Imaginary unit Facts for Kids

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.

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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 \$