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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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taxicab geometry
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binary
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wordplay
albgebra
sets
multiplaction squares
cryptic crossnumbers
area
averages
folding tube maps
sums
rectangles
irreducible numbers
people maths
coordinates
geometry
expansions
triangle numbers
perimeter
combinatorics
lines
2d shapes
routes
pentagons
doubling
balancing
proportion
graphs
colouring
division
chalkdust crossnumber
planes
arrows
the only crossnumber
polynomials
geometric mean
coins
factorials
grids
neighbours
square grids
sport
clocks
cube numbers
numbers
multiples
circles
xor
geometric means
symmetry
ellipses
rugby
percentages
mean
products
median
sequences
matrices
probability
ave
cryptic clues
christmas
digital products
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trigonometry
quadrilaterals
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consecutive numbers
spheres
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dodecagons
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