numbers

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'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.

The first video in a series on extremely large numbers, introducing the topic with examples. Later parts go deep into the fast-growing hierarchy.

Tom Scott explains for Computerphile how floating point numbers work, why moving the binary point is fast, and why it handles money badly.

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.

3Blue1Brown explains why plotting primes in polar coordinates produces spiral patterns, using modular arithmetic, Dirichlet's theorem and rational approximations of pi.

3Blue1Brown explores infinite sums from convergent to divergent, with an introduction to the 2-adic metric, themed on the cycle between discovering and inventing mathematics.

Vsauce's Michael Stevens takes viewers past infinity using transfinite numbers, ordinals and cardinals, with input from mathematician Hugh Woodin. It is a fast-paced, mind-bending tour of different sizes of infinity.

A PBS Infinite Series episode explaining the mathematics of quantum computers, including how qubits store complex numbers and how quantum computation differs from classical computation. Hosted by Kelsey Houston-Edwards.

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.

3Blue1Brown shows how pi emerges from prime numbers via complex numbers and sums of squares, culminating in a formula for approximating pi.

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.

Vsauce explores math through card tricks and big numbers, including an animation of how enormous 52 factorial, the number of deck orderings, really is.

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.

A short 3Blue1Brown video explaining Euler's formula e^(i*pi) = -1 through dynamics, treating velocity as the thing that moves a point to a new position. It builds geometric intuition for why an exponent with i produces rotation.

3Blue1Brown illustrates how enormous the 256-bit search space is, showing how infeasible it is to find a hash or key by guessing and checking.

3Blue1Brown builds intuition for Euler's formula, e^(pi i) = -1, using the core ideas of group theory. It treats exponentiation geometrically as rotation and scaling instead of repeated multiplication.

Mathologer uses continued fractions to show why the golden ratio is the most irrational number, comparing it with root 2, e and pi and pointing to where this shows up in nature.

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.

Mathologer explains epicycles and complex Fourier series, inspired by an animation of Homer Simpson drawn with epicycles. It covers the shape of the Moon's orbit around the Sun and how to make your own epicycle drawings.

Roy Beck's short gumball presentation using World Bank and Census data to argue that letting people immigrate to the US cannot solve world poverty because the numbers are so large. It is made by NumbersUSA, an immigration-restriction advocacy group.

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.

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.

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.

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.

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.

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.

Mathologer examines how identities like 1+2+3+... = -1/12 arise in Ramanujan's work, starting from his 1913 letter to Hardy. The video explains the rules for summing infinite series and why such results are meaningful but not naive sums.

An 11-minute animated introduction to P vs. NP and the wider landscape of complexity classes, inspired by the Complexity Zoo wiki. It assumes no background in computer science.

3Blue1Brown visualizes the Riemann zeta function and the analytic continuation that extends it beyond where its series converges, building toward the Riemann hypothesis.

An older 3Blue1Brown video on why topology matters, built around the inscribed rectangle problem. It walks through how tori and Mobius strips appear in the solution. The channel later replaced it with an updated version.

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.

An animated 11-minute tour of the whole field of mathematics as a single map, showing how pure and applied math and their sub-fields relate to each other.

MinutePhysics, with 3Blue1Brown, explains Bell's theorem using a Venn diagram paradox and polarizing filters. It shows why no local hidden variable theory can reproduce quantum correlations.