# Imaginary unit

square root of negative one, used to define extra-dimensional complex numbers

In math, imaginary unit, or ${\displaystyle i}$, 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 ${\displaystyle i={\sqrt {-1}}}$ (i.e., the principal root of ${\displaystyle -1}$), where ${\displaystyle i}$ satisfies the property ${\displaystyle i\times i=i^{2}=-1}$.[1][2]

The reason why ${\displaystyle i}$ 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 ${\displaystyle -1}$). Though the problem is solvable, the square root of ${\displaystyle -1}$ 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 ${\displaystyle i}$ , but that is not needed. The square roots of ${\displaystyle i}$  can be written as: ${\displaystyle \pm {\sqrt {i}}=\pm {\frac {\sqrt {2}}{2}}(1+i)}$ , a result which can be shown as follows:

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

## 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 ${\displaystyle n}$  is any integer):

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

## References

1. "Compendium of Mathematical Symbols". Math Vault. 2020-03-01. Retrieved 2020-08-10.
2. Weisstein, Eric W. "Imaginary Unit". mathworld.wolfram.com. Retrieved 2020-08-10.