The imaginary constant

Imaginary and complex numbers are based around a single letter, which effectively acts as an unknown constant. This is i.

The value of i

i is equal to the square root of 1. This can be written in either of the following forms:

i=\sqrt{-1}

i^2=-1

Simplifying imaginary numbers

The second form has some interesting use-cases. Because we know that i^2=-1, we can rewrite any power of i as a simple real number, or as just a coefficient of i.

Example: Simplify i^3

• i^3=i^2\times i
• i^3=-1\times i
• i^3=-i

Example: Simplify i^4

• i^4=i^2\times i^2
• i^4=-1\times-1
• i^4=1

Example: Simplify -7i^2

• -7i^2=-7\times i^2
• -7i^2=-7\times-1
• -7i^2=7