numberphile

Numberphile's James Grime explains Cantor's result that some infinities are bigger than others, including why the reals outnumber the integers. It is a short, accessible introduction to countable and uncountable infinity.

Numberphile with Simon Pampena on how humans misunderstand randomness, including coin-flip sequences, run lengths and the gambler's fallacy.

Numberphile's James Grime introduces quaternions, a type of number beyond the complex numbers, building up from real and complex numbers and explaining why they are useful.

Numberphile's Tony Padilla and Matt Parker discuss Graham's number, once the largest number used in a mathematical proof, and how enormous it is.

Numberphile with Dr Holly Krieger shows how the digits of pi surprisingly emerge from the Mandelbrot set. A short, accessible math explainer.

Numberphile with Tom Crawford of Oxford explains the Reynolds number, the second of three videos on Navier-Stokes and fluid flow, with demonstrations of laminar versus turbulent behavior.

A Numberphile video with Dr James Grime on Sloane's Gap, the observation that integers in the OEIS are not evenly represented and some numbers are underrepresented. It explains the clusters of 'interesting' numbers.

Matt Parker on Numberphile explains the strange properties of spheres in higher dimensions, such as how they poke out of their bounding boxes. It uses Pythagoras to build intuition for 4D and beyond.

Numberphile with Daniel Erman on the Josephus problem, where people in a circle are eliminated in turn. It shows how the survivor's position follows a neat pattern tied to powers of two and binary.

Cliff Stoll shows and explains his two Friden EC-132 electronic calculators for Numberphile. The video covers how these early machines worked, including their unusual acoustic delay line memory.

Numberphile's James Grime discusses highly composite numbers, nicknamed anti-primes, which have more factors than any smaller number, using 5040 as the central example. It touches on Plato and Ramanujan.

Numberphile with Tadashi Tokieda on the Trinity Hall Prime, a prime number whose digits form a picture of the college's coat of arms, left as a gift by a departing mathematician.

Numberphile explains the long and short scales for naming large numbers, covering millions, billions, trillions, milliards and the Greek myriad, with James Grime and Tony Padilla.

Numberphile's James Grime makes the case for base 12 (dozenal) over base 10, showing how twelve's many divisors make arithmetic and everyday fractions easier.

Numberphile's Ben Sparks works through the physics of a car crash, showing how kinetic energy grows with the square of speed and why stopping distance climbs so quickly as velocity increases.

Numberphile with Tim Browning on the sum-of-three-cubes problem and why 33 was the lowest unsolved case. The description notes a later solution for 33.

Matt Parker talks Numberphile through the different kinds of numbers: algebraic, transcendental, computable, normal, and beyond. It shows how few numbers we can actually name or describe.

Numberphile interviews cryptographer Ron Rivest about end-to-end verifiable voting. He explains how homomorphic encryption and receipts let a voter confirm their ballot was counted without being able to prove how they voted to anyone else.

Numberphile's Ben Sparks shows why the golden ratio is the most irrational number, using seeds, continued fractions and rational approximations. He also explains why plants use it for spacing leaves and seeds.

Numberphile's Dr James Grime explains the flaw in the Enigma machine, namely that a letter could never be encrypted to itself, and how that helped the Allies break the code. It touches on the Bombe and Alan Turing.

Numberphile talks with Grant Sanderson of 3Blue1Brown about a darts puzzle with a shrinking bullseye. Solving it takes the volumes of high-dimensional spheres, and the answer turns up Euler's number.

Numberphile interviews Grant Sanderson of 3blue1brown about Bertrand's paradox, where different reasonable ways of picking a random chord of a circle give different probabilities.

Mathologer examines the infamous Numberphile claim that 1+2+3+... equals -1/12, explaining what is wrong with the calculation and how it connects to real math like the zeta function and analytic continuation.

A Numberphile video in which physicists Tony Padilla and Ed Copeland show a seemingly astounding result that 1+2+3+4+... equals -1/12. The argument is informal and has drawn criticism for glossing over what the result actually means.

Numberphile visits YouTube to learn why view counts on new videos used to freeze at 301. It explains that YouTube holds counts while it verifies views aren't from bots or fakes.

3Blue1Brown's animated visual introduction to the Fourier Transform, showing how a signal breaks into frequency components.

3Blue1Brown's first chapter on deep learning explains what neurons and layers are and the math underlying a basic neural network, using handwritten digit recognition with visual animations.

Freya Holmer explains how Bezier curves work, from the helper-line construction to their uses in animation, text rendering and shapes. It uses visuals from her Unity Shapes plugin and was a SoME1 contest winner.

Freya Holmer's 74-minute video essay on splines, covering Bezier curves, B-splines, derivatives and the different levels of continuity (C0, C1, C2), built with custom animation tools. She notes she did not cover non-uniform splines.

3Blue1Brown explains the math and mechanics behind Bitcoin visually: ledgers, digital signatures, proof of work and the blockchain, building up a decentralized payment system from first principles.

An animated introduction to geometric algebra, presented as a language that can describe most of physics. It was made with manim and covers multivectors and how they generalize familiar tools like cross products and quaternions.

3Blue1Brown walks through a very hard Putnam competition problem and its elegant solution, emphasizing the visual intuition and the thought process behind solving it.

A 10-minute 3Blue1Brown chapter explaining the determinant as the factor by which a linear transformation scales area or volume. It builds the idea geometrically with animations before any formulas.

Veritasium's explainer on Godel's incompleteness theorems and the limits of mathematical proof, tracing set theory, Hilbert's program and the road to Turing and modern computing.

3Blue1Brown explains fractal dimension and shows why many fractals are not strictly self-similar. Self-similarity turns out to be a special case of the broader idea of fractional dimension.

Vsauce's roughly 24 minute explainer on the Banach-Tarski paradox, building from infinity and set theory to the decomposition of a sphere into pieces that form two spheres.

3Blue1Brown visualizes high-dimensional spheres to explain a surprising puzzle about how a sphere fits in a box as dimensions grow. It builds geometric intuition for counterintuitive properties of high-dimensional space.

3Blue1Brown's chapter on eigenvectors and eigenvalues in the Essence of Linear Algebra series. It builds a visual intuition for what they mean, why an eigenbasis is useful, and ends with a puzzle.

3Blue1Brown shows how the uncertainty principle arises from Fourier transforms. A wave packet cannot be narrow in both position and frequency, so the principle is a property of waves rather than of measurement.

3Blue1Brown builds visual intuition for quaternions, showing how to think about this four-dimensional number system using stereographic projection and rotations in 3D space. It is part one of a two-part series with an interactive companion site.