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# Mathematicians Use Numbers Differently From The Rest of Us

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- Published on Jun 5, 2023 veröffentlicht
- 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 brilliant.org/veritasium to start your free 30-day trial, and the first 200 people get 20% off an annual premium subscription.

If you're looking for a molecular modeling kit, try Snatoms - a kit I invented where the atoms snap together magnetically: snatoms.com

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References:

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 a...

Excellent series on p-adic numbers: • p adic numbers. Part 1...

Great videos by James Tanton: @JamesTantonMath

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Special thanks to our Patreon supporters:

Emil Abu Milad, Tj Steyn, meg noah, Bernard McGee, KeyWestr, Amadeo Bee, TTST, Balkrishna Heroor, John H. Austin, Jr., john kiehl, Anton Ragin, Diffbot, Gnare, Dave Kircher, Burt Humburg, Blake Byers, Evgeny Skvortsov, Meekay, Bill Linder, Paul Peijzel, Josh Hibschman, Mac Malkawi, Juan Benet, Ubiquity Ventures, Richard Sundvall, Lee Redden, Stephen Wilcox, Marinus Kuivenhoven, Michael Krugman, Sam Lutfi.

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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

Renaud Ally3 months ago^{+4375}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

Geniuz3 months ago^{+51}@Leeroy Jenkins revanced youtube moment, ads blocked, sponsored segments skpped automatically

Hyperreks3 months ago^{+5}@Geniuz yessir

Aaditya3 months ago^{+16}It's all thanks to Patreons and sponsors

Rafael D3 months ago^{+4}@Leeroy Jenkins I'm not paying time with time. Ads are not for everyone just like adblockers.

idontwantanid3 months ago^{+15}@Leeroy Jenkins but that's a trivial amount of time; I think the point was that there's high-quality content that's as close to free as makes no difference. You can let the ad play while you brush your teeth or watch another YT video in another tab.

AndypapandyMonth ago^{+338}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

Jaxon ShainMonth ago^{+5}Its interesting to think of lack of depth as an insult. Why would this be so?

AndypapandyMonth ago^{+10}@Jaxon Shain Perhaps insult was not the ideal word to use. What I meant to say is that he goes deeper than alot of similar content creators, which I find enjoyable.

Jaxon ShainMonth ago^{+5}@Andypapandy yea I enjoy the depth as well, especially now as it is so easy to use A.I to make content.

Rohan GeorgeMonth ago^{+8}what a mature comment section

Aaron Factor3 months ago^{+369}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.

Fabssiee3 months ago^{+6}That sounds so cool

Aaron Factor3 months ago^{+22}@Fabssiee my professor called it “Martian math”. We used symbols in place of numbers that went in a specific operation just as our numbers system in base 10 would. This way we had no prior knowledge to scaffold learning with and it was as if we were children getting exposed to numbers for the first time

Aaron Factor3 months ago^{+15}@floppathebased1492 sorry for my poor explanation. We first learned how to do the various orders of math operations using different bases I.e. multiplication addition division. After that, she removed the numbers completely and instead used a random order of symbols and shapes likes triangles with slashes, then a star,etc. the point was that each of our numbers is itself a system that we have to learn and rely on. I want to say we were using base seven and she had 6 different Symbols before they’d move place value and repeat like our traditional number system. I hope that makes more sense and sorry for the confusion.

TecateGPT2 months ago^{+3}I'm currently on that journey of learning the "why" in math.

Like for example I tried explaining to my gf how we can put

- * - = +

But in a real life example and God that was difficult to do lol.

I've tried looking online and all I got were proofs.

But the best example I've gotten so far was.

Suppose you record someone walking, you're using a tape player to capture the footage.

You allow that tape to play forward, person walks forward (+)

You then reverse that tape (-)

Person walking forward (+)

Now walking backward(-)

Now allow that person walking backward (-)

Play the tape backward again (-)

Person walks forward (+).

I have no other examples but a tape player really because I've never known an object to be "negative".

I feel like negative numbers exist in 4 Dimensional objects and time is one of them.

Which is why we can sort of have the power to "control time" by recording footage and playing it back

Dávid Čano2 months ago^{+1}@Aaron Factor hey very late comment lol but is there any chance you have some pointers for what you are talking about somewhere online? Or in a book or something? Or at least least similar?

