Imaginary unit

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

The reason $i$$i$ was created was to answer a polynomial equation, $x^{2}+1=0$$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$$i$, but that is not needed. The square root of $i$$i$ can be written as: ${\sqrt {i}}=\pm {\frac {\sqrt {2}}{2}}(1+i)$${\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}}(1+i)\right)^{2}\$	$=\left(\pm {\frac {\sqrt {2}}{2}}\right)^{2}(1+i)^{2}\$$=\left(\pm {\frac {\sqrt {2}}{2}}\right)^{2}(1+i)^{2}\$
	$=(\pm 1)^{2}{\frac {2}{4}}(1+i)(1+i)\$$=(\pm 1)^{2}{\frac {2}{4}}(1+i)(1+i)\$
	$=1\times {\frac {1}{2}}(1+2i+i^{2})\quad \quad (i^{2}=-1)\$$=1\times {\frac {1}{2}}(1+2i+i^{2})\quad \quad (i^{2}=-1)\$
	$={\frac {1}{2}}(2i)\$$={\frac {1}{2}}(2i)\$
	$=i\$$=i\$

Powers of i

The powers of $i$$i$ follow a predictable pattern:

$i^{-3}=i$$i^{-3}=i$

$i^{-2}=-1$$i^{-2}=-1$

$i^{-1}=-i$$i^{-1}=-i$

$i^{0}=1$$i^{0}=1$

$i^{1}=i$$i^{1}=i$

$i^{2}=-1$$i^{2}=-1$

$i^{3}=-i$$i^{3}=-i$

$i^{4}=1$$i^{4}=1$

$i^{5}=i$$i^{5}=i$

$i^{6}=-1$$i^{6}=-1$

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

$i^{4n}=1$$i^{4n}=1$

$i^{4n+1}=i$$i^{4n+1}=i$

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

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

Related pages

• Complex number
• Imaginary number
• Real number
• Euler's Identity