Complex & Imaginary Numbers

The Hook

You can't square root a negative... until now.

What is it?

Numbers involving 'i', where i = √-1. Written as a + bi.

Time Travel (History)

Raphael Bombelli invented them in 1572. He was the first to realize that 'impossible' numbers could solve real cubic equations.

Cheat Code

i² = -1. Memorize this. It is the key to everything.

Practice This Now!

The Field Guide

In Plain English: For centuries, √-4 was an error message. Mathematicians finally said, 'Let's just pretend there IS an answer, call it i, and see what happens.' It worked. A complex number has a Real part (normal math) and an Imaginary part (i math).

In The Real World: Electrical Engineering. AC circuits (like the outlet in your wall) are modeled using complex numbers to track voltage and phase.

How To Do It

Treat 'i' like a variable (x) for adding/subtracting.
Combine Real parts with Real parts.
Combine Imaginary parts with Imaginary parts.
If multiplying, remember to swap i² for -1 immediately.

⚠

Booby Trap!

Thinking i² is 1. It is NEGATIVE 1. This sign flip changes the whole answer.

Real World Challenge

"In a circuit, Impedance Z1 = 3 + 2i and Z2 = 4 - 5i. What is the total impedance (sum)?"

The Logic: Add reals: 3+4=7. Add imaginaries: 2i - 5i = -3i. Result: 7 - 3i.

Training Drills

The Cycle

i^1 = i i^2 = -1 i^3 = -i i^4 = 1

Repeats every 4 powers.