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# Complex Numbers Formulas

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

## 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.

Let z1=x1+iy1 and z2=x2+iy2 then

z1+z2=(x1+x2)+i(y1+y2)

## Subtraction

z1-z2=(x1-x2)+i(y1-y2)

## Multiplication

(z1.z2)=(x1+iy1).(x2+iy2)

## 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 z1 and z2

1. (z1 + z2)2 = z12 + z22 + 2z1z2
2. (z1 – z2)2 = z12 + z22 – 2z1z2
3. (z1 + z2)3 = z13 + z23 + 3z1z22+ 3z12z2
4. (z1 – z2)3 = z13 – z23 + 3z1z22 -3z12z2
5. z12 – z22 = (z1 + z2 )(z1 – z2)

## Euler’s formula

• For any real number x, eix = 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 ei?, 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