The Imaginary Unit

The imaginary unit $i$ is defined as:
$i = \sqrt{-1} \qquad i^2 = -1$

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), $7i$ (purely imaginary).

Complex numbers plotted on the complex plane (real axis horizontal, imaginary axis vertical)

Powers of $i$ cycle every 4:
- $i^1 = i$
- $i^2 = -1$
- $i^3 = -i$
- $i^4 = 1$
- $i^5 = i$ (cycle repeats)

Powers of i cycle every 4: i, -1, -i, 1

To find $i^n$ : divide $n$ by 4 and use the remainder.

Example 1

Simplify $i^{27}$

$27 \div 4 = 6$ remainder $3$

$i^{27} = i^3 = -i$

-i

Theory

Operations with Complex Numbers

Addition/Subtraction: Combine real parts and imaginary parts separately.
$(a + bi) + (c + di) = (a+c) + (b+d)i$

Multiplication: Use FOIL, then replace $i^2$ with $-1$ .
$(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$

Example 1

Multiply $(3 + 2i)(1 - 4i)$

FOIL: $3(1) + 3(-4i) + 2i(1) + 2i(-4i)$

$= 3 - 12i + 2i - 8i^2$

$= 3 - 10i - 8(-1) = 3 - 10i + 8 = 11 - 10i$

11 - 10i

1 more section including examples, practice problems, and step-by-step solutions.

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