Gregory Nelson2 months ago^{+3}I work in computer science, and in watching this video, I realized that computers store integer quantities using a similar strategy in a binary string, only for a finite string (since you can't store an infinite amount of data). Typical integers are stored in 32 or 64 bit registers (there is also a short int, or integer, that is only 2 bytes or 16 bit registers). The first register is usually reserved as the 'sign bit', but for all intense and purposes, this can be considered as the overflow register to assume up to infinity for the purposes of this discussion. For illustration purposes, let's assume only 8 bit registers with the first being a sign bit and the remaining being the magnitude part. Each bit from the right to the left represents a power of 2, with the right most being on the order of 2⁰ * (0 or 1).

0 would be: 0 | 000 0000

1 would be: 0 | 000 0001

2 would be: 0 | 000 0010

127 would be: 0 | 111 1111 (the maximum positive signed integer value that can be stored in a singly byte, or 8 bit registers)

However, adding just 1 more to this value would roll this result and would yield:

0 | 111 1111

0 | 000 0001

--------------------

1 | 000 0000 (the minimum negative value possible, -128 in this case)

Similarly, subtracting a value n from 0 rolls the sign bit to reveal the negative integer representation of that n magnitude (it 'magically' barrows a '2', or a binary 10, from the next order from the 'infinity' beyond the sign bit range):

0 | 000 0000 (0)

0 | 000 0001 (1)

--------------------

1 | 111 1111 (-1)

0 | 000 0000 (0)

0 | 000 0110 (6)

---------------------

1 | 111 1010 (-6)

This same methodology extends out to Short Int (2 byte), (Normal) Int (4 byte), and Long Int (8 byte), just with a lot more bit registers available and allowing for a much greater range in integer numbers.

Bonus, binary is already a base-2 counting system that makes it a P-adic by default.

Velton Gooden Jr.3 months ago^{+158}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.

tquasa16 days agoI’m going to go into calculus for both of us, bro, wish me luck

Saadusmani7812 days ago@tquasagood luck👍👍😃, get good grades there.

Istaqlal2 months ago^{+27}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. ❤

Rohan GeorgeMonth ago^{+1}gotta be big brain, whole new level of understanding, and its an area of study that is completely new to us, i felt just like this.

John Chessant3 months ago^{+2313}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!

green3 months ago^{+16}absolutely

gw3 months ago^{+59}Waiting for 3B1B to pop up somewhere

Austin Hamilton3 months ago^{+36}Like half of his videos are math related lol

Arnab Biswas3 months ago^{+14}Next-up I want him to look at parker square.

Pedro Paulo Wanderley Júnior3 months ago^{+9}Yeah...not math Clip-Sharer to me, but a sleep helper Clip-Sharer. 😂

Just Another Percussionist2 months ago^{+34}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.

CdFMaster3 months ago^{+51}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.

Hasan Long18 days ago^{+6}Bravo! You covered in 30 minutes what took me semesters to master in my youth. I am totally inspired.

Ardavan FarahvashMonth ago^{+9}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.

Ian R2 months ago^{+6}Derek, you have literally been the person teaching me the most since I found Clip-Share. 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.

Cycling Geologist3 months ago^{+1564}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.

trout3 months ago^{+123}It makes you feel like your learning but it's basically entertainment because you will forget anything about it the next day.

Cycling Geologist3 months ago^{+44}@trout well with my adhd, I essentially forget the previous sentence because I’m having such a hard time following.

Burstein von Knackerthrasher3 months ago^{+18}@Cycling Geologist Do you remember leaving either of these comments several hours later? I'm here to remind your brain

JS3 months ago^{+33}@Cycling Geologist Derek might say we're not visual learners, but I can see the infinity triple cylinders in my head. It's stuck there forever in endless loop to remind this video topic. No idea how to use this knowledge, since this whole video was like a rocket engineer teaching a toddler how to build a hypothetical navigation system. But give me an endless pile of cylinders, and I'll build stacks of 0, 1, 2.

Michael Westen3 months agoyeah because you probably finished 12 years of school and didn't get shot in the process

John Robinson5 days ago^{+2}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

Sandeep Raj Betanapalli10 days ago^{+2}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…

Curiously Crispy3 months ago^{+19}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!!!!

TheShamansQuestionMonth ago^{+3}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.)

tacoboy 22114 days ago^{+1}I liked math in school since i liked puzzles but this video had me enthralled like nothing before this is literally 3D math

Jesse Justice3 months ago^{+4165}I can’t focus for 5 mins at school but can watch a full 30 minutes video from you no problem

masey3 months ago^{+27}Lol

Kexerino3 months ago^{+306}This is something you choose to do, school isn't

MiruPikachu / 猫咪酱3 months ago^{+106}Prob because you knows that you're gonna learn smth that you actually wants to know about, rather than listening to random shits that your teacher's gonna teach you ;-;

SubZero3 months ago^{+28}It's probably all the answers before and also probably because this channel has a better teacher than the one at your school. I felt the same when I was in school as well and realized that quite a few of my teachers were just not right for me

Saurabh Kumar Singh3 months ago^{+102}You can't . This 30 video was uploaded 10 mins ago, and clearly you didn't watch it for 30 minutes before coming to the comments, ie lost your focus .

