Overview

Complex numbers extend the real line to a plane. Most AMC/AIME problems start with algebraic manipulation and then switch to geometry when convenient.

Key Ideas

• $z = a + bi$ with $i^2 = -1$.
• Powers of $i$ repeat every $4$.
• Real part is $a$, imaginary part is $b$.

Core Skills

Separate Real and Imaginary Parts

Write $z=a+bi$ and equate real and imaginary parts to solve equations.

Use $i^2=-1$ Early

Simplify powers of $i$ before multiplying out to avoid sign mistakes.

Convert Between Forms

Switch between algebraic and geometric interpretations when helpful.

Worked Example

Compute $i^{2023}$.

Powers of $i$ cycle with period $4$. Since $2023 \equiv 3 \pmod 4$, $i^{2023} = i^3 = -i$.

More Examples

Example 1: Add and Multiply

Compute $(3+2i)(1-4i)$.

$3-12i+2i-8i^2 = 11-10i$.

Example 2: Solve a Simple Equation

Solve $z+2i=3-4i$.

$z=3-6i$.

Example 3: Powers of $i$

Compute $i^{58}$.

$58\equiv 2\pmod 4$, so $i^{58}=-1$.

Strategy Checklist

• Reduce powers of $i$ modulo $4$.
• Keep real and imaginary parts separate.
• Check arithmetic with $i^2=-1$.

Common Pitfalls

• Forgetting the period $4$ for powers of $i$.
• Treating the imaginary part as $bi$ instead of $b$.
• Dropping the $i^2=-1$ sign change.

Practice Problems