# 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, 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

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

I get the inverse of the exact same thing.

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

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

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.

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

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

Thousandth like I’m so amazing

@Boom Bam wow u are

Ikr

@Prashant Singh Noble was set up for only practical application originally thus Einstein did not receive a Noble for Special or General Relativity as it took awhile for the practical use to explode (atomic bomb and many other things) They basically bent their rules later to give Einstein a Noble for something not that big but it was practical.

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

And how they’ll change now as more discoveries are made

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?

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

@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

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.

@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.

@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.

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

@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.

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…

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

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.

question about dividing by zero maybe

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.

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.

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.

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

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

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.

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.

You must have gone to a good school

I was also taught the same way

Nice Quinn!! I think you’ll appreciate calculus too when you get there, it’s very illuminating

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

Even after learning it in high school, it still sometimes blow my mind with how much sense it makes

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

Amazing video!

Don’t read my profile…

@Yeltsa kcir ok i wont

I’m still lost on how Euler’s formula works. I wish videos like this wouldn’t skip steps in their explanations so people could actually follow along with the math they’re presenting instead of assuming we all get advanced mathematics perfectly already. Far too many holes for me to follow the math in this.

Pero madre mía que haces aquí compañero

I dont know what you just said but you sound smart so I give you like lol