complex numbers

Sabine Hossenfelder asks whether complex numbers are just a convenient tool or necessary in physics. She discusses a recent paper arguing that quantum mechanics cannot be done with real numbers alone.

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.

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.

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.

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.

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.

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 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.

WonderWhy surveys complex international borders, including enclaves like Vatican City, San Marino and Lesotho, the enclave-within-enclave town of Baarle, and the India-Bangladesh Cooch Behar region.

CityNerd's Ray Delahanty tallies the personal costs of driving and then the externalities that car dependency imposes on everyone else, with numbers for each. It is an urbanist case that driving is heavily subsidized and costs society far more than drivers pay.

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.

A short Healthcare Triage episode by Aaron Carroll arguing that milk gets an undeserved pass in efforts to improve diets. It examines claims about calcium and bone strength and concludes you don't need to drink milk.

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.

Veritasium explores how synchronization arises spontaneously in nature, with metronomes, fireflies and bridges, drawing on Steven Strogatz's work on chaos and complexity.

CGP Grey explains the history of the US Social Security card and number, and why they became a de facto national ID despite never being meant for it. He also covers why America resists a proper identity card.

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.

Real Engineering looks at the numbers for running California on renewables, focusing on the large energy storage needed. It argues about the cost of batteries and the role of other sources.

Polylog explains how algorithms can be turned into circuits and encoded as satisfiability problems, which lets you effectively run them backwards. It uses this to motivate breaking RSA, general reductions to SAT, and what P vs NP is really asking.

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.

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.

A short Globe and Mail animated explainer on why renting can leave you as well off as buying a home. It walks through a rent-versus-buy comparison using Toronto numbers and the opportunity cost of a down payment. The video notes it dates from 2015 and the housing market has changed since.

Matt Parker of Stand-up Maths explains why people keep making the same mistake with millions and billions, using a Bloomberg cash calculator example. He looks at the unit slip behind the repeated arithmetic error.

A short infographic video on the distribution of wealth in America, contrasting what people believe, what they consider ideal, and the actual numbers. It highlights how top-heavy US wealth is.

A Veritasium video on the logistic map, showing how a simple population equation leads to chaos and bifurcation. It connects the idea to convection, neuron firing and the Mandelbrot set, with interactive animations.

NASA's animated explanation of the Curiosity rover's Mars landing, which relied on a complex automated sequence taking about seven minutes. It features NASA engineers discussing the risks before the 2012 landing.

Scott Manley explains the plumbing of rocket engines, including pressure-fed, gas generator, expander and staged combustion cycles. He shows why more complex designs exist and what they gain.

Jared Owen's 3D animated breakdown of how a Brunswick GS-X bowling pinsetter works, showing its mechanisms for sweeping, lifting and resetting pins and balls.

Eager Space examines the practicality of putting data centers in orbit, walking through three scenarios with cost calculations and the launch cost reductions that would be required.

Military History Visualized charts ship production by the United States and Japan from December 1941 to September 1945. The growing numbers show why Japan, despite a stronger starting navy, could not win a long war.

PBS Space Time, together with It's Okay to be Smart and PBS Eons, asks how a universe drifting toward disorder produced something as complex as life. It presents the idea that the laws of physics, through thermodynamics, may actually favor the emergence of life.

Engineering Explained compares the lifecycle emissions of keeping an old car with those of buying a new, more efficient one. It weighs manufacturing emissions against emissions from driving, and covers gas versus electric cars.

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.

A video arguing that exercise matters for the brain as much as the body, linking movement, the nervous system and BDNF to learning and memory. It draws on research connecting complex movement to brain development.

Tom Scott explains YouTube's 11-character video IDs, which are base-64 encoded 64-bit numbers, and whether YouTube could ever run out of them. It covers counting systems and base 64 briefly.

Casey Neistat's short New York Times op-doc investigating how accurate restaurant and packaged-food calorie labels really are. He gets items from chains like Chipotle, Starbucks and Subway measured by nutrition researchers and compares them to the printed numbers.