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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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squares
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addition
crossnumbers
shapes
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integration
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balancing
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chalkdust crossnumber
square roots
digital clocks
angles
parabolas
trigonometry
geometry
determinants
mean
cryptic clues
elections
combinatorics
sum to infinity
rectangles
regular shapes
digits
planes
triangles
matrices
scales
circles
ave
median
taxicab geometry
graphs
perimeter
crosswords
even numbers
area
binary
quadratics
products
dice
complex numbers
2d shapes
prime numbers
algebra
division
sport
quadrilaterals
arrows
surds
triangle numbers
coordinates
coins
shape
indices
star numbers
cryptic crossnumbers
decahedra
geometric mean
range
ellipses
irreducible numbers
clocks
square numbers
books
menace
cards
calculus
functions
sums
means
lines
3d shapes
numbers
spheres
digital products
sets
the only crossnumber
gerrymandering
consecutive integers
pascal's triangle
chess
dominos
averages
advent
multiples
folding tube maps
albgebra
wordplay
time
partitions
volume
routes
expansions
floors
bases
factors
rugby
number
tiling
remainders
logic
palindromes
integers
hexagons
probabilty
sequences
fractions
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multiplication
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