 3z_{1}z_{2}^{2}+ 3z_{1}^{2}z_{2} - (z
_{1}– z_{2})^{3}= z_{1}^{3}– z_{2}^{3}+ 3z_{1}z_{2}^{2}-3z_{1}^{2}z_{2} - z
_{1}^{2}– z_{2}^{2}= (z_{1}+ z_{2})(z_{1}– z_{2})

## Euler’s formula

- For any real number x, e
^{ix}= cos x + i sin x - Let z be a non zero complex number; we can write z in the polar form as,

z = r(cos ? + i sin ?) = r e^{i?}, where r is the modulus and ? is argument of z. - $z \times e^{i \alpha} = re^{i \theta} \times e^{i \alpha} = re^{i(\alpha + \theta)}$

## De Moivre’s theorem

De Moivre’s theorem states following cases

- Case I It states that for any integer n,

$(cos \theta + i sin \theta)^n= cos (n \theta) + i sin (n \theta)$ - Case II if n is of the form p/q where p, q are integers and q > 0

then

$(cos \theta + i sin \theta)^n =cos (2k \pi+ \theta) \frac {p}{q} + sin (2k \pi + \theta)\frac {p}{q}$

Where k=0,1,2,…q-1

## Cube Root of Unity

- Cube Root of Unity are $1,\omega, \omega ^2$
- $\omega = \frac {-1+i \sqrt {3}}{2}$
- So cube roots are 1, $ \frac {-1+i \sqrt {3}}{2}$ and $ \frac {-1-i \sqrt {3}}{2}$

**Properties of Cube roots of Unity**

- $z^3 -1 =(z-1)(z-\omega)(z-\omega ^2)$
- $\omega$ and $\omega ^2$ are roots of the equation $z^2 +z + 1=0$
- Sum of the roots is $1+ \omega + \omega ^2 =0$

Hope you find this list of Complex Numbers Formulas Useful and helpful

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