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.

The first part of Welch Labs' series on imaginary numbers, presenting them as the natural extension of the number line into two dimensions. It covers the history and the math with strong visuals.

Veritasium tells the history of how imaginary numbers arose from the quest for a general solution to the cubic equation, including the duels and secrecy among Renaissance mathematicians. It argues math had to give up the requirement that it mirror reality.

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.

A Numberphile interview with Donald Knuth about how he came to write the first book on surreal numbers after a lunch with John Conway. He tells the story casually in about 14 minutes.

Physicist Geoffrey West speaks at an FQXi conference on time about complexity and scaling laws. He covers how metabolism, growth and lifespan scale across organisms and how the same mathematics applies to cities and companies.

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.

Richard Cook's short Velocity 2013 talk on resilience in complex adaptive systems, arguing systems operate near the edge of failure and that the surprise is how rarely they fail. He presents a model of how pressures push systems toward failure boundaries.

Dr. Richard Cook, a physician and patient safety researcher, presents his ideas on how complex systems fail, at Velocity 2012. He argues for designing for resiliency rather than trying to exhaustively handle every failure mode.

A high-quality recording of President Eisenhower's January 1961 farewell address, including his warning about the unwarranted influence of the military-industrial complex. The uploader marks the key passage at 8:41.

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.

Rich Hickey's Strange Loop 2011 keynote separates 'simple' (not intertwined) from 'easy' (familiar or near at hand) and argues that software complexity comes from complecting things. It offers a vocabulary for judging designs and abstractions.

Freya Holmer's Dutch Game Day 2023 talk asking why vectors can't be multiplied, building from complex numbers to a single axiom and arriving at geometric algebra, quaternions and multivectors.

Joseph Tainter lectures at a 2010 sustainability conference on why complex societies such as Rome collapse. His thesis is that rising complexity brings diminishing returns until societies cannot afford to maintain it.

The first lecture of Francis Su's Harvey Mudd real analysis course, covering the construction of the rational numbers. It begins a full lecture series that follows a standard analysis text.

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.

Leonard Susskind lectures at Stanford on how gravity and quantum information theory have merged into a new way of thinking about physics. He covers entanglement and complexity, with applications from black holes to strange metals and the foundations of quantum mechanics.

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.

A long Lex Fridman interview with Stephen Wolfram on complexity, cellular automata, computational irreducibility and the Wolfram Physics Project.

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.

A three-minute clip of Dwight D. Eisenhower's farewell address of January 17, 1961. In it he warns of the unwarranted influence of the military-industrial complex.

Tom Murphy's satirical SIGBOVIK 2007 presentation, delivered under a pseudonym, proving that generalized Super Mario Bros. is NP-complete. It is a short, deadpan take on a complexity theory result.

Lex Fridman interviews University of Glasgow chemist Lee Cronin about the origin of life, aliens, complexity and consciousness. The conversation centers on Cronin's assembly theory.

A 3Blue1Brown live lockdown math episode asking what Euler's formula e^(pi i) is actually saying. It builds intuition through exponentials, rotation and vectors landing on the unit circle.

Rick Beato analyzes a pop song he argues is among the most harmonically complex ever, breaking down its chords and key changes. It is a music theory deep dive aimed at general music fans.

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.

Rich Hickey's Strange Loop 2011 keynote distinguishing 'simple' (not intertwined) from 'easy' (familiar or near at hand). It argues that software complexity comes from complecting things and that we should pick simple constructs.

Rich Hickey's keynote at RailsConf 2012, a close sibling of his 'Simple Made Easy' talk. He distinguishes simple (not intertwined) from easy (familiar), and argues that software complexity comes from complecting things, with Rails-specific examples including large gem collections.

Technology Connections takes a long, detailed look at how a 1970s electromechanical pinball machine works, from relays to stepper motors and mechanical logic. It is the first part of a series.

Scott Meyers' keynote at DConf 2014, drawing on 25 years of C++ experience to offer the D language community a plea about what it should avoid. It centers on the accumulated complexity and pitfalls of C++ as a cautionary tale.

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.

Milton Friedman explains why the US tax code is enormously complex and why meaningful reform is unlikely. He argues that politicians profit from the complexity, since tax breaks are products they sell and the code works as a tool of reward and punishment. The clip is about 14 minutes from his lecture series.

Owen Maitzen's animated hour-long introduction to Hackenbush, a simple combinatorial game, and how its strategy leads to surreal numbers and the roots of the number system.

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.

Lex Fridman interviews Donald Knuth about algorithms, complexity, The Art of Computer Programming, and his work habits. The conversation covers his approach to verifying algorithms and how he organises his time.

Jon Gjengset spends about three hours livecoding complex changes in a Rust codebase with agentic AI tools, testing his hypothesis that AI is weak at non-pattern work. He documents where Claude Code helps and where it struggles.

An excerpt from Richard Dawkins's lecture on Climbing Mount Improbable demonstrating how a complex eye can evolve through incremental steps. It answers the creationist irreducible complexity argument.