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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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sport
polygons
cryptic clues
complex numbers
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probabilty
irreducible numbers
medians
pascal's triangle
spheres
dates
digits
scales
prime numbers
speed
star numbers
the only crossnumber
cards
surds
coins
matrices
routes
range
addition
even numbers
geometry
dice
factorials
digital clocks
tiling
taxicab geometry
cryptic crossnumbers
axes
square roots
median
coordinates
christmas
parabolas
digital products
consecutive integers
area
integration
planes
pentagons
balancing
indices
elections
quadratics
2d shapes
hexagons
expansions
differentiation
rectangles
volume
games
triangle numbers
time
albgebra
money
tournaments
proportion
3d shapes
doubling
books
partitions
shapes
averages
factors
grids
shape
remainders
perfect numbers
geometric means
bases
symmetry
graphs
crosswords
advent
triangles
perimeter
odd numbers
unit fractions
quadrilaterals
sets
floors
crossnumbers
colouring
tangents
circles
dominos
regular shapes
determinants
squares
binary
square grids
calculus
angles
menace
wordplay
clocks
sums
sequences
trigonometry
dodecagons
chocolate
probability
integers
neighbours
geometric mean
cube numbers
ellipses
rugby
fractions
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cubics
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arrows
numbers
folding tube maps
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chalkdust crossnumber
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