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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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consecutive integers
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chalkdust crossnumber
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dominos
decahedra
logic
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volume
tiling
cryptic crossnumbers
numbers
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digits
doubling
wordplay
coins
indices
graphs
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multiplaction squares
spheres
tournaments
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2d shapes
routes
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shape
lists
star numbers
sum to infinity
albgebra
even numbers
sequences
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unit fractions
geometry
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prime factors
percentages
cryptic clues
rectangles
christmas
square grids
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arrows
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division
ave
binary
dice
ellipses
digital products
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