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Puzzles
i to the power of i
If \(i=\sqrt{-1}\), what is the value of \(i^i\)?
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By
Euler's formula
, \(i=e^{i\frac{\pi}{2}}\). This means that:
$$i^i=(e^{i\frac{\pi}{2}})^i\\ =e^{i^2\frac{\pi}{2}}\\ =e^{-\frac{\pi}{2}}$$
It is notable that this is a real number
Extension
What is \(-i^{-i}\)?
Tags:
numbers
,
complex numbers
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taxicab geometry
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albgebra
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circles
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factorials
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ave
clocks
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scales
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binary
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addition
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averages
dominos
colouring
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lines
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perimeter
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pentagons
sport
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multiples
dice
differentiation
algebra
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