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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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star numbers
integers
sequences
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spheres
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dice
taxicab geometry
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perimeter
proportion
geometry
games
rugby
balancing
unit fractions
wordplay
grids
determinants
cards
tangents
lines
even numbers
differentiation
number
bases
factors
consecutive numbers
2d shapes
chocolate
probabilty
fractions
algebra
routes
cryptic crossnumbers
partitions
geometric mean
perfect numbers
elections
triangle numbers
palindromes
consecutive integers
multiples
shapes
sets
circles
pentagons
volume
time
digital clocks
addition
christmas
geometric means
the only crossnumber
functions
people maths
books
remainders
sport
chess
binary
coins
colouring
square numbers
cube numbers
albgebra
dates
decahedra
complex numbers
sum to infinity
odd numbers
trigonometry
chalkdust crossnumber
menace
floors
pascal's triangle
numbers
percentages
triangles
3d shapes
symmetry
rectangles
means
matrices
indices
dodecagons
gerrymandering
range
tournaments
surds
median
division
multiplication
quadratics
axes
tiling
money
dominos
regular shapes
squares
arrows
advent
quadrilaterals
digital products
scales
calculus
digits
crossnumber
planes
folding tube maps
parabolas
shape
speed
area
crossnumbers
clocks
angles
logic
products
irreducible numbers
expansions
graphs
coordinates
square roots
ave
crosswords
probability
averages
integration
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hexagons
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