Overview

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

Key Ideas

• $(a+bi) \pm (c+di) = (a\pm c) + (b\pm d)i$.
• $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$.
• Solve equations by isolating $z$ and rationalizing with conjugates.

Core Skills

Multiply Carefully

Use FOIL and replace $i^2$ with $-1$ immediately.

Solve Linear Equations

Isolate $z$ and use conjugates to rationalize denominators.

Equate Parts

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

Worked Example

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

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

More Examples

Example 1: Multiply

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

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

Example 2: 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 3: 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 $a=1$, $b=3$.

Strategy Checklist

• Use FOIL and reduce $i^2$ immediately.
• Rationalize denominators.
• Equate real and imaginary parts to solve.

Common Pitfalls

• Forgetting to rationalize the denominator.
• Dropping the $i^2 = -1$ step in multiplication.
• Mixing real and imaginary parts after expansion.

Practice Problems