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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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a random puzzle
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Most recent collections
Advent calendar 2023
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List of all puzzles
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routes
surds
digits
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taxicab geometry
sport
sums
cryptic clues
unit fractions
odd numbers
multiplication
trigonometry
2d shapes
partitions
pascal's triangle
probabilty
square roots
people maths
money
gerrymandering
triangles
logic
lines
elections
means
dominos
digital products
perimeter
numbers
bases
square numbers
integration
decahedra
expansions
wordplay
geometry
symmetry
quadrilaterals
star numbers
parabolas
cube numbers
matrices
squares
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chocolate
shape
advent
cryptic crossnumbers
coordinates
speed
even numbers
cards
multiples
arrows
clocks
geometric means
number
sequences
coins
planes
menace
doubling
algebra
rectangles
irreducible numbers
scales
rugby
percentages
crossnumbers
addition
angles
grids
functions
median
prime numbers
factors
sum to infinity
differentiation
ave
the only crossnumber
axes
perfect numbers
tournaments
remainders
sets
chess
dates
triangle numbers
books
dodecagons
hexagons
range
quadratics
floors
crosswords
tangents
complex numbers
consecutive integers
spheres
circles
area
chalkdust crossnumber
graphs
products
probability
games
folding tube maps
geometric mean
proportion
crossnumber
balancing
dice
binary
averages
combinatorics
ellipses
polygons
fractions
volume
consecutive numbers
indices
time
mean
christmas
albgebra
cubics
3d shapes
regular shapes
digital clocks
integers
tiling
pentagons
determinants
shapes
division
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factorials
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