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 $-7i^2$

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