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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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triangles
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christmas
geometric means
remainders
odd numbers
taxicab geometry
perfect numbers
neighbours
averages
ellipses
cubics
books
addition
sport
products
squares
star numbers
symmetry
regular shapes
consecutive integers
quadratics
fractions
decahedra
ave
multiplication
digital clocks
number
logic
numbers
hexagons
chess
dominos
algebra
tangents
matrices
lines
probability
dodecagons
numbers grids
dates
chocolate
grids
balancing
integers
indices
sums
folding tube maps
square roots
integration
irreducible numbers
coins
unit fractions
perimeter
partitions
calculus
polygons
surds
cryptic clues
multiples
polynomials
floors
digits
advent
games
expansions
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geometric mean
quadrilaterals
pascal's triangle
money
angles
planes
tournaments
division
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palindromes
menace
square grids
proportion
tiling
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gerrymandering
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