Jonathan ClarkMonth ago^{+5}I love how the far end of infinity wraps back around to zero and negative numbers. Thanks for the mind-blowing video!

JDMonth ago^{+10}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

Vic Jang3 months ago^{+1}The amount of new knowledge to me in this video is insane. I’ve never heard about p-adics and I’m so surprised and overwhelmed to receive so much new info from a Clip-Share video. Great job, Derek. The quality of your video keeps getting more and more astonishing.

Chuck Malloch3 months ago^{+5}I love these videos. Even in my eighth decade, I love learning, and am fascinated by your videos. Thank you!

Mensa Swede3 months ago^{+2}I’d love to see mathologer’s take on this. It feels like a different “proof” could be created that shows the “adic”-ness properties are false for non-prime adic’s. The cardinal value of any number should have properties that are independent of the base used to write the number down.

Rodney Jenkins3 months ago^{+3738}I'm jealous that Derek gets a personal lecture from such an amazing mathematician.

Lone Starr3 months ago^{+250}I had the pleasure of meeting Alex Kontorovich in person several times by now, on conferences and a summer school. Had 2 or 3 chats with him. As far as I can tell, he really is like he comes across in these videos. And he gives the best talks, by a long shot, even when they're intended for a professional audience and not for a general one like in this video. He has a way of conveying his enthusiasm that is truly unique and exhilarating. It fills you up with passion, like you have to go and prove some theorems, now! What an awesome guy, really.

Nitin Sharma3 months ago^{+55}@Lone Starr Now, he'll be jealous of you too:)

VoteScientist3 months ago^{+15}An operation that only works in base 10 is not Mathematics it's just Arithmetic.

KC3 months ago^{+10}we are too since he's credited as coauthor of the video :)

Lone Starr3 months ago^{+7}@Nitin Sharma That was the intention behind telling him, lol.

Hash Tag25 days agoI like it how mathematicians just create weird problems and challenge other mathematicians to solve and expect this will one day be useful for solving real world problems, and they actually do.

DrAshente3 months ago^{+26}I am someone who can't stand maths. But it's not because of maths, it's because of the school system. I find videos like this fascinating and I like watching them. Since I left school, I enjoy watching them even more. And it turns out that I don't suck at maths as much as I used to get taught. There just weren't any topics that interested me.

Weltschmerz ist oft häufig…3 months ago^{+1}What topics in Maths interest you?

madhavi malati2 months agoThis is so true. I am in school… it’s not that maths itself is boring it’s just the school system.

Mr TonyMonth ago^{+3}I have read the book "Fermat's Last Theorem - by Simon Singh" he also elaborates on maths a little but the story around this problem, how it came into existence and how it took 358 Years to solve it, is what intrigued me. This is why we are the king species, it took us 358 years 5 generations to solve this but we did it and for a guy (Fermat) this was just a note on his Journal. While reading I didn't know if Fermat was right so it was a mystery every page.

TheWaterDragonMonth ago6 Years ago i started studying Physics. At the end of my First semester and the start of the second one this was included... 6 Years after I first heard about it, I'm finally able to understand it

Thank you so much

Stevo3 months ago^{+4}I had a professor who went over some of this stuff and I felt like I understood it well enough to accept it, but could never re-explain it to anyone else, at least not in a convincing manner; it almost always ended in arguments (lol). You have an excellent way of explaining it and using visual aids to step through it. Thanks!

Also, curious, is "10-adic" synonymous with "base10"? I've never heard "10-adic" before this video, always "base x".

Entropie -3 months ago^{+4}When talking about base 10 it only has to do with how we represent numbers as strings of digits, this is purely cosmetic and has no effect on the underlying structure.

When talking 10-adic numbers the choice of 10 does affect the underlying structure, it is mathematically different than talking about 2-adic for example.

Generally it is most natural to represent p-adic numbers in base p.

So p-adic (usually) implies base p but a number being in base p does not mean that it is a p-adic number, it could be a real number represented in base p instead.

John Yin11 days agoExpanding on this a bit, as Derek has proven, the statement "x(x-1)=0 implies x=0 or 1" is true for the real numbers no matter which base you write it in. However, this is not true for 10-adic numbers. This speaks to the "cosmetic" and "structure" @entropie-3622 was referring to.

