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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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addition
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range
routes
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square grids
division
trigonometry
axes
differentiation
rugby
cube numbers
cryptic crossnumbers
triangles
star numbers
regular shapes
unit fractions
prime factors
parabolas
elections
colouring
median
functions
people maths
products
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shapes
dates
lists
quadratics
hexagons
3d shapes
chocolate
mean
perimeter
speed
angles
perfect numbers
number
time
numbers
multiplication
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clocks
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scales
advent
consecutive numbers
books
tangents
square numbers
sum to infinity
digital clocks
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neighbours
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ave
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factors
powers
folding tube maps
doubling
tournaments
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means
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ellipses
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pascal's triangle
multiplaction squares
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