Complex Numbers: Introduction

The imaginary unit i is defined as: [formula] A complex number has the form a + bi where a is the real part and b is the imaginary part. Examples: 3 + 2i, -1 + 4i, 5 (real), 7i (purely imaginary). Powers of i cycle every 4: - i^1 = i - i^2 = -1 - i^3 = -i - i^4 = 1 - i^5 = i (cycle repeats) To find i^n: divide n by 4 and use the remainder.

Example: Simplify i^{27}

Addition/Subtraction: Combine real parts and imaginary parts separately. [formula] Multiplication: Use FOIL, then replace i^2 with -1. [formula]

Example: Multiply (3 + 2i)(1 - 4i)