Leo Leahy3 months ago^{+1763}As an engineer and video editor, I am absolutely mind-blown by the production quality of this video. I can't even imagine the number of hours put into the editing alone. It's amazing that content like this is available for free. Not that your other videos aren't great as well, but this was something else.

Gwyn Sea3 months ago^{+11}I don't know what income 2.2m views on youtube achieves... do you?

Starry Port3 months ago^{+3}the video editing has been done by AI tho...?

baad lyrics3 months ago^{+26}1000 views are about one dollar but that depends heavily on how advertiser friendly the content is.. this channel prolly gets way more than one dollar per 1000 views but lets calculate it with 1 dollar to stay on the safe side.. so 2200000 / 1000 = 2200 dollars. But like i said, its prolly more like 3 or 4 k. But the big money isnt in views, the big money is in sponsorships.. for a standard 60 seconds sponsorship on this kind of video and channel the sponsor prolly pays in the ballpark of 10k-50k or so for it. Should be about 10 - 50 dollars per 1000 views from a sponsor.. so in this case if we calculate with 15 bucks times 2200000 / 1000 that would be 33000 dollars but could be quite a bit more for this good of a channel thats very advertiser friendly

Sam Borgman3 months ago^{+6}@Gwyn Sea 1 million is $1k. But it massively fluctuates seemingly randomly. Big channels make lot more money in other ways than they make from YT ad revenue.

Manoj Bhakar PCM3 months ago^{+25}its done by python software named "manim". i also made these type of videos to teach my students basic operations.

Piotr KawałekMonth agoThis video is fantastic. I am very impressed with your math videos Derek, the topics are so well-presented.

Courtney BMonth ago^{+2}I’ve never seen math explained this way. This is like beautiful art. Amazing and mind blowing! Thank you very much for sharing.

TheAdvertisementMonth ago^{+1}11:44 Haha, my calculus teacher talked endlessly about FLT. It's always a treat to learn a bit more math to understand the process of finding it.

Pugz3 months ago^{+16}This seems incredibly similar to how signed numbers are stored in computers

Yotam Avidar_ConstantiniMonth ago^{+1}I would love to learn more about that

PugzMonth ago@yotamavidar_constantini9222 The way signed numbers work is actually really clever. It allows the computer to use addition and subtraction without having to switch the order for positive and negative numbers in the background. Signed integers start with a sign bit and the following bits store the value itself. This allows the computer to carry from or to the sign bit when needed, which then switches the number between positive and negative. Understanding how this works also explains integer overflow very well. A quirk about how the carrying works (I'm guessing it's so any positive number is read the same signed and unsigned) is that it also "carries" from and to the bit before the signed bit, which is not part of the number's assigned memory and therefore does not change.

An example of 4-bit signed integers from 5 to -5:

0 101 = 5

0 100 = 4

0 011 = 3

0 010 = 2

0 001 = 1

0 000 = 0

1 111 = -1

1 110 = -2

1 101 = -3

1 100 = -4

1 011 = -5

Arthur Reitz3 months ago^{+1}Personally, I think that the fact that p-adic numbers works is less crazy than complex numbers.

einargs3 months ago^{+914}As someone who does computer science, it was extremely cool to suddenly make the connection to how we represent negative numbers using two's complement.

Pranav P S3 months ago^{+18}I see the connection, but they are finite in length... So how does it work?

Wala saurus3 months ago^{+100}@Pranav P S A negative number is actually represented in computers by the inverse of a number + 1 for example -3 would be 11111101. This is why signed integers can only represent half of the positive numbers that a unsigned integer can represent. You still can only represent 256 different values in 1 byte, and since half of them are negative, it goes from -128 to 127 instead. Since you invert a number to get two's complement, you can tell whether or not its negative by looking at the leftmost digit: if its 1, its a negative number, otherwise its 0 and therefore positive. The only difference between and unsigned vs signed integer is how the computer looks at that leftmost bit.

R. C3 months ago^{+32}@Wala saurus I always thought it was a brilliant way to represent negatives. It also allows tons of algorithm tricks to work with positive and negative numbers in a fast and efficient way.

Leif3 months ago^{+8}@Wala saurus what do youbmean by inverse of a number plus 1 sorry ? Why couldn't the negative of an integer in binary jist be the same as positive integer but with a negative sign? Seems clearer and more efficient to me? I'm guessing it is partly because the negstive sign means something else in binary so the computer would misinterpret it? Why don't they just change that then?

