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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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sequences
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combinatorics
neighbours
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floors
arrows
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albgebra
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integration
parabolas
tournaments
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hexagons
powers
coordinates
triangles
dice
geometric means
consecutive integers
polynomials
angles
means
probability
surds
chocolate
even numbers
functions
indices
cube numbers
christmas
cryptic crossnumbers
coins
range
tiling
unit fractions
percentages
digital clocks
proportion
folding tube maps
balancing
numbers
chess
gerrymandering
products
multiples
taxicab geometry
dates
dodecagons
graphs
time
quadrilaterals
pentagons
rugby
multiplication
probabilty
sum to infinity
squares
remainders
rectangles
2d shapes
palindromes
circles
perimeter
colouring
scales
differentiation
cryptic clues
odd numbers
ellipses
determinants
star numbers
spheres
3d shapes
expansions
factorials
triangle numbers
algebra
axes
number
complex numbers
money
doubling
speed
numbers grids
sets
square numbers
geometric mean
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books
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mean
decahedra
chalkdust crossnumber
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