Identities

The usual algebraic identities still work on complex numbers —

$i$

just behaves like any other symbol, as long as you fold

$i^2$

back into

$-1$

wherever it shows up.

$(a+bi)^2 = a^2 - b^2 + 2abi$

$(a+bi)(a-bi) = a^2 + b^2$

That second one is the difference-of-squares identity — except since

$(bi)^2 = -b^2$

, it turns into a sum of squares instead.

Example:

$(2+3i)^2 = 4 - 9 + 12i = -5 + 12i$

.