Complex Numbers

The imaginary unit

Powers of i

i = \sqrt{- 1}, i^{2} = - 1, i^{3} = - i, i^{4} = 1

Any power of i

i^{4 k + r} = i^{r}

Square root of a negative

\sqrt{- a} = i \sqrt{a}

a > 0

Add or subtract

(a + b i) \pm (c + d i) = (a \pm c) + (b \pm d) i

Multiply

(a + b i) (c + d i) = (a c - b d) + (a d + b c) i

Conjugate

z = a + b i \Rightarrow \bar{z} = a - b i

Number times its conjugate

(a + b i) (a - b i) = a^{2} + b^{2}

Divide

\frac{a + b i}{c + d i} = \frac{(a c + b d) + (b c - a d) i}{c^{2} + d^{2}}

Absolute value (modulus)

| a + b i | = \sqrt{a^{2} + b^{2}}

Sum of squares

a^{2} + b^{2} = (a + b i) (a - b i)

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