The Complex Number System

The Imaginary Unit

$i = \sqrt{-1} \quad \text{so} \quad i^2 = -1$

Complex Numbers

A complex number has the form: $a + bi$ where $a$ is the real part and $b$ is the imaginary part.

Examples: $3 + 2i$, $-1 - 4i$, $5$ (real), $7i$ (pure imaginary)

Operations with Complex Numbers

Addition and Subtraction

Combine like terms: $(3 + 2i) + (1 - 5i) = 4 - 3i$ $(6 - i) - (2 + 3i) = 4 - 4i$

Multiplication (FOIL)

$(2 + 3i)(4 - i) = 8 - 2i + 12i - 3i^2 = 8 + 10i + 3 = 11 + 10i$

Powers of $i$

$i^1 = i, \quad i^2 = -1, \quad i^3 = -i, \quad i^4 = 1$

The pattern repeats every 4 powers. For $i^n$, find $n \mod 4$.

Complex Conjugates

The conjugate of $a + bi$ is $a - bi$.

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

Division

Multiply by the conjugate of the denominator: $\frac{3 + 2i}{1 - i} = \frac{(3+2i)(1+i)}{(1-i)(1+i)} = \frac{3+3i+2i+2i^2}{1+1} = \frac{1+5i}{2} = \frac{1}{2} + \frac{5}{2}i$

Solving Equations with Complex Solutions

$x^2 + 4 = 0 \implies x^2 = -4 \implies x = \pm 2i$

$x^2 - 6x + 13 = 0$ $x = \frac{6 \pm \sqrt{36-52}}{2} = \frac{6 \pm \sqrt{-16}}{2} = \frac{6 \pm 4i}{2} = 3 \pm 2i$

Key insight: Complex solutions to polynomials with real coefficients always come in conjugate pairs.