Leif3 months ago^{+2}@Wala saurus and what donyou mean by invert a number to get two's complement? What is the complement of a number? Like negative and positive you mean? Never heard it referred to that way..

Tejbir SinghMonth ago^{+2}I couldn't follow it all. But it was still fascinating, and I can still appreciate how well the concept was taught.

blankcomments_3 months ago^{+2}Hey very nice video. I would be very interested in a video about perfectoid spaces. I guess there is a lot of visualization, animation etc. possible, to make them understandable, to a brought community. A lot examples and use cases can be shown and explained. This would be a nice follow up to this video, as p-adic numbers have to be comprehend in order to comprehend perfectoid spaces. I let suprise myself! Hope you keep it up!

Chris McConnellMonth ago^{+1}Thank you for the amazing content Derek.

I'm sure that I learned something, I'm not however sure of what it is that I learned.

Sultan Of Mad Olives3 months agoExcellent explanation! I’d heard Fermat’s theorem was solved- but this is the first I’ve seen a good explanation of how.

Alfred NyakindaMonth agoAs a science journalist who isn't either a professional scientist or mathematician, I applaud the accessibility and scientific analysis of your videos.

That being said, I have developed an interest in orbital mechanics. I am curious whether virtual numbers played a role in determining the escape speed of objects with mass.

Beyond escape speed, two objects would be slowing down to infinity, (relative to each other) but never actually reach zero velocity and reverse their motion, (apparently; according to mathematics).

Terry Bollinger3 months ago^{+1221}As a computer scientist, your comparison of p-adic numbers to two's complement negative numbers was extremely helpful for getting this topic to finally "click" in my head. Thanks!

Edward Chan3 months ago^{+4}Me too!

Ray Lopez3 months ago^{+45}Don't want to be negative but as a computer scientist it should all add up...

Terry Bollinger3 months ago^{+6}@Ray Lopez I _think_ that was a pun?... :)

Luis Sierra3 months ago^{+3}@Ray Lopez nice

Deji Adegbite3 months ago@Ray Lopez Intended pun?

Peter ChurchyardMonth agoMost of the numbers we deal with are really just an edge condition. As a programmer solutions to problems usually have values that are on the edges and have to be tested for specifically.

circuitgamer77Month agoI'm only seven minutes in, but I was thinking this felt similar to negitive numbers in the binary used in computers, and then you introduced 9's compliment... I'm excited to see what other patterns show up :)

THUNDER THUMBS2 months ago^{+4}16:05 One easy way for people to visualize the conversion of numbers from one base to another could be how they are represented as binary(basically base 2). So just instead of how we can easily convert binary(base 2) to denary(base 10) we can convert other bases into denary pretty easily without having to complicate it. I don't have many views on this point tho as I am just a student who just about nearly started learning about computer science.

njanoloMonth agohope he’s getting rewarded sufficiently for all these great lectures 👏🏼👏🏼👏🏼

inam101Month agoA lot of loves for Veritasium. Thanks a lot for making such a complex and important concept very easy to digest.

Zachary Vogt3 months ago^{+747}This video is the perfect example of encouraging the audience to rise to the level of the content (the exact opposite of talking down to the audience.) Very inspiring.

KC3 months ago^{+34}It’s comforting to me that there are people out there that understand this stuff. It’s not me…but I’m glad they exist.

Jin Kee3 months ago^{+17}Subtract the subject matter from my attention span and you get a p-atic number

Jin Kee3 months ago^{+6}Repent and believe in Jesus Christ didn't you watch the video? If you had an infinitely loving being that loved one more person it would become murderous. Explains a lot, actually.

Leif3 months agoWait the first part he said can't be right..yoibcantnget 1/3 by multiplyojgna bunch of numbers greater than 1 by 3. That will necessarily be much greater than 1. Why did he say this then when it's clearly wrong?

Derek H.3 months ago^{+3}@Leif 3:26, It would be wrong if you ever stopped writing any digits to the left, but as long as that sequence of numbers is infinite, that makes all leading decimal places zero. Carrying from the one's place to the tenth's place, to the hundred's etc. are all finite operations which match your intuition. You could imagine it as "carrying the leftover numbers to the the infinite's place". --That's a bit of a non-sensical phrase since there is no infinite's place, but the point is there are no leftover that actually contribute to the finite number you get as an answer. There is no higher value decimal place that isn't just a leading zero.

I'm not sure if it is "technically" legal for our most common number system to even mix an infinity (...666667) with a finite (3), but if you accept that premise and follow along with him anyways, it actually shows how you can discover a new number system which acts as a NEW self consistent mathematical model with amazing implications and practical applications.

