 # Imaginary unit facts for kids

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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 Imaginary unit Facts for Kids. Kiddle Encyclopedia.