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