Uhkay3 months agoThis was one of the most excited I have felt in learning something new in the past few years. This is incredible, thank you so much for making free content like this.

(I had all notifications on and still was not recommended the past several videos so I resubscribed for all video notifications, a heads up if you had the same problem.)

Ashley Rose3 months ago^{+1}I have a problem that I don't think is anecdotal to my viewing setup. In this video, on the calculation screens, there seems to be small, repetitive, artificially introduced artifacts in the white background. If this is a technical glitch in your recording setup, hopefully this distraction can be fixed in future videos. Given the quality of your videos, I can't imagine that you would squander so much of your quality and integrity to employ such a manipulative, low-brow tactic. Your videos are top notch, and your channel doesn't need to exploit vulnerabilities in our cognition and nervous system for the sake of prolonging engagement. Even during the simplest of mathematical presentations, I don't think we need artificial forms of stimulation to retain our attention and interest.

Jeremy Bognon28 days agoI started understanding this when I started thinking about binary. It makes so much sense now, it's not the exact same but it's the idea

Roasted Boiii2 months agostarted watching this video and just had a flashback of my advanced quantitative reasoning math class. been over 10 years since ive had that class and for some reason everything makes sense to me cause i had answers that didnt make sense but seeing this made it all make sense. thank you

Mark CampanelliMonth agoThanks for a reasonably accessible description of the p-adic numbers!

faridrb3 months ago^{+912}I took a graduate course on p-adics in university and it felt like all I did was manipulating symbols on paper without understanding what is happening. This video finally made me understand what is going on.

Bryan Castro3 months ago^{+65}The essence and beauty of mathematics is to understand, and it is pretty common to find people in academia who teach soulless mathematics. Something must be done, because learning so abstract and difficult concepts without the proper background and motivation is pointless, or so I believe.

Jadon Marshall3 months ago^{+10}@Bryan Castro that’s something I feel I struggle with I think math is so cool but I also don’t necessarily understand the magnitude of why this kinda crazy stuff manipulating infinity is important

Steve Barnes3 months ago^{+3}Unfortunately, in NZ, P addicts result from the use of pure methamphetamine. (p = pure)

But, seriously. The use of base 3 in computing is not that hard, ground, positive to ground, negative to ground. Gound being 0 to source voltage +or-.

Joseph Van Name3 months ago^{+3}How can you learn more from a video on Clip-Share than a graduate course?

Steve Barnes3 months ago^{+8}@Joseph Van Name "...can you learn more from a video on Clip-Share than a graduate course?"

Yes. There is a difference between studying and learning.

M KMonth agoThis video is an excellent insight into how my mind works. Please continue making such beautiful videos about numbers.

Jonas Hellborg23 days agoThank you, that gave me some insights in how a set of numbers can help to solve other problems. But watching this first time on Sunday morning was brutal. :-)

leksu3 months ago^{+3}can you please a lot more on math?? your explanation and animation comes together so well.

Abel Tshimbalanga3 months ago^{+2}This is crazy....I especially love the analogy at the end.

I'm so glad that I took discreet mathematics to even be able to understand some of this

ilsegugioMonth agoas a mathematician, some of your videos are so neat they genuinely disturb me 😂

Magguu3 months ago^{+689}I couldn't imagine someone could do a youtube video on this topic. Complete with graphics and engaging commentary. It takes a very special level of film making skill plus top notch scientific knowledge to do such a thing. I am a phd in maths. At one time sixteen years back, i was entranced by p-adics. Used to organize student level lectures on it. Slowly my interest wore off and i moved on.

Thanks for reminding those days again.

Xyvaz Krown3 months ago^{+10}if you want maths topics with graphics and engaging commentary, I'd reccomend 3blue1brown

Barbara Houk3 months ago^{+13}Yes I have viewed 3blue 1brown on several topics. So wish Clip-Share was functioning when I was much younger. I watch now with a badly damaged brain (bacterial meningitis with 2 strokes in 2005). I follow only somewhat. I miss my capabilities prior to this event. But reaching and stretching helps loss at a slower rate.

Druskee3 months agoThe most amazing thing is that there was no film involved ;)

Magguu3 months ago^{+2}@Druskee lol ... I mean movie making, ... Writing, dialogues, camera placement, post production, editing, graphics, etc. I have great respect for people who are good at this. My efforts in this have proved to be laughing stocks.

As far as I can see, Derek is an unique person who combines movie making skill with scientific aptitude to such perfection. His videos about relativistic effects of electric current are ample proof.

