Imaginary Numbers

Q: What is the difference between imaginary numbers and complex numbers?

A complex number has the form $a + bi$, where $a$ is the real part and $bi$ is the imaginary part. A pure imaginary number is the special case where $a = 0$, so it looks like $bi$. Every imaginary number is a complex number, but not every complex number is imaginary — for example, $3 + 2i$ is complex but not purely imaginary.

Q: What are the powers of i?

The powers of $i$ cycle every four steps: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$, and then the pattern repeats. To find any power of $i$, divide the exponent by 4 and use the remainder to identify which of these four values it equals. For instance, $i^{13} = i^1 = i$ because 13 divided by 4 has remainder 1.

Imaginary Numbers
Pure Imaginary Numbers

Complex numbers with no real part, such as 5i.

Nested diagram showing number sets: natural, whole, integers, rational, algebraic, real numbers, with complex and pure...

Key Formula

i = \sqrt{-1} \quad \text{so that} \quad i^2 = -1

Where:

$i$ = The imaginary unit, defined as the square root of −1
$bi$ = A pure imaginary number, where b is any nonzero real number

Worked Example

Problem: Simplify

\sqrt{-36}

Step 1: Rewrite the negative under the square root using the definition of

i

\sqrt{-36} = \sqrt{36 \cdot (-1)}

Step 2: Separate the square root into two factors.

\sqrt{36 \cdot (-1)} = \sqrt{36} \cdot \sqrt{-1}

Step 3: Evaluate each part. The square root of 36 is 6, and the square root of −1 is

i

\sqrt{36} \cdot \sqrt{-1} = 6i

Answer:

\sqrt{-36} = 6i

Another Example

This example shows that multiplying two imaginary numbers together produces a real number (not an imaginary one), which surprises many students. It also reinforces the key rule that $i^2 = -1$ .

Problem: Multiply the imaginary numbers

3i

and

4i

Step 1: Multiply the real coefficients and the imaginary units separately.

(3i)(4i) = 3 \cdot 4 \cdot i \cdot i = 12i^2

Step 2: Replace

i^2

with

-1

, using the fundamental property of the imaginary unit.

12i^2 = 12(-1) = -12

Step 3: Notice that the product of two imaginary numbers is a real number.

(3i)(4i) = -12

Answer:

(3i)(4i) = -12

, a real number.

Frequently Asked Questions

What is the difference between imaginary numbers and complex numbers?

A complex number has the form

a + bi

, where

a

is the real part and

bi

is the imaginary part. A pure imaginary number is the special case where

a = 0

, so it looks like

bi

. Every imaginary number is a complex number, but not every complex number is imaginary — for example,

3 + 2i

is complex but not purely imaginary.

Do imaginary numbers actually exist or are they just made up?

Imaginary numbers are as mathematically valid as negative numbers or irrational numbers. The name 'imaginary' is historical and somewhat misleading — it was coined by René Descartes as a dismissive label. Imaginary numbers are essential in physics, engineering, and signal processing, where they model real-world phenomena like alternating current and wave behavior.

What are the powers of i?

The powers of

i

cycle every four steps:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

, and then the pattern repeats. To find any power of

i

, divide the exponent by 4 and use the remainder to identify which of these four values it equals. For instance,

i^{13} = i^1 = i

because 13 divided by 4 has remainder 1.

Imaginary Numbers vs. Real Numbers

	Imaginary Numbers	Real Numbers
Definition	Numbers of the form $bi$ where $b \neq 0$ and $i = \sqrt{-1}$	All numbers on the standard number line (positive, negative, zero, fractions, irrationals)
Square	The square of a pure imaginary number is always negative	The square of a real number is always zero or positive
Number line	Plotted on the vertical axis of the complex plane	Plotted on the horizontal axis (the traditional number line)
Example	$5i,\; -3i,\; \frac{1}{2}i$	$7,\; -3,\; \sqrt{2},\; \frac{1}{2}$
Combined form	A subset of complex numbers (where real part = 0)	A subset of complex numbers (where imaginary part = 0)

Why It Matters

You first encounter imaginary numbers when solving quadratic equations with negative discriminants, such as

x^2 + 4 = 0

, which has no real solutions but does have imaginary solutions

x = \pm 2i

. Imaginary and complex numbers are foundational in advanced algebra, precalculus, and the study of polynomials — the Fundamental Theorem of Algebra guarantees every polynomial has roots in the complex numbers. Beyond math class, electrical engineers use imaginary numbers daily to analyze alternating-current circuits, and physicists rely on them in quantum mechanics.

Common Mistakes

Mistake: Writing

\sqrt{-a \cdot -b} = \sqrt{a} \cdot \sqrt{b}

when both

a

and

b

are positive. For example, claiming

\sqrt{-4} \cdot \sqrt{-9} = \sqrt{36} = 6

Correction: The rule

\sqrt{ab} = \sqrt{a}\sqrt{b}

only works when at least one of the numbers is non-negative. Convert to

i

form first:

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

Mistake: Forgetting that

i^2 = -1

and leaving answers with

i^2

in them, or incorrectly simplifying

i^2

1

Correction: Always replace

i^2

with

-1

. This is the defining property of

i

and affects the sign of your answer.

Related Terms

Complex Numbers — Numbers of the form a + bi that include imaginary numbers
Real Part — The real component of a complex number (zero for imaginary numbers)
Real Numbers — All numbers on the number line; complement imaginary numbers
Natural Numbers — The counting numbers; a subset of real numbers
Integers — Whole numbers and their negatives; part of the real number system
Rational Numbers — Ratios of integers; another subset of real numbers
Algebraic Numbers — Roots of polynomials; includes both real and imaginary numbers
Whole Numbers — Non-negative integers; a subset of the real number system