Complex and Imaginary Numbers

Powers of $i$

$i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Pattern repeats every 4.

$i^{23} = i^{4(5)+3} = i^3 = -i$.

Complex Numbers

$a + bi$ where $a$ = real part, $b$ = imaginary part.

Operations

Addition: $(3 + 2i) + (1 - 5i) = 4 - 3i$

Multiplication: $(2 + 3i)(1 - i) = 2 - 2i + 3i - 3i^2 = 2 + i + 3 = 5 + i$

Conjugate: Conjugate of $a + bi$ is $a - bi$.

$(a + bi)(a - bi) = a^2 + b^2$ (always real).

Division: Multiply by conjugate of denominator. $\frac{3 + i}{1 - 2i} \times \frac{1 + 2i}{1 + 2i} = \frac{3 + 6i + i + 2i^2}{1 + 4} = \frac{1 + 7i}{5}$