mscroggs
.co.uk
mscroggs
.co.uk
blog
puzzles
academic
talks
contact
subscribe
↓ ☰ ↓
subscribe
Loading image...
Puzzles
i to the power of i
If \(i=\sqrt{-1}\), what is the value of \(i^i\)?
Show answer & extension
Hide answer & extension
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
,
puzzles about
numbers
, or
a random puzzle
.
Archive
Show me a random puzzle
Most recent collections
Advent calendar 2025
Advent calendar 2024
Advent calendar 2023
Advent calendar 2022
List of all puzzles
Tags
rectangles
dodecagons
indices
polynomials
square numbers
median
range
quadrilaterals
complex numbers
division
colouring
geometric mean
digital clocks
chocolate
money
geometry
3d shapes
ellipses
dominos
multiplication
perimeter
logic
multiples
coins
means
prime numbers
neighbours
speed
parabolas
people maths
factors
circles
geometric means
grids
dice
cryptic crossnumbers
lines
bases
tournaments
games
palindromes
doubling
spheres
integration
sport
square grids
chess
balancing
floors
probability
numbers
square roots
the only crossnumber
folding tube maps
medians
surds
cards
graphs
sums
tangents
clocks
regular shapes
dates
time
irreducible numbers
cryptic clues
matrices
xor
consecutive numbers
calculus
quadratics
prime factors
combinatorics
mean
consecutive integers
area
odd numbers
ave
even numbers
factorials
tiling
pentagons
sets
digital products
powers
trigonometry
advent
volume
multiplaction squares
chalkdust crossnumber
functions
wordplay
sum to infinity
christmas
polygons
unit fractions
albgebra
cube numbers
coordinates
2d shapes
axes
fractions
integers
binary
digits
planes
addition
arrows
crossnumbers
perfect numbers
proportion
squares
probabilty
menace
books
decahedra
algebra
hexagons
lists
expansions
determinants
shape
sequences
crosswords
remainders
percentages
angles
number
triangles
products
averages
rugby
cubics
elections
shapes
pascal's triangle
taxicab geometry
scales
routes
triangle numbers
differentiation
partitions
gerrymandering
symmetry
star numbers
Archive
Show me a random puzzle
▼ show ▼
▲ hide ▲
Most recent collections
Advent calendar 2025
Advent calendar 2024
Advent calendar 2023
Advent calendar 2022
List of all puzzles
Tags
range
menace
decahedra
circles
polygons
chess
wordplay
triangle numbers
dominos
odd numbers
functions
numbers
prime factors
probabilty
angles
factorials
area
taxicab geometry
perimeter
cubics
remainders
cards
binary
probability
albgebra
tournaments
number
chalkdust crossnumber
determinants
coordinates
gerrymandering
digital products
arrows
shapes
balancing
axes
dodecagons
pascal's triangle
digital clocks
consecutive numbers
dice
ave
proportion
coins
the only crossnumber
geometric mean
polynomials
percentages
speed
square numbers
bases
median
christmas
chocolate
consecutive integers
crosswords
neighbours
volume
means
averages
triangles
perfect numbers
books
elections
lists
2d shapes
games
surds
sum to infinity
algebra
unit fractions
regular shapes
multiplication
geometry
fractions
ellipses
xor
digits
3d shapes
multiplaction squares
grids
complex numbers
money
sets
scales
star numbers
rectangles
integers
floors
medians
multiples
graphs
hexagons
integration
combinatorics
quadrilaterals
differentiation
time
palindromes
colouring
square roots
powers
tangents
expansions
sequences
people maths
calculus
matrices
tiling
rugby
pentagons
advent
logic
indices
crossnumbers
routes
parabolas
folding tube maps
trigonometry
geometric means
lines
irreducible numbers
cryptic clues
clocks
planes
products
division
symmetry
square grids
cryptic crossnumbers
cube numbers
sport
even numbers
partitions
quadratics
prime numbers
shape
squares
sums
mean
addition
dates
factors
doubling
spheres
© Matthew Scroggs 2012–2026