This Problem Broke Math (and led to quantum physics)

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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. ??? 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 ??? Special thanks to Patreon supporters: Luis Felipe, Anton Ragin, Paul Peijzel, S S, Benedikt Heinen, Diffbot, Micah Mangione, Juan Benet, Ruslan Khroma, Richard Sundvall, Lee Redden, Sam Lutfi, MJP, Gnare, Nick DiCandilo, Dave Kircher, Edward Larsen, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Ruslan Khroma, Robert Blum, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson,Ron Neal 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