# Imaginary unit

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In math, imaginary units, or ${\displaystyle i}$, 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 ${\displaystyle i={\sqrt {-1}}}$, which has the property ${\displaystyle i\times i=i^{2}=-1}$.

The reason ${\displaystyle i}$ was created was to answer a polynomial equation, ${\displaystyle x^{2}+1=0}$, which normally has no solution as the value of ${\displaystyle x^{2}}$ 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 ${\displaystyle i}$, but that is not needed. The square root of ${\displaystyle i}$ can be written as: ${\displaystyle {\sqrt {i}}=\pm {\frac {\sqrt {2}}{2}}(1+i)}$.
This can be shown as:

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

## Powers of i

The powers of ${\displaystyle i}$ follow a predictable pattern:

${\displaystyle i^{-3}=i}$
${\displaystyle i^{-2}=-1}$
${\displaystyle i^{-1}=-i}$
${\displaystyle i^{0}=1}$
${\displaystyle i^{1}=i}$
${\displaystyle i^{2}=-1}$
${\displaystyle i^{3}=-i}$
${\displaystyle i^{4}=1}$
${\displaystyle i^{5}=i}$
${\displaystyle i^{6}=-1}$

This can be shown with the following pattern where n is any integer:

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