SimplyKyle!3 months ago@RTA Gaming could you link/give the title of that very video?

Blood Lichen3 months ago^{+15}The more I learn about math, the more it feels like we are trying to solve a really big puzzle without even half the pieces. We are missing something, I hope we find out what it is, and whatever it is not being something we cannot comprehend.

kazedcat2 months ago^{+1}Sorry to break it to you but we already know that the pieces will never be complete. That is the whole point of Gödels incompleteness theorem. There will always be questions that cannot be answered.

Joji Joestar20 days ago@kazedcat this does not stop people from discovering new things in mathematics. It's like saying "we don't have the fuel to discover 100% of the ocean, don't bother exploring coral reefs".

kazedcat20 days ago@Joji Joestar Yes It does not stop mathematicians from exploring but it is useful to know the limits before you explore the Pacific with a rubber boat and only 3 days of supply.

Sacha Durand Saint Omer18 days agoWow, I am amazed by the quality of the content, and the presentation is top notch!

blankcomments_3 months ago^{+2}Would be interested in a video about perfectoid spaces. Great content. Keep it up!

Jesse Ferland3 months ago^{+5}The graphic representation really pulled it all together for me. It was hard to understand how we could just ignore all those places to the left.

Jon Molnar3 months ago^{+553}This is weirdly similar to the way computers (typically) encode negative integers using 2's complement notation, where ...1111 (in binary) is how you represent -1. In computers this works because you run out of bits eventually and the carry gets thrown away. That's functionally the same as the digits just going on forever, so computers are kind of using 2-adic integers. Neat!

Bernd the Brick3 months ago^{+45}I know that computers use negative numbers like that. But now I understand why it works.

xilefx3 months ago^{+17}that what I was thinking and I don't get any of this actually - happy to see my unconcious getting it

Pif de Mestre3 months ago^{+7}yes, that works the same way (although on computer you are generally limited to 32 bits or 64 bits)

Sharky3 months ago^{+11}that's what I said! this reminds me of the fast square root solution! its almost like a p-adic solution in constrained bit depth

Lightinthedark3 months ago^{+38}Who would have thought that overflow errors exist in real life 😅

Lucas Cobham2 months agoA couple of days ago, I stumbled upon a very similar concept reading about how computers (binary in nature, o 2-adic I guess) divide numbers. Just like the number 12 = 2x10^0 + 1x10^1, the number 0,12 = 1x10^-1 + 2x10^-2. And “the same” can be done in binary, placing a decimal place at the leftmost position, and then having the bits represent negative powers of two. The mind blowing discovery was that fractions that were not periodic in base 10 (like 1/10) were periodic in base 2 (and this happens since 10 isn’t a power of 2). This made me think that there could be infinitely many number systems in which the fractions that were periodic were entirely different. Fascinating

Luciano CasaroliMonth agoThere are various branches of math which at first glance, even after being shown the proof, are almost unbelievable.

Nishi2 months ago^{+1}From a CS background it's quite intuitive how we calculate 1's complement and 2's complement(negetive of a number) , i believe we work in 2 adic number system. The concept of distance between two p-adic numbers too , like most significant bits and least significant bits (the pyramid visualization helped a lot )... Great video 🙌 ....

Nirbhay Kumar Chaubey25 days agoSame here

Roccolnikov3 months agoThis is some of the best content I have ever seen on this website. Please keep it up! And thank you!

yamen razojMonth agoDerek keep going, your videos are very interesting and useful! :)

Jaswant3 months ago^{+740}As a Maths graduate, I really appreciated this being taught so well. I remember learning them for the first time and they looked so counter-intutive at that time.

KorigamiK3 months ago^{+4}If you don’t mind me asking, what do people do after they graduate with a degree in mathematics. What will your work in the job be?

Le YASEP3 months ago^{+1}Why even bother with intuition in maths ? 😀

Persona3 months ago^{+8}@KorigamiK Anything from math research to biology research to investment banking. The last pays more of course. You'll find expertise in math is desired in all sorts of places. There's even math used in the art world, like for image reconstruction.

Edward Song3 months ago^{+8}@KorigamiK If you graduate with undergraduate maths degree you can do almost any job. More importantly you will have enough background for a master in many subjects. However, if you wish to do pure maths further you will end up in academia.

Edward Song3 months ago^{+7}@Le YASEP Intuition is very important in mathematics. They are needed for understanding old mathematics and creating new mathematics.

