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