🌀 Complex Numbers Without the Misery

🌀 Complex Numbers Without the Misery

School said the square root of −1 could not exist. Yet engineers use it to work out how much current a motor will draw, and the radio chip in your phone uses it to decode Wi-Fi. This book shows why this "impossible" number turned out to be one of the most useful ideas in math, and why it's really just a turn.

WHAT'S INSIDE:

📜 Why √−1 was needed — the cubic formulas of the 1500s that forced it into existence

🔄 i and its powers — a cycle of four that repeats forever

➕ a + bi — adding and multiplying, like keeping money in two currencies

➗ Conjugates and division — the partner number that makes division work

🗺️ The complex plane — every complex number is a point on a map

↪️ Multiplication as rotation — why multiplying by i is nothing more than a quarter turn

📏 Modulus and argument — the distance and direction of every point

✨ Euler's formula — why e^(iπ) + 1 = 0 is not magic, just a half turn written down

⚠️ A "5 Common Mistakes" chapter — including forgetting that i² is −1

🧮 A Formulas & Quick Reference appendix, with a worked example

✏️ 10 fully worked examples with every step shown

✅ 11 practice problems with complete answers at the back

WHO THIS IS FOR:

Anyone who thought "imaginary numbers" were something math teachers made up. Anyone who wants to know why engineers and phone makers can't do without them.

No prior knowledge of complex numbers required. 21 pages.