This Problem Broke Math (and led to Quantum Physics)

A general solution to the cubic equation was long considered impossible, until we gave up the requirement that math reflect reality. This video is sponsored by Brilliant. The first 200 people to sign up via https://brilliant.org/veritasium get 20% off a yearly subscription.

Thanks to Dr Amir Alexander, Dr Alexander Kontorovich, Dr Chris Ferrie, and Dr Adam Becker for the helpful advice and feedback on the earlier versions of the script.

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References:
Some great videos about the cubic:

500 years of not teaching the cubic formula. – https://youtu.be/N-KXStupwsc

Imaginary Numbers are Real – https://youtu.be/T647CGsuOVU

Dunham, W. (1990). Journey through genius: The great theorems of mathematics. New York. – https://ve42.co/Dunham90

Toscano, F. (2020). The Secret Formula. Princeton University Press. – https://ve42.co/Toscano2020

Bochner, S. (1963). The significance of some basic mathematical conceptions for physics. Isis, 54(2), 179-205. – https://ve42.co/Bochner63

Muroi, K. (2019). Cubic equations of Babylonian mathematics. arXiv preprint arXiv:1905.08034. – https://ve42.co/Murio21

Branson, W. Solving the cubic with Cardano, – https://ve42.co/Branson2014

Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural. arXiv preprint arXiv:1308.2181. – https://ve42.co/Rothman

Vali Siadat, M., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations. Math Horizons, 28(1), 12-15. – https://ve42.co/Siadat21

Merino, O. (2006). A short history of complex numbers. University of Rhode Island. – https://ve42.co/Merino2006

Cardano, G (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3

Bombelli, R (1579) L’Algebra https://ve42.co/Bombelli

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Written by Derek Muller, Stephen Welch and Petr Lebedev
Animation by Fabio Albertelli, Jakub Misiek, Iván Tello and Jesús Rascón
Mathematical animations done with Manim – thanks Grant Sanderson and the Manim community!
SFX by Shaun Clifford
Filmed by Derek Muller and Emily Zhang
Edited by Derek Muller and Petr Lebedev
Additional video supplied by Getty Images
Music from Epidemic Sound
Additional Music By Jonny Hyman
Produced by Derek Muller, Petr Lebedev and Emily Zhang