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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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division
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percentages
cards
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probabilty
digital clocks
dodecagons
planes
mean
powers
multiplication
cryptic clues
coins
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averages
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remainders
integration
number
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sum to infinity
wordplay
chocolate
surds
multiples
palindromes
polynomials
lists
range
unit fractions
colouring
perfect numbers
routes
algebra
chalkdust crossnumber
bases
geometric means
crosswords
square roots
chess
taxicab geometry
products
graphs
logic
regular shapes
time
triangle numbers
sums
symmetry
sport
cubics
sets
integers
albgebra
ave
determinants
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gerrymandering
partitions
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angles
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median
doubling
addition
folding tube maps
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