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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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gerrymandering
odd numbers
coordinates
determinants
consecutive integers
dominos
chess
speed
expansions
irreducible numbers
squares
people maths
sport
numbers
quadratics
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powers
matrices
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advent
floors
taxicab geometry
spheres
unit fractions
area
bases
doubling
triangles
hexagons
coins
differentiation
dates
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cards
binary
cryptic clues
calculus
elections
grids
time
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decahedra
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folding tube maps
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perimeter
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pentagons
colouring
ave
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