# Complex Number Formula

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.

Complex number extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.

### Here are formulas for Complex Number

#### Equality of Complex Numbers Formula

$\LARGE a+bi=c+di\Leftrightarrow a=c\:\:and\:\:b=d$

$\LARGE (a+bi)+(c+di)=(a+c)+(b+d)i$

#### Subtraction of Complex Numbers

$\LARGE (a+bi)-(c+di)=(a-c)+(b-d)i$

#### Multiplication of Complex Numbers

$\LARGE (a+bi)\times(c+di)=(ac-bd)+(ad+bc)i$

#### Multiplication Conjugates

$\LARGE (a+bi)(a+bi)=a^{2}+b^{2}$

#### Division of Complex Numbers

$\LARGE \frac{(a+bi)}{(c+di)}=\frac{a+bi}{c+di}\times\frac{c-di}{c-di}=\frac{ac+bd}{c^{2}+d^{2}}+\frac{bc-ad}{c^{2}+d^{2}}i$

#### Powers of Complex Numbers

1.  $i^{n}$ = i, if n = 4a+1, i.e. one more than the multiple of 4.

Example – $\large i^{1}=i\:;\:i^{5}=i\:;\:i^{9}=i\:; i^{4a+1}\:;$

2. $i^{n}$= -1, if n = 4a+2, i.e. two more than the multiple of 4.

Example – $\large i^{2}=-1\:;\:i^{6}=-1\:;\:i^{10}=-1\:; i^{4a+2}\:;$

3. $i^{n}$= -i, if n = 4a+3, i.e. three more than the multiple of 4.

Example – $\large i^{3}=-i\:;\:i^{7}=-i\:;\:i^{11}=-i\:;i^{4a+3}\:;$

4. $i^{n}$= 1, if n = 4a, i.e. the multiple of 4.

Example – $\large i^{4}=1\:;\:i^{8}=1\:;\:i^{12}=1\:;i^{4a}\:;$

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