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