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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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combinatorics
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dates
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chess
routes
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balancing
tiling
folding tube maps
xor
functions
quadratics
ave
unit fractions
proportion
dodecagons
addition
shapes
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fractions
floors
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sport
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decahedra
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integers
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products
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