Infinity
Hilbert’s Hotel: You’re Missing the Best Bit!
Infinity, enumerating the rationals and elegant solutions to innocuous problems.
Analysis
Arithmetic
The Fundamental Theorem of Arithmetic
Over 2,300 years ago Euclid proved the Fundamental Theorem of Arithmetic. Now it is our turn.
Analysis
Cantor’s Diagonal Argument
“Diagonalization seems to show that there is an inexhaustibility phenomenon for definability similar to that for provability” — Franzén…
Analysis
Why is 1 + 2 + 4 + 8 + … = -1?
On calculating infinite divergent series sums
Calculus
The Fundamental Theorem of Calculus
The beginner’s guide to proving the Fundamental Theorem of Calculus, with both a visual approach for those less keen on algebra, and an…
Analysis
Euler’s Log-Sine Integral
Let’s discover the ingenious, yet simple, solution to Euler’s Log-Sine integral
Analysis
Pythagorean Triples and Other Beasts
I never thought that I would write math stories in English, but I also never thought that I would live in the UK either.
Calculus
Solving the Dirichlet Integral
The Dirichlet integral was first solved by the brilliant Leonhard Euler, it then achieved fame with Dirichlet’s work on Fourier series…
Geometry
Why Trisecting the Angle is Impossible
An accidental introduction to field theory.
Analysis
Cantor Pairing Function
Represent an ordered pair of integers with a single integer.
Analysis
An Elementary Proof of the Fundamental Theorem of Algebra
Using only elementary methods we will prove the Fundamental Theorem of Algebra. Along the way, we’ll befriend some key ideas from…
Calculus
Solving the Gaussian Integral
Lord Kelvin wrote of this integral: “A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville…