Table of Contents

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

So,

Examples

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

To add 2 complex numbers, add each part separately

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

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

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	a^² + b^²
a – bi	a + bi	a^² + b^²

So, to divide a + bi by c – di:

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

Examples

Reciprocal of Number

Table of Factors and Multiples

Common Factor

Simplifying Square Roots