Imaginary Unit

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The Imaginary Unit, or i, used in mathematics to pull together the real number system and the complex number system. Its definition depends on how it is used.

The reason i was created was to answer a polynomial equation, x2 + 1 = 0, which normally has no solution.

[edit] 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 \

[edit] Powers of i

The powers of i follow a predictable pattern:


i − 3 = i
i − 2 = − 1
i − 1 = − i
i0 = 1
i1 = i
i2 = − 1
i3 = − i
i4 = 1
i5 = i
i6 = − 1

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

i4n = 1
i4n + 1 = i
i4n + 2 = − 1
i4n + 3 = − i

[edit] See Also