Intro to complex numbers (article) | خان اکیڈیمی

In the real number system, there is no solution to the equation

x^{2} = - 1

. In this lesson, we will study a new number system in which the equation does have a solution.

The backbone of this new number system is the number

i

, also known as the imaginary unit.

$i^{2} = - 1$ ‍
$\sqrt{- 1} = i$ ‍

By taking multiples of this imaginary unit, we can create infinitely many more new numbers, like

3 i

i \sqrt{5}

, and

- 12 i

. These are examples of imaginary numbers.

However, we can go even further than that and add real numbers and imaginary numbers, for example

2 + 7 i

and

3 - \sqrt{2} i

. These combinations are called complex numbers.

Defining complex numbers

A complex number is any number that can be written as

a + b i

, where

i

is the imaginary unit and

a

and

b

are real numbers.

\begin{array}{ccc} a & + & b i \\ ↑ & ↑ \\ Real & Imaginary \\ part & part \end{array}

a

is called the $real$ ‍ part of the number, and

b

is called the $imaginary$ ‍ part of the number.

The table below shows examples of complex numbers, with the real and imaginary parts identified. Some people find it easier to identify the real and imaginary parts if the number is written in standard form.

Complex Number	Standard Form $a + b i$ ‍	Description of parts
$7 i - 2$ ‍	$- 2 + 7 i$ ‍	The real part is $- 2$ ‍ and the imaginary part is $7$ ‍.
$4 - 3 i$ ‍	$4 + (- 3) i$ ‍	The real part is $4$ ‍ and the imaginary part is $- 3$ ‍
$9 i$ ‍	$0 + 9 i$ ‍	The real part is $0$ ‍ and the imaginary part is $9$ ‍
$- 2$ ‍	$- 2 + 0 i$ ‍	The real part is $- 2$ ‍ and the imaginary part is $0$ ‍

Check your understanding

Problem 1

What is the real part of $13.2 i + 1$ ‍?

Problem 2

What is the imaginary part of $21 - 14 i$ ‍?

Problem 3

What is the real part of $17 i$ ‍?

Classifying complex numbers

We already know what a real number is, and we just defined what a complex number is. Now let's go back and give a proper definition for an imaginary number.

An imaginary number is a complex number $a+bi$ ‍ where $a=0$ ‍.

Similarly, we can say that a real number is a complex number $a+bi$ ‍ where $b=0$ ‍.

From the first definition, we can conclude that any imaginary number is also a complex number. From the second definition, we can conclude that any real number is also a complex number.

In addition, there can be complex numbers that are neither real nor imaginary, like

4 + 2 i

\begin{matrix} Complex numbers \\ \begin{array}{lcr} 4 + 2 i & - 3 - \sqrt{5} i \end{array} \\ \begin{matrix} Real numbers \\ \begin{array}{lcr} \sqrt{5} & - 12.2 & 3 \end{array} \end{matrix} \\ \begin{matrix} Imaginary numbers \\ \begin{array}{lcr} \sqrt{5} i & - 12.2 i & 3 i \end{array} \end{matrix} \end{matrix}

Reflection question

Is the following statement true or false?

Any complex number is either real or imaginary.

Examples

In the table below, we have classified several numbers as real, pure imaginary, and/or complex.

	$\begin{aligned} Real \\ (b = 0) \end{aligned}$ ‍	$\begin{aligned} Imaginary \\ (a = 0) \end{aligned}$ ‍	$\begin{aligned} Complex \\ (a + b i) \end{aligned}$ ‍
$\begin{aligned} 7 + 8 i \\ (7 + 8 i) \end{aligned}$ ‍			X
$\begin{aligned} \sqrt{3} \\ (\sqrt{3} + 0 i) \end{aligned}$ ‍	X		X
$\begin{aligned} 1 \\ (1 + 0 i) \end{aligned}$ ‍	X		X
$\begin{aligned} - 1.3 i \\ (0 + (- 1.3) i) \end{aligned}$ ‍		X	X
$\begin{aligned} 100 i \\ (0 + 100 i) \end{aligned}$ ‍		X	X

Notice that in the table, all of the numbers listed are complex numbers! This is true in general!

Now you try it!

Problem 4

What type of number is $- 2 + 3 i$ ‍?

Problem 5

What type of number is $10.2$ ‍?

Problem 6

What type of number is $- 17 i$ ‍?

Why are these numbers important?

So why do we study complex numbers anyway? Believe it or not, complex numbers have many applications—electrical engineering and quantum mechanics to name a few!

From a purely mathematical standpoint, one cool thing that complex numbers allow us to do is to solve any polynomial equation.

For example, the polynomial equation

x^{2} - 2 x + 5 = 0

does not have any real solutions nor any imaginary solutions. However, it does have two complex number solutions. These are

1 + 2 i

and

1 - 2 i

As we continue our study of mathematics, we will learn more about these numbers and where they are used.

Course: Algebra 2 > Unit 2

Intro to complex numbers