The Imaginary Unit

Consider the equation $x^{2} + 4 = 0$ . Rearranging gives $x^{2} = - 4$ , and since no real number squared gives a negative result, we say there are no real solutions. Mathematicians in the 16th century pushed past this barrier by introducing a new mathematical object: the imaginary unit.

Imaginary Unit

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

This single definition unlocks an entirely new number system. Despite the name "imaginary", these are rigorous mathematical objects — just as valid as the real numbers you've always worked with.

Using $i$ , we can now write solutions to equations like $x^{2} = - 4$ :
$x = \pm - 4 = \pm 2 i$

Note

Higher powers of $i$ cycle with period 4:

$i^{1} = i$
$i^{2} = - 1$
$i^{3} = - i$
$i^{4} = 1$

This cycle repeats, so $i^{5} = i$ , $i^{6} = - 1$ , and so on. To simplify $i^{n}$ , divide $n$ by 4 and use the remainder.

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 $i^{2} = - 1$ . The value $a$ is called the real part, written $Re (z)$ , and $b$ is called the imaginary part, written $Im (z)$ .

This is called the Cartesian form (or rectangular form) of a complex number. The set of all complex numbers is denoted $C$ .

Special cases:

When $b = 0$ : $z = a$ is a purely real number. Every real number is a complex number.
When $a = 0$ : $z = bi$ is a purely imaginary number.
When $a = 0$ and $b = 0$ : $z = 0$ , the only number that is both real and purely imaginary.

Analogy

Think of $a + bi$ like coordinates on a map: the real part $a$ tells you how far east/west you are, and the imaginary part $b$ tells you how far north/south. Just as a location needs two coordinates, a complex number needs two components.

Warning

The imaginary part of $z = a + bi$ is the real number $b$ , not $bi$ . For example, if $z = 3 + 5 i$ , then $Im (z) = 5$ , not $5 i$ . This is a very common source of error.

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AHL 1.12—Complex numbers – Cartesian form and Argand diag

The Imaginary Unit

Complex Numbers in Cartesian Form

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