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