Definition

 i² = -1

Given two real numbers a and b we will define the complex number z as

z = a + bi

The number a is called the real part of z and the number b is called the imaginary part of z.

A complex number that has a zero real part,

z = 0 + bi = bi

is called a pure imaginary number.

Addition and multiplication with complex numbers

z₁ + z₂ = (a + c) + (b + d)i

z₁z₂ = (ac - bd) + (ad + cb)i

Complex Conjugate

Given the complex number z = a + bi, the complex conjugate is denoted by and is defined to be:

Some properties of complex conjugate: 

Modulus of a complex number

Given a complex number z = a + bi, the modulus is denoted by    and is defined by:

Notice that if z is a real number (i.e. ) then, 

What is logarithm? A logarithm answers the question: How many of one number do we multiply to get another number?  More...