Mathematicians Use Numbers Differently From The Rest of Us

Published 2023-06-06
There's a strange number system, featured in the work of a dozen Fields Medalists, that helps solve problems that are intractable with real numbers. Head to to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.

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Koblitz, N. (2012). p-adic Numbers, p-adic Analysis, and Zeta-Functions (Vol. 58). Springer Science & Business Media.

Amazing intro to p-adic numbers here:    • 1 Billion is Tiny in an Alternate Uni...  
Excellent series on p-adic numbers:    • p adic numbers. Part 1 of 3: 10-adic ...  
Great videos by James Tanton: @JamesTantonMath

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Written by Derek Muller and Alex Kontorovich
Edited by Trenton Oliver
Animated by Mike Radjabov, Ivy Tello, Fabio Albertelli and Jakub Misiek
Filmed by Derek Muller
Additional video/photos supplied by Getty Images & Pond5
Music from Epidemic Sound & Jonny Hyman
Produced by Derek Muller, Petr Lebedev, & Emily Zhang

All Comments (21)
  • @RenaudAlly
    These are literally scientific documentaries of the highest quality at this point. It's amazing that I'm able to watch this stuff for no cost at all. Thank you so much Veritasium
  • @andy07070
    The level of quality in these videos is sublime. You never insult the audiences, by not going as deep as is required. Excellent work as always
  • @aaronfactor6838
    I studied this in my teaching program. We did this to better help us understand the “Why” in math. So many steps in math we just are taught and accept, but many people can’t understand the actual mathematical reasoning that allows us to complete that step. We studied different math operations in various bases to quite literally re-teach ourselves math. We even used symbols instead of numbers. It was a very eye opening experience.
  • @Istaqlal
    My brain just turned to mushed slime and left out of my ears, this needs to be a 10 hour course to be understood even to the slightest level. He just used algebra, geometry, number theory, series, and everything else and expects us to get a hang of it. There is more stuff in this video than in my whole 12th grade mathematics textbook. Whatever little i understood tho, completely blew my mind. Love all the videos. I do stubbornly wanna pursue math and physics further, but i think my puny brain will never grasp such grand concepts. Let alone be able to contribute anything further to the progress. ❤
  • @veltongoodenjr
    You've got to be doing something right when I was absolutely surprised when the video had ended, and I was longing to know more. I'm by no measure a "math person", and yet I was able to follow along for the larger part of the video with the calculations and by means of the visualization and explanation make connections between things I've up until now had no idea were related. Had I the time, and were my goals for the future ever so slightly different, I'd probably plunge right into the world of mathematics just because of how fascinating what this revealed was. Even though, this has not in fact moved me to such extreme action, it has come rather close in that I will now forever see the mathematical concepts discussed in this video in a different way.
  • @JustAnotherPerc
    I could barely learn some of the stuff in my 7th grade algebra class within a whole hour.
    However with this 30 minute video, I learnt a whole new math system.
  • @johnchessant3012
    I don't normally think of Veritasium as a math youtuber, but with videos on Newton's calculation of pi, Godel's incompleteness theorem, discrete Fourier transform, logistic map, Penrose tiling, Hilbert's hotel paradox, and various probability puzzles, he definitely should be. I mean, this video alone (p-adic numbers, Fermat's last theorem, Hensel lifting) would be an extremely ambitious topic even for a math-focused channel, and he and Alex Kontorovich did a great job with it!
  • @CdFMasterVideo
    Can I just say how much I love that you kept the same background music and outro animation for all theses years? One can get lost through life then come back to this channel years later and still feel at home. It's an underrated quality.
  • @manOmanyTrades
    Bravo! You covered in 30 minutes what took me semesters to master in my youth. I am totally inspired.
  • It's been a while since a topic in mathematics captured my imagination so much. There is something about the p-adics that feels wrong, but also something that feels so compelling and so deep.
    What a wonderful introduction and brilliantly done.
  • Your delivery of this content was absolutely sublime. You somehow took an incredibly complex topic and simplified it through examples and explanation so a relative novice can grasp it. Thankyou so much. This really made my day
  • @IanRobinett
    Derek, you have literally been the person teaching me the most since I found YouTube. Shortly after is Destin at SmarterEveryDay, but you two give me more knowledge than I've ever wanted in so many fields. I HAAAAATE most of the subjects you cover on the surface, but when you break them down into applicable and project-oriented and realistic applications, it makes me realize my disdain for things like Mathematics and Science, is because of the academic application, versus what it means in real life. You two are truly those who have expanded my mind to forget my hatred for the academia part, and realize that it can directly apply to the "fun stuff" as well. I guess it proves the difference between "AP" and "GT" students... Same intelligence, just different applications. Regardless, this video was amazing, and thank you for the visual and practical applications.
  • @CyclingGeo
    I’m a geologist so my maths is questionable at best.

    I find it utterly fascinating how well I can follow along with this, yet still be completely bewildered and confused.
  • Today I felt like exactly, I am seeing a new world of mathematics…. It’s a wonderful explanation. As a data scientist and a embedded software developer I would say this level of mathematics is required very high level of intelligence to understand in a single shot. Thank you for sharing this extremely valuable piece of research work. I owe you a party for sure. 😊 keep doing…
  • @chrispycryptic
    This video was quite literally an 'AHA!' moment for me. It's like when you are trying to put together a difficult puzzle and have found every single edge piece except one. It is that feeling when you finally find that little bastard 3 days later and get to experience the satisfaction of gazing upon the complete frame. Such beautiful content! Thanks so much Derek, you rock!!!!
  • This is the kind of clarity and explanation we need in university maths classes. So much of the time we are left to our own devices to interpret the logic of abstract claims like the "size" of a number. Textbooks usually state the mathematical relation. I fully get how hard it is to describe these things conceptually to a general population but it's so useful and it makes these things appreciated more. Looking at p-adics still freaks me out and I don't quite see them as stars but I can at least see how viewing a series as a different category of number altogether makes sense for why series are used in proofs so often to break down some what simple rational number or variable. (I'm not explaining myself properly because I know the convergence of infinite sums is useful. It's more understanding how the parts inside work and what those mean, or just another way to visualise infinite series.)
  • @tacoboy2218
    I liked math in school since i liked puzzles but this video had me enthralled like nothing before this is literally 3D math
  • I love how the far end of infinity wraps back around to zero and negative numbers. Thanks for the mind-blowing video!
  • @jessejustice454
    I can’t focus for 5 mins at school but can watch a full 30 minutes video from you no problem
  • @RoguePsychMan
    I majored in math as an undergrad at a top engineering school. I recently found my notebooks from the classes and it’s a whole new language