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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
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geometric mean
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division
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shape
numbers
time
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determinants
functions
square grids
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cryptic crossnumbers
proportion
books
sum to infinity
volume
consecutive integers
irreducible numbers
factors
prime numbers
christmas
differentiation
colouring
wordplay
ave
expansions
square roots
balancing
2d shapes
palindromes
perfect numbers
sets
squares
cubics
star numbers
gerrymandering
tangents
partitions
area
pentagons
ellipses
percentages
algebra
circles
sequences
coordinates
dominos
clocks
polygons
perimeter
dodecagons
crossnumbers
scales
probability
cube numbers
number
symmetry
multiplication
graphs
grids
dates
triangle numbers
complex numbers
multiples
quadratics
neighbours
powers
spheres
even numbers
routes
unit fractions
integers
consecutive numbers
calculus
digital clocks
coins
albgebra
odd numbers
sport
angles
numbers grids
games
chess
sums
speed
advent
medians
dice
money
probabilty
products
addition
arrows
means
hexagons
range
crosswords
quadrilaterals
mean
cards
rugby
chocolate
lines
menace
elections
digits
regular shapes
bases
geometric means
cryptic clues
doubling
geometry
digital products
averages
indices
triangles
folding tube maps
rectangles
3d shapes
logic
median
square numbers
shapes
planes
axes
fractions
binary
taxicab geometry
pascal's triangle
decahedra
floors
tournaments
the only crossnumber
surds
matrices
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