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:

$i^{4n} = 1$

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

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

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

Related pages

• Complex number
• Real number
• Euler's Identity