Imaginary unit

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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[change | edit source]

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[change | edit source]

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[change | edit source]