Purely Imaginary Number — Definition, Formula & Examples

A purely imaginary number is a number that can be written as

bi

, where

b

is a nonzero real number and

i = \sqrt{-1}

. It has no real part — it lives entirely on the imaginary axis of the complex plane.

A complex number $z = a + bi$ is purely imaginary if and only if $a = 0$ and $b \neq 0$ , where $a, b \in \mathbb{R}$ and $i^2 = -1$ .

Key Formula

z = bi \quad \text{where } b \in \mathbb{R},\; b \neq 0

Where:

$z$ = The purely imaginary number
$b$ = A nonzero real number (the imaginary part)
$i$ = The imaginary unit, defined by i² = −1

How It Works

Every complex number has the form

a + bi

, with a real part

a

and an imaginary part

b

. When the real part equals zero, you are left with just

bi

, which is purely imaginary. On the complex plane, purely imaginary numbers sit on the vertical (imaginary) axis. Squaring a purely imaginary number always gives a negative real number:

(bi)^2 = -b^2

Worked Example

Problem: Determine whether z = 0 + 6i is purely imaginary, and find z².

Check the form: Write z in standard form a + bi. Here a = 0 and b = 6, so the real part is zero and the imaginary part is nonzero.

z = 0 + 6i = 6i

Classify: Since a = 0 and b ≠ 0, z is purely imaginary.

Square it: Compute z² using the rule i² = −1.

z^2 = (6i)^2 = 36i^2 = 36(-1) = -36

Answer: z = 6i is purely imaginary, and z² = −36.

Why It Matters

Purely imaginary numbers appear when you solve equations like

x^2 + 9 = 0

, which has solutions

x = \pm 3i

. They are essential in electrical engineering, where impedance of ideal inductors and capacitors is purely imaginary, and in physics when analyzing oscillations and wave behavior.

Common Mistakes

Mistake: Calling 0 a purely imaginary number.

Correction: Zero has the form 0 + 0i. Since b = 0, it does not qualify as purely imaginary. Zero is the only complex number that is both real and imaginary in some conventions, but it is not purely imaginary.