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What Are Complex Numbers?

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There are no real numbers that solve the equation $x^{2} = - 1$ . No matter what you do, neither a positive number multiplied by itself, nor a negative number multiplied by itself, will give you a negative number as the product. So in order to take square roots of negative numbers, mathematicians have introduced something called complex numbers. Complex numbers are based on the imaginary unit $i$ .

Theory

Imaginary Numbers

The imaginary unit

i

is defined as

i^{2} = - 1

. Imaginary numbers are multiples of the imaginary unit. Every imaginary number can be written as

z = a i,

where $a \in ℝ$ .

A complex number consists of two parts: A real part and an imaginary part. The real part of a complex number is a real number on the real number line—just an ordinary number like you’re used to. The imaginary part of a complex number is the real number in front of the imaginary unit

i

. When you write complex numbers, you can choose to split the number into its real part and imaginary part. The number is then written in Cartesian form, which is in contrast to polar form.

Theory

Cartesian Form

In Cartesian form, complex numbers are written in the form

z = a + b i,

where $a, b \in ℝ$ . Here, $a$ is the real part of $z$ , and $b$ is the imaginary part of $z$ .

The real part of a complex number

z

is often denoted by

Re (z)

. The imaginary part of a complex number

z

is often denoted by

Im (z)

. The set of all complex numbers is denoted by

ℂ

Example 1

Find $Re (z)$ and $Im (z)$ for the complex number $z = 5 - 3 i \in ℂ$

The real part of a complex number is always the number without $i$ . Here you have $Re (z) = 5$ . The imaginary part of a complex number is always the number in front of $i$ . Note that $i$ is not part of the number’s imaginary part. Here you have $Im (z) = - 3$ .

Both imaginary and real numbers are contained in the set of complex numbers. Imaginary numbers can be thought of as complex numbers without a real part, and real numbers can be thought of as complex numbers without an imaginary part. The set of real numbers is therefore contained in the set of complex numbers:

The set of real numbers contained in the set of complex numbers.

Think About This

Despite the name imaginary numbers, they are not more artificial than for instance negative numbers. Complex and imaginary numbers have important applications within many branches of mathematics and the natural sciences, such as physics.

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