Complex Numbers

By the end of this lesson, you will be able to define imaginary and complex numbers, perform arithmetic (add, subtract, multiply, divide) with complex numbers, find complex conjugates, and solve quadratics with complex solutions using the quadratic formula.

The Big Idea

The equation $x^2 = -1$ has no real solution — but it does have a solution in a larger number system. By defining $i = \sqrt{-1}$, we extend the real numbers to the complex numbers: numbers of the form $a + bi$ where $a$ and $b$ are real. This extension is not just mathematical convenience — complex numbers are essential in electrical engineering, quantum mechanics, signal processing, and virtually all of advanced mathematics.

The Imaginary Unit

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

Powers of $i$ cycle with period 4:

Power	Value
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$
$i^5$	$i$ (cycle repeats)

Finding $i^n$: divide $n$ by 4; use the remainder.

Complex Number Form

A complex number is $z = a + bi$ where:

• $a$ is the real part ($\text{Re}(z)$)
• $b$ is the imaginary part ($\text{Im}(z)$)
• Pure real: $b = 0$; pure imaginary: $a = 0$

Arithmetic with Complex Numbers

Addition/Subtraction: combine real parts, combine imaginary parts.

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

Multiplication: FOIL, then replace $i^2$ with $-1$.

$(3 + 2i)(1 - 5i) = 3 - 15i + 2i - 10i^2 = 3 - 13i + 10 = 13 - 13i$

Division: multiply numerator and denominator by the complex conjugate of the denominator.

$$\frac{3 + 2i}{1 - i} = \frac{(3+2i)(1+i)}{(1-i)(1+i)} = \frac{3+3i+2i+2i^2}{1+1} = \frac{3+5i-2}{2} = \frac{1+5i}{2} = \frac{1}{2} + \frac{5}{2}i$$

Complex Conjugate

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

Key property: $(a+bi)(a-bi) = a^2 + b^2$ — always a real number.

This is why we use the conjugate to rationalize complex denominators.

Example 1 — Powers of $i$

Simplify: $i^{35}$.

$35 = 4 \times 8 + 3$, remainder 3 → $i^{35} = i^3 = -i$