Complex Numbers

Real numbers and imaginary numbers are each examples of complex numbers.

For example, we can combine the real number 5 and the imaginary number 4i to form the complex number, 5 + 4i.

Definition — Complex Number

A complex number is a number that can be written in the form a + bi, where a and b are real numbers and

• The real number a is called the real part of the complex number.

• The real number b is called the imaginary part of the complex number.

Note:

• In the complex number a + bi, the imaginary part is b, not bi.
• In a complex number a + bi, a and b can be any real numbers, including zero.

A complex number whose imaginary part, b, is 0, is called a real number.

For example, the complex number 6 + 0i can be written as 6. The number 6 is a real number.

This means that the real numbers are a subset of the complex numbers.

Note:

All real numbers are complex numbers. However, not all complex numbers are real numbers.

A complex number whose imaginary part, b, is not 0, is called an imaginary number.

For example, the complex number 6 + 4i is an imaginary number.

The imaginary numbers are a subset of the complex numbers.

A complex number whose real part, a, is 0, is called a pure imaginary number.

For example, the complex number 0 - 2i can be written as -2i. The number -2i is a pure imaginary number.

Next we will study the arithmetic of complex numbers. That is, we will learn how to add, subtract, multiply, and divide them.

Before we do so, it will be helpful to define what it means for one complex number to be equal to another complex number.

Definition — Equality of Complex Numbers

Two complex numbers are equal if their real parts are equal and their imaginary parts are equal.

That is, a + bi = c + di if a = c and b = d.

Here, a, b, c, and d are real numbers.

3 + 5i = (7 - 4) + 5i since 3 = 7 - 4

8 + 2i = 8 + (1 + 1)i since 2 = 1 + 1

Note:

8 + 7i = 7i + 8 since the real part of each complex number is 8 and the imaginary part of each is 7.