Operation with Complex Numbers

So far, we have dealt with real numbers such as:

In this section, we shall look at a new concept called an imaginary number. Consider the square root of 2. We know that this yields the non-repeating decimal

Now, what is the square root of -2? You might think that there is no solution to the square root of a negative number. However, this is not true! In fact, this is where the imaginary number comes into play. The concept of an imaginary number stems from the imaginary unit, denoted by the letter i, and is represented by the following derivation:

Thus, the square root of -2 is simply

As a matter of fact, we can add real and imaginary numbers together. This structure of numbers leads us to the idea of a complex number.

Standard Form of Complex Numbers

The standard form of complex numbers is

where

• Re (z) = a is the real part of the complex number z

• Im (z) = b is the imaginary part of the complex number z

This is also denoted by

Real and Imaginary Numbers

There are two important subclasses of complex numbers: for a complex number z = a + bi

• If Im (z) = 0, then z = a is a real number

• If Re (z) = 0, then z = bi is said to be purely imaginary

Why are Complex Numbers Important?

Complex numbers have a range of applications. For instance, they are widely used in the field of electrical engineering and quantum mechanics. Complex numbers also help us solve polynomial equations that do not have any real solutions: have a look at Graph and Solve Quadratic Equations which explains how to do this.

We can conduct basic arithmetic operations with complex numbers such as addition, subtraction, multiplication, and division.

Operations with Complex Numbers; Addition and Subtraction

In this section we will explain the most important operations you should be able to perform with complex numbers:

• Addition and subtraction of complex numbers
• Scalar multiplication
• Multiplication and division of complex numbers

Addition and Subtraction of Complex Numbers

To add complex numbers, simply add the corresponding real and imaginary parts. The same rule applies when subtracting complex numbers.

Let z₁ and z₂ be two complex numbers with z₁ = a + bi and z₂ = c + di, where a, b, c, and d are real numbers.

Addition of Complex Numbers Formula

Distributing the positive sign in the second term (to both the real and imaginary parts) and collecting like terms, we obtain

Subtraction of Complex Numbers Formula

Distributing the negative sign in the second term (to both the real and imaginary parts) and collecting like terms, we obtain

Scalar Multiplication of Complex Numbers

To multiply a complex number by a scalar, simply multiply both the real and imaginary parts by the scalar separately.

Let z = a + bi be a complex number and c be a scalar, where a, b and c are real numbers.

Scalar Multiplication of Complex Numbers Formula

Multiplication of Complex Numbers

Multiplying complex numbers is exactly the same as the binomial expansion technique: apply the FOIL method and combine like terms.

Multiplication of Complex Numbers Formula

This is how the FOIL method works, step-by-step.

Let z₁ and z₂ be two complex numbers with z₁ = a + bi and z₂ = c + di, where a, b, c, and d are real numbers. To multiply them

1. Write both in the standard form.
2. Perform the binomial expansion.
3. Combine like terms.

Noting that i² = -1, we obtain

Simplifying this, we get

Division of Complex Numbers

If you have a fraction of complex numbers, multiply the numerator and denominator by the complex conjugate of the denominator.