Introduction to Imaginary Numbers

This is not a math tutorial. See Why am I writing about math?

Terminology (fuzzy)

The imaginary unit is $i$, where $i^2 = -1$.

An imaginary number is a number in the form $bi$, where $b$ is a real number ($\mathbb{R}$) and $b \neq 0$.

A complex number is a number in the form $a + bi$, where both $a$ and $b$ are real numbers.

The imaginary unit

Technically $i$ is both the imaginary unit and an imaginary number. $i$ can be expressed in the form $bi$, where $b = 1$:

The powers of i cycle

$i$ raised to the power of whole numbers $({0, 1, 2, 3...})$ has a cycle of $4$:

• $i^0 = 1$
• $i^1 = i$
• $i^2 = -1$ (the definition of $i$)
• $i^3 = -i$ ($i^3 = i^2 \times i = -1 \times i = -i$)
• $i^4 = 1$ (back to the beginning of the cycle)

An imaginary number to the power of zero is a real number

Following the rule that any (non-zero) number to the power of 0 equals 1:

An imaginary number is also a complex number

Related to:

• notes / Introduction to complex numbers
• notes / Real and complex numbers in space

A complex number is a number in the form $a + bi$, where both $a$ and $b$ are real numbers. It follows that:

bi plotted on the Cartesian plane

It follows from an imaginary number also being a complex number that $bi$ can be plotted on the Cartesian plane at the point $(0, b)$.