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Imaginary unit




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In math, imaginary units, or [math]i[/math], 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 [math]i = \sqrt{-1}[/math], which has the property [math]i \times i = i^2 = -1[/math].

The reason [math]i[/math] was created was to answer a polynomial equation, [math]x^2 + 1 = 0[/math], 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 [math]i[/math], but that is not needed. The square root of [math]i[/math] can be written as: [math] \sqrt{i} = \pm \frac{\sqrt{2}}{2} (1 + i) [/math].
This can be shown as:

[math]\left( \pm \frac{\sqrt{2}}{2} (1 + i) \right)^2 \ [/math] [math]= \left( \pm \frac{\sqrt{2}}{2} \right)^2 (1 + i)^2 \ [/math]
[math]= (\pm 1)^2 \frac{2}{4} (1 + i)(1 + i) \ [/math]
[math]= 1 \times \frac{1}{2} (1 + 2i + i^2) \quad \quad (i^2 = -1) \ [/math]
[math]= \frac{1}{2} (2i) \ [/math]
[math]= i \ [/math]

Powers of i

The powers of [math]i[/math] follow a predictable pattern:

[math]i^{-3} = i[/math]
[math]i^{-2} = -1[/math]
[math]i^{-1} = -i[/math]
[math]i^0 = 1[/math]
[math]i^1 = i[/math]
[math]i^2 = -1[/math]
[math]i^3 = -i[/math]
[math]i^4 = 1[/math]
[math]i^5 = i[/math]
[math]i^6 = -1[/math]

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

[math]i^{4n} = 1[/math]
[math]i^{4n+1} = i[/math]
[math]i^{4n+2} = -1[/math]
[math]i^{4n+3} = -i[/math]

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