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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
If you enjoyed this puzzle, check out
Sunday Afternoon Maths IV
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complex numbers
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digital products
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addition
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the only crossnumber
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shape
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christmas
symmetry
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dice
albgebra
volume
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trigonometry
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pascal's triangle
integers
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clocks
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perfect numbers
differentiation
means
binary
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remainders
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bases
perimeter
indices
rectangles
factors
number
planes
regular shapes
numbers
crossnumbers
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algebra
probability
parabolas
3d shapes
quadrilaterals
mean
elections
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cryptic crossnumbers
digits
wordplay
chocolate
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