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):

$i^{4n}=1$

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

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

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

Related pages

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