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.

Algebra with complex numbers is the same as real algebra, except you keep $i^2 = -1$ and collect real and imaginary parts.

Conjugates turn complex expressions into real numbers and are essential for solving equations and simplifying fractions.

Key Ideas

• $z = a + bi$ with $i^2 = -1$.
• Powers of $i$ repeat every $4$.
• Real part is $a$, imaginary part is $b$.
• $(a+bi) \pm (c+di) = (a\pm c) + (b\pm d)i$.
• $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$.
• Solve equations by isolating $z$ and rationalizing with conjugates.
• If $z = a+bi$, then $\overline{z} = a-bi$.
• $z\overline{z} = a^2 + b^2 = |z|^2$.
• Use conjugates to rationalize denominators.

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. Especially when using FOIL.

Convert Between Forms

Switch between algebraic and geometric interpretations when helpful.

Solve Linear Equations

Isolate $z$ and use conjugates to rationalize denominators.

Equate Parts

If $z=w$, then real and imaginary parts must match separately.

Rationalize Denominators

Multiply by the conjugate to remove $i$ from denominators.

Convert to Real Equations

Use $z+\overline{z}=2\Re(z)$ and $z\overline{z}=|z|^2$ to switch to real variables.

Use Modulus Identities

If $|z|$ is fixed, relate $a$ and $b$ via $a^2+b^2$.

Worked Examples

Example 1

Compute $i^{2023}$.

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

Example 2

If $z + 6i = iz$, find $z$.

Rearrange: $z - iz = -6i$, so $z(1-i) = -6i$. Then $z = \frac{-6i}{1-i} = \frac{-6i(1+i)}{(1-i)(1+i)} = \frac{6-6i}{2} = 3-3i$.

Example 3

Let $z + \overline{z} = 6$ and $z\overline{z} = 13$. Find $z$.

Write $z = a+bi$. Then $2a=6$ so $a=3$. Also $a^2+b^2=13$ so $b^2=4$ and $b=\pm 2$. Thus $z = 3 \pm 2i$.

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 $i^{58}=-1$.

Example 4: Multiply

Compute $(1+2i)(3-5i)$.

$3-5i+6i-10i^2 = 13+i$.

Example 5: Solve for $z$

Solve $(2-i)z = 5+7i$.

$z = \frac{5+7i}{2-i} = \frac{(5+7i)(2+i)}{5} = \frac{3+19i}{5}$.

Example 6: Match Real/Imag

If $(a+bi)(1-i)=4+2i$, find $a$ and $b$.

Expand: $(a+b) + (b-a)i = 4+2i$, so $a+b=4$ and $b-a=2$. Thus $a=1$, $b=3$.

Example 7: Rationalize

Simplify $\frac{3+2i}{1-4i}$.

Multiply by $1+4i$: $\frac{(3+2i)(1+4i)}{1+16} = \frac{ -5+14i}{17}$.

Example 8: Modulus

If $|z|=5$ and $\Re(z)=3$, find $\Im(z)$.

$a^2+b^2=25$ gives $b=\pm 4$.

Example 9: Real Product

If $z=2-3i$, compute $z\overline{z}$.

Strategy Checklist

• Reduce powers of $i$ modulo $4$.
• Keep real and imaginary parts separate and equate real and imaginary parts when solving equations.
• Check arithmetic with $i^2=-1$. Especially when using FOIL.
• Rationalize denominators, use the conjugate to eliminate $i$ in denominators.
• Replace $z$ with $a+bi$ to solve real equations.
• Use $|z|^2$ to relate real and imaginary parts.

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.
• Forgetting to rationalize the denominator.
• Mixing real and imaginary parts after expansion.
• Mixing up $z+\overline{z}$ with $2|z|$.
• Forgetting that $\overline{z}$ changes the sign of the imaginary part only.
• Dropping the denominator after multiplying by the conjugate.

Practice Problems