Easy

The Imaginary Unit and i²

Simple Explanation

No real number squares to a negative number — squaring anything real always gives zero or a positive result. To solve equations like x²=−1 anyway, mathematicians defined a brand-new number, i, specifically so that i² = −1. Every other imaginary and complex number is built from this one definition.

Why Do We Need It?

This single definition opens up an entire new number system that can solve equations the real numbers alone cannot — and it turns out to be genuinely essential in electrical engineering, physics, and signal processing, not just an abstract curiosity.

Formula

The Imaginary Unit

i² = −1 (powers of i cycle: i, −1, −i, 1, i, −1, ...)

i is defined so that squaring it gives −1. Every higher power of i repeats this same cycle of four values.

i: — the imaginary unit, defined by i² = −1

When to use it: Whenever i² (or any higher power of i) appears in an expression and needs to be simplified.

Worked Example

Simplify a high power of i

Simplify i⁷.

Why Does This Work?

Since i²=−1 by definition, i³=i²·i=−1·i=−i, and i⁴=i²·i²=(−1)(−1)=1 — and once a power returns to exactly 1, multiplying by i again just restarts the same sequence. This is why the pattern repeats forever in blocks of 4.

Real-Life Example

Alternating current (AC) circuit analysis

Electrical engineers need to solve equations describing AC circuits that have no solution using real numbers alone.

The imaginary unit i (often written j in electrical engineering, to avoid confusion with electric current) is the essential building block that makes these circuit calculations possible.

Practice

What is i¹⁰?

Easy

Common mistake

Assuming i² equals +1 out of habit from ordinary square roots — by definition, i² = −1, never +1.

Quick Review

i is defined so that i² = −1.
Powers of i cycle every 4 values: i, −1, −i, 1, ...
To simplify iⁿ, use the remainder of n divided by 4.