# Imaginary unit

In math, imaginary units, or $i$ , are numbers that can be represented by equations but refer to values that could not physically exist in real life. The mathematical definition of an imaginary unit is $i={\sqrt {-1}}$ , which has the property $i\times i=i^{2}=-1$ .

The reason $i$ 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 not be represented by a physical quantity of any objects in real life.

## Square root of i

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

 $\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:

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