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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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prime numbers
irreducible numbers
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people maths
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ave
crosswords
logic
time
planes
symmetry
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determinants
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regular shapes
cube numbers
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2d shapes
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multiplication
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consecutive integers
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lines
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division
palindromes
area
circles
digital products
polygons
christmas
pascal's triangle
dominos
the only crossnumber
consecutive numbers
probability
mean
coordinates
chess
rugby
binary
scales
trigonometry
menace
sum to infinity
colouring
axes
sums
angles
surds
differentiation
routes
complex numbers
pentagons
folding tube maps
sets
advent
perimeter
remainders
indices
spheres
number
unit fractions
cubics
tiling
chalkdust crossnumber
combinatorics
dice
integration
partitions
tournaments
crossnumbers
quadratics
ellipses
dodecagons
books
decahedra
expansions
even numbers
polynomials
3d shapes
dates
percentages
addition
coins
elections
tangents
geometry
crossnumber
sequences
median
parabolas
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gerrymandering
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