Algebraic Operations

A complex number $z = a + bi$ has a real part $a$ and an imaginary part $b$, where $i^2 = -1$. Addition, subtraction, and multiplication all follow ordinary algebra rules, treating $i$ like any other variable -- except that whenever $i^2$ shows up, it gets replaced with $-1$.

Addition and subtraction

Combine the real parts together and the imaginary parts together:

$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$

Example: $(10 - 7i) + (-1 - 5i) = (10 - 1) + (-7 - 5)i = 9 - 12i$

Multiplication

Expand like two binomials, then simplify using $i^2 = -1$:

$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

Example: $(2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i^2 = 2 - 5i + 12 = 14 - 5i$

The questions below mix all three operations -- watch for sign errors when distributing a minus sign across a subtraction, and remember to fold $i^2$ back into the real part before reading off your final answer.