Ze Chen3 months ago^{+4}As an average joe I understood probably less than 1% of this but the way you present makes me keep watching it 😅

NEWTON'S WORLD2 months ago^{+1}Hmmm no one is average 😅😅😅😅

FarmerGilesMonth agoMind-opening and inspiring, brilliantly articulated and demonstrated. Marvelous! 👍👏🤛

Wasseem Zaher3 months agoFantastic video. Thank you so much for your work, keep them coming.

Jakob Linowitzki26 days ago^{+1}The quality of these videos is insanely high. Thank you very much!

Akira Sendoh3 months agoOutstanding video. Thank you for the effort of simplifying this.

GameWorldRS3 months ago^{+1062}This feels eerily similar to how negative numbers are stored in computers: using 2's compliment. You sort of touched on this with the 9's compliment but basically the larger the number is the closer to zero it is from the negative side. When computer memory overflows it flips from signifying the largest possible positive value to signifying the largest possible negative value. As you increase further your values become less and less negative until you overflow again (this time for real) and get back to 0.

These p-adic numbers almost feel like we overflow infinity and go back to negatives / fractions.🤯

David Durant3 months ago^{+258}Having a computer science background I immediately thought "the structure of universe has an integer overflow problem!"

Rational Coder3 months ago^{+15}@David Durant Bro me too

GetMoGaming3 months ago^{+27}Unhandled Exception: "Mind" is missing

Sed D3 months ago^{+24}Also though this will eventual lead to 2’s complement arithmetic, but didn’t(

Bacon The Vainglorious3 months ago^{+3}Doesn’t it feel spiritual too? This sort of look at the numbers and take mods of numbers has been a numerology thing for as long as I’ve seen it. The whole thing where the final digits mean larger adjustments than the previous ones implies an inflection point somewhere maybe. Say a number that’s written out as 11111……11111. What could it be used for? Or one that’s -11111….1111. Would that even make sense? A “non dual” number, that is both positive and negative. Or what if we had a third sign other than +-, like # or something. I’m no expert but love thinking abt it

eroi bior3 months agoamazing that content like this is available for free. Not that your other videos aren't great as well, but this was something else.

Xiox Y17 hours agoYou should had put p-adic numbers in the title. This video would have been easier to find and for people like us who are starting to work with these numbers, this is a great introduction.

MR. nemo3 months agoat 25:00, the formula for converging series with abs(lambda)1, ie lambda=3 for triadics,jjj if you compute n-->infinity (1-3^n)/(1-3) => -infinity/-2. it works because you pretend the numerator in the formula is simply 1 from the start. rather, since you redefine infinity as a small thing, going inwards instead than outwards, n-->infinity (1-3^n)/(1-3) => 1/(-2).

visualizing triadics as a branching tree is cool, but since the 3's place is left, we usually would read it as something larger. that's where we started from, with decimals and 10-adics. so it's very conveniente that the tree branches are not diverging, like a tree's foliage is larger than the trunk. here, suddenly, foliages are confined in the restricting cone that is tangent to all the external edges of each floor of the fractal tri-piramid. that's why it works. to make it work, you must suddenly change the geometry of the visualisation, that was assumed from the start to be branching out, not in. in other words it worked because you betrayed the initial axioms, or you arrived at a paradoxical point where you can .after all, why go left and have the 3's places represent a smaller thing than the 1's place? it's the same as going right and adding decreasing powers of 3 positionally. it's like if you try to join two halves of a deck of playing cards, as to make it whole and all cards aligned face up, but the more you close the gap, the harder is to make them unite, the cards always end up half face up, half down. you don't understand why it doesn't work, you're trying to close the gap by approaching the two halves of the deck. then it hits you, one of the halves is reversed. what you should do to solve that is to flip the deck above and put it on the bottom one. what derek did is to paint the bottom deck's faces like the back, and paint the backs like the fronts. it artificially solves the problem by changing the question

edit: i see why the expanding series of powers of 3's would make finer adjustments. i'm just wondering if "in the world of p-adics, what we think is big is actually small" is something that is really true and was observed; or if it's a corollary side-effect, something that functionally is produced from what we have arbitrarily set as conditional. in other words, it's a measurement issue. you measure with an instrument you have. are you 1.80m tall because 1m is something in nature, or beause we decided so? what if 1m is redefined as what we now call 2m, would you be 90cm tall now?

PubliusPi3 months agoOnce again, this is the best content on the internet. Thank you for sharing with us!

Fill AshThrow'n'out3 months agoAre p-adic numbers used for calculations in quantum computers? Btw I'm very impressed how complex mathematics could be. Never heard of anything like this before. Even more impressive is that even the ancient greeks already thought about it!

greatbadger3 months ago^{+425}Fascinating topic! I am so glad it got