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