Overview

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

Key Ideas

• 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

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 Example

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: Rationalize

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

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

Example 2: Modulus

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

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

Example 3: Real Product

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

$2^2+(-3)^2=13$.

Strategy Checklist

• 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

• 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