The Imaginary Unit

Consider the equation $x^{2} + 4 = 0$ . Rearranging gives $x^{2} = - 4$ , and there is no real number whose square is negative. For centuries, mathematicians dismissed such equations as unsolvable — until 16th-century mathematicians decided to simply define a new kind of number.

Imaginary Unit

The imaginary unit $i$ is defined by the property:
$i^{2} = - 1$
Equivalently, $i = - 1$ .

This one definition unlocks an entirely new number system. Although called "imaginary", these are rigorous mathematical objects — not placeholders or tricks.

Note

Higher powers of $i$ follow a repeating cycle of period 4:
$i^{1} = i, i^{2} = - 1, i^{3} = - i, i^{4} = 1, i^{5} = i, \dots$
To simplify $i^{n}$ , divide $n$ by 4 and use the remainder.

Example

Simplify $i^{23}$ .

Step 1: Divide the exponent by 4: $23 = 4 \times 5 + 3$ , so the remainder is 3.

Step 2: Use the cycle: $i^{23} = i^{3} = - i$ .

Complex Numbers in Cartesian Form

Complex Number

A complex number is any expression of the form
$z = a + bi$
where $a, b \in R$ , and $i$ is the imaginary unit. The set of all complex numbers is denoted $C$ .

The two components have specific names:

Real part: $Re (z) = a$
Imaginary part: $Im (z) = b$ (note: this is a real number — it is the coefficient of $i$ )

Note

The real and imaginary parts can be any real numbers, including zero:

When $b = 0$ : $z = a$ is a purely real number. Every real number is also a complex number.
When $a = 0$ : $z = bi$ is a purely imaginary number.
The real numbers $R$ are a subset of $C$ .

Example

Identify the real and imaginary parts of the following:

Complex number	$Re (z)$	$Im (z)$
$z = 3 + 5 i$	$3$	$5$
$z = - 2 - 7 i$	$- 2$	$- 7$
$z = 4$	$4$	$0$
$z = - 3 i$	$0$	$- 3$

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AHL 1.12—Complex numbers introduction

The Imaginary Unit

Complex Numbers in Cartesian Form

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