Math Core

The equation $x^2 = -1$ has no real solution, because the square of every real number is zero or positive. Mathematicians handled this the same way they once handled $x + 5 = 2$ (by inventing negative numbers) and $x^2 = 2$ (by accepting irrational ones): they extended the number system. The result, the complex numbers, is what makes every quadratic equation solvable.

The imaginary unit

Define a new number $i$ whose square is $-1$ .

Definition

Imaginary unit

The imaginary unit $i$ is the number with

i^2 = -1, \qquad\text{so}\qquad i = \sqrt{-1}.

For any positive real number $a$ , the square root of $-a$ is defined as $\sqrt{-a} = i\sqrt{a}$ .

With this definition, square roots of negative numbers become ordinary expressions you can simplify.

\sqrt{-25} = i\sqrt{25} = 5i, \qquad \sqrt{-7} = i\sqrt{7}, \qquad \sqrt{-50} = i\sqrt{25 \cdot 2} = 5i\sqrt{2}.

It is customary to write $i\sqrt{7}$ or $\sqrt{7}\,i$ rather than $\sqrt{7}i$ , so that nobody mistakes the $i$ for being under the radical.

Check the first one: $(5i)^2 = 25 i^2 = 25(-1) = -25$ . So $5i$ really is a number whose square is $-25$ . (So is $-5i$ , just as both $5$ and $-5$ square to $25$ . The symbol $\sqrt{-25}$ means the one with the positive coefficient, $5i$ .)

The word "imaginary" is a historical accident. These numbers are no less legitimate than negative numbers, and they are essential in electrical engineering, signal processing and quantum physics.

Complex numbers

Combining real numbers with multiples of $i$ gives a whole new set of numbers.

Definition

Complex number

A complex number is a number of the form $a + bi$ , where $a$ and $b$ are real numbers. This is its standard form.

$a$ is the real part.
$b$ is the imaginary part (the real number multiplying $i$ , not $bi$ ).

Some examples:

number	standard form	real part	imaginary part
$3 - 4i$	$3 + (-4)i$	$3$	$-4$
$6i$	$0 + 6i$	$0$	$6$
$-2$	$-2 + 0i$	$-2$	$0$
$\dfrac{1 + i}{2}$	$\dfrac{1}{2} + \dfrac{1}{2}i$	$\dfrac{1}{2}$	$\dfrac{1}{2}$

Every real number is a complex number with imaginary part $0$ . A complex number with real part $0$ and $b \ne 0$ , such as $6i$ , is called pure imaginary. So the real numbers sit inside the complex numbers, the way the integers sit inside the rational numbers.

Worked example: Writing in standard form

Write $\dfrac{6 - \sqrt{-12}}{2}$ in the form $a + bi$ .

First simplify the radical: $\sqrt{-12} = i\sqrt{4 \cdot 3} = 2i\sqrt{3}$ . Then divide each term by $2$ :

\frac{6 - 2i\sqrt{3}}{2} = 3 - i\sqrt{3}.

The real part is $3$ and the imaginary part is $-\sqrt{3}$ .

Equal complex numbers

Two complex numbers are equal only when their real parts are equal and their imaginary parts are equal. This turns one equation with complex numbers into two equations with real numbers.

Worked example: Matching parts

Find real numbers $x$ and $y$ so that $2x + (y - 1)i = 8 + 5i$ .

Match real parts: $2x = 8$ , so $x = 4$ . Match imaginary parts: $y - 1 = 5$ , so $y = 6$ .

Powers of i

Multiplying by $i$ over and over cycles through just four values.

i^1 = i, \qquad i^2 = -1, \qquad i^3 = i^2 \cdot i = -i, \qquad i^4 = (i^2)^2 = 1.

After $i^4 = 1$ the pattern repeats: $i^5 = i^4 \cdot i = i$ , $i^6 = -1$ , and so on.

Powers of i repeat every 4

To find $i^n$ for a whole number $n$ , divide $n$ by $4$ and use the remainder:

remainder	0	1	2	3
$i^n$	$1$	$i$	$-1$	$-i$

Worked example: A large power

Simplify $i^{23}$ .

$23 = 4 \cdot 5 + 3$ , so $i^{23} = (i^4)^5 \cdot i^3 = 1^5 \cdot (-i) = -i$ .

A product rule that breaks

For nonnegative real numbers, $\sqrt{a}\cdot\sqrt{b} = \sqrt{ab}$ . This rule is not true when both numbers are negative.

Common mistake

$\sqrt{-4} \cdot \sqrt{-9} \ne \sqrt{36}$ . Always rewrite square roots of negatives using $i$ before multiplying:

\sqrt{-4} \cdot \sqrt{-9} = (2i)(3i) = 6i^2 = -6.

Multiplying under one radical first would give $\sqrt{(-4)(-9)} = \sqrt{36} = 6$ , which has the wrong sign.

The complex plane

Real numbers fill a number line. Complex numbers need two dimensions, because each has two independent parts. In the complex plane, the horizontal axis is the real axis and the vertical axis is the imaginary axis, and $a + bi$ is plotted at the point $(a, b)$ .

In the complex plane, a + bi is the point (a, b). The dashed segment from 0 to 3 + 4i has length 5.Open in grapher →

The distance from $0$ to $a + bi$ is called its absolute value (or modulus), and the Pythagorean theorem gives

|a + bi| = \sqrt{a^2 + b^2}.

For example, $|3 + 4i| = \sqrt{9 + 16} = 5$ . For a real number ( $b = 0$ ) this is just $\sqrt{a^2} = |a|$ , the usual absolute value.

Tip

Complex numbers can't be put in order the way real numbers can: it makes no sense to say $2 + i < 3i$ . Absolute value is how you compare their sizes: $|2 + i| = \sqrt{5}$ is smaller than $|3i| = 3$ .

Practice

Practice 1

Simplify $\sqrt{-49}$ .

Practice 2

Write $\sqrt{-18}$ in the form $k\,i\sqrt{2}$ . What is $k$ ?