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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
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clocks
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square roots
range
planes
balancing
routes
surds
menace
mean
neighbours
powers
number
crosswords
lines
hexagons
sport
arrows
binary
spheres
palindromes
people maths
angles
dates
cryptic clues
multiplication
probability
determinants
squares
volume
scales
tiling
factorials
chess
multiplaction squares
decahedra
albgebra
percentages
folding tube maps
quadratics
sets
taxicab geometry
means
star numbers
prime numbers
logic
axes
matrices
sums
graphs
money
numbers
cube numbers
colouring
cryptic crossnumbers
pascal's triangle
consecutive integers
algebra
regular shapes
prime factors
division
probabilty
combinatorics
perimeter
geometric mean
trigonometry
chalkdust crossnumber
differentiation
elections
consecutive numbers
rugby
complex numbers
unit fractions
cubics
dominos
cards
addition
sequences
tangents
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integers
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