# Complex Numbers

## Imaginary number

Imaginary number (when squared give a negative result)

i is the unit of imaginary number which is square root of -1

i = √-1, because i2=-1

So, (7i)2 = -7

## Definition of Complex Number

A Complex Number is a combination of a Real Number and an Imaginary Number

Example

4 + 7i, where 4 is Real Number, and 7i is Imaginary Number

## Either Part Can Be Zero

• a Complex Number has a real part and an imaginary part.
• Either part can be 0

So,

• all Real Numbers are Complex numbers
• all Imaginary Numbers are Complex Numbers.
Examples
 Real Part Imaginary Part Complex Number Combination 4 7i 4 + 7i Pure Real 4 0i 4 Pure Imaginary 0 7i 7i

We often use z for a complex number. And Re() for the real part and Im() for the imaginary part, like this:

z = a + bi, where Re(z) = a, Im(z) = b

(a+bi) + (c+di) = (a+c) + (b+d)i

## Multiplying 2 complex numbers

To multiply 2 complex numbers, each part of the first complex number gets multiplied by

each part of the second complex number

(a+bi)(c+di) = ac + adi + bci + bdi2

Because i2 = -1, so

(a+bi)(c+di) = ac – bd + adi + bci

## Dividing 2 complex numbers

To divide 2 complex numbers, multiply both top and bottom by the conjugate of the bottom.

Note: A conjugate is where we change the sign in the middle

 Complex number conjugate Multiplying By the Conjugate a + bi a – bi a2 + b2 a – bi a + bi a2 + b2

So, to divide a + bi by c – di:

Or, to divide a + bi by c + di:

Examples
 Application Result z1 Given Re(z1) = 4, Im(z2)=7 4+7i z2 Given Re(z1) = 4, Im(z2)=-7 4-7i z1+z2 (4+4) + (7-7)i 8 + 0i = 8 z1 x z2 4×4 – 7x(-7) + 4x(-7)i + 4x7i 44 + 0i = 44 z1 / z2    