This Problem Broke Math (and led to quantum physics)

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 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.

Some great videos about the cubic:

500 years of not teaching the cubic formula. —

Imaginary Numbers are Real —

Dunham, W. (1990). Journey through genius: The great theorems of mathematics. New York. —

Toscano, F. (2020). The Secret Formula. Princeton University Press. —

Bochner, S. (1963). The significance of some basic mathematical conceptions for physics. Isis, 54(2), 179-205. —

Muroi, K. (2019). Cubic equations of Babylonian mathematics. arXiv preprint arXiv:1905.08034. —

Branson, W. Solving the cubic with Cardano, —

Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural. arXiv preprint arXiv:1308.2181. —

Vali Siadat, M., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations. Math Horizons, 28(1), 12-15. —

Merino, O. (2006). A short history of complex numbers. University of Rhode Island. —

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

Bombelli, R (1579) L’Algebra

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, Alex Kontorovich, 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

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43 Responses

  1. Dr. Trefor Bazett says:

    Was expecting cool math, didn’t expect the crazy history story, but it was my favorite part:D

    • Hypervious says:

      I get the inverse of the exact same thing.

    • Henry says:

      Dr. Derek Mueller should create a multimedia book teaching all these. it’s so refreshing and entertaining to learn math.

    • A B says:

      I love this channel and I really tried to watch this whole video. Until I realized how dumb I am and that this video should have been in japanese or something due to how horrible I am at math lol

    • We Are in the Matrix says:

      The two 1’s in 1=1 can refer to the same thing or to two different similar things. However, the two 1’s in 1+1 can only refer to two different similar things. Why?

      Numerical identity should be sufficient where qualitative identity is the requirement.

    • Voidisyinyang Voidisyinyang says:

      check out math professor Louis Kauffman on why the imaginary numbers are noncommutative as primordial time.

  2. DoS - Domain of Science says:

    This was a fascinating insight into the origins of the mathematics that’s so familiar. Wonderful. Thanks Derek!

  3. Håkon Flatøy says:

    I love the historic bit. To know where things and ideas came from, how they grew by the years.

  4. Philip ROSE says:

    This is a faultless presentation of one of the most inspiring naratives in history, maths and physics. Congratulations! You have set a new paradigm YT. Could you do the same for Dirac’s equation?

    • West Explains Best says:

      I am seriously contemplating showing this video to my Alg 2 class. Visual demonstration of completing the square and math history? Too good!

    • Linzzz says:

      @West Explains Best I was just saying to someone that I sorely wish the history of all of this had been taught to me back when I was learning it. I went on to study math in college, but I still wish that someone earlier on had showed us the humanity in math, the bickering scientists and the disbelief/hope that a solution would ever exist.

      I think it would be awesome for you to show it

  5. Erik Hare says:

    Gauss called them “Lateral Numbers.” I believe we would have a much easier time with them if we used his term. Also, there may well be lateral dimensions, but I may have simply watched way too much Doctor Who.

    • RedRocket4000 says:

      @Mystixor They are still imaginary numbers in the vast number of cases. As taught in math class all imaginary use in real world applications can actually be calculated without imaginary numbers but reaching the solution that way can be extremely complicated.

      This was the early 70’s and for some reason advanced levels of physics were not covered.

      Fun part is Dark Matter and Dark Energy could indicate our understanding of Quantum mechanics is very wrong as hinted in using a imaginary number. In this case current use of complex numbers might by like Newtons laws false but but still very useful for any practical doing anything the exceptions not effecting us at our level. Relativity replaced all of Newton it just that relativistic equations that replaced Newton are more complex than is needed for most things we do on earth and in some things in Space so we stick to Newton in these cases. In Space though distances are great enough that the tiny at smaller scale errors of Newton and thus effect nothing practical in space the differences become huge and you must use Relativity as the differing speeds of time in particular means don’t adjust for relativity GPS gets more and more inaccurate over time as the fact that time passes differently for the satellite and here on earth throw off the location.

    • Mystixor says:

      @RedRocket4000 What makes you so sure about imaginary numbers being condemned to being solely the path to but never the solution itself. Quantum mechanics is not only counter- but unintuitive. You cannot go into it with classical intuition. Imaginary numbers there are just as real as real numbers are to us. Maybe it is indeed the wrong way of looking into the problem. But your comparison to Newtonian gravity does not make a point here.

    • mcilrain says:

      “Lateral dimensions” sounds like the British English version of “parallel dimensions”, like elevator vs lift.

    • Anderpanders says:

      @Mystixor seeing as quantum colors are like a tripole kinda situation, could there be even more imaginary layers to mathematics?

      Also fun thing about the tripole thing. It’s like magnetism but instead of two poles there’s three and they attracts each other. In electro magnetism there are two poles that attract each other, and in gravity there is only 1 pole that attracts itself.

      You seem to know stuff so I just wanted to share my ramblings, sorry.

    • Joshua Smith says:

      I was about to say, if it’s orthogonal to a real dimension (as in 3D) it’s a 4th dimension. Maybe that’s why gravity is so weak, it’s leaking out into the lateral dimension. Oops, PBS Spacetime is leaking into my thoughts again…

  6. AsianSushi AMV 烈 says:

    It’s crazy to think that a simple question shook the entire math world. All it takes is for one person to look at a problem in a different way – makes you wonder what the next “simple question” will be

    • Derek Anderson says:

      the simple question is already here, in physics form “the theory of everything” Combining both quantum physics and relativistic physics. I have no doubt when solved we will look back and scoff at the simplicity of the solution.

    • Rosyid Haryadi says:

      question about dividing by zero maybe

    • Chessmapling says:

      Even today’s math problems continue to be solved this way. It made me realize math truly is so much more beautiful than how it’s taught in schools.

    • We Are in the Matrix says:

      Here’s one… the two 1’s in 1=1 can refer to the same thing or to two different similar things. However, the two 1’s in 1+1 can only refer to two different similar things. Why?

      Numerical identity should be sufficient where qualitative identity is the requirement.

  7. The Sigma Enigma says:

    Instead of letter grades A through D, 8th graders should get a grade placement based on which century of Italian mathematics they most closely align with.

    • Vigilant Cosmic Penguin says:

      Guys, we can reform standardized testing now! We found the perfect scoring system!

    • dd9988771030 says:

      Congratulations! You have scored “Roman Republic” in Math!

    • Julian Herbert says:

      If you think 8th graders are learning about imaginary numbers, solving cubic equations, or quadratics for that matter you either don’t remember primary school or were an exceptionally gifted child. My guess is the concept of a variables is introduced in 7 or 8th grade, probably putting 8th grades some where in the dark ages. Probably where they belong from what I’ve seen , haha.

  8. Quinn Blumenthal says:

    One of the things that I’ve most appreciated about my Algebra 2 class is that they actually taught me how to mathematically derive the quadratic formula, by completing the square, so that I know what’s actually going on. It also helped me a lot to understand why imaginary numbers need to exist, by understanding the principals behind the x³ function that was unsolvable before hand.

  9. Gazehound says:

    Just had my mind blown learning that “complete the square” is literal.

  10. Shadoune666 says:

    Woooow that was amazing, to learn the story of imaginary numbers and actually understanding Euler’s formula.
    Amazing video!

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