# Imaginary unit

In math, imaginary unit, or $i$ , is a number that can be represented by equations, but refers to a value that can only exist outside of real numbers. The mathematical definition of the imaginary unit is $i={\sqrt {-1}}$ (i.e., the principal root of $-1$ ), where $i$ satisfies the property $i\times i=i^{2}=-1$ .

The reason why $i$ was created was to answer a polynomial equation, $x^{2}+1=0$ , which normally has no solution (as the value of $x^{2}$ would have to equal $-1$ ). Though the problem is solvable, the square root of $-1$ could hardly be represented by a physical quantity of objects in real life.

## Square root of i

It is sometimes assumed that one must create another number to show the square roots of $i$ , but that is not needed. The square roots of $i$ can be written as: $\pm {\sqrt {i}}=\pm {\frac {\sqrt {2}}{2}}(1+i)$ , a result which can be shown as follows:

 $\left(\pm {\frac {\sqrt {2}}{2}}(1+i)\right)^{2}\$ $=\left(\pm {\frac {\sqrt {2}}{2}}\right)^{2}(1+i)^{2}\$ $=(\pm 1)^{2}{\frac {2}{4}}(1+i)(1+i)\$ $=1\times {\frac {1}{2}}(1+2i+i^{2})\quad \quad (i^{2}=-1)\$ $={\frac {1}{2}}(2i)\$ $=i\$ ## Powers of i

The powers of $i$ follow a predictable pattern:

$i^{-3}=i$ $i^{-2}=-1$ $i^{-1}=-i$ $i^{0}=1$ $i^{1}=i$ $i^{2}=-1$ $i^{3}=-i$ $i^{4}=1$ $i^{5}=i$ $i^{6}=-1$ This can be shown with the following pattern (where $n$ is any integer):

$i^{4n}=1$ $i^{4n+1}=i$ $i^{4n+2}=-1$ $i^{4n+3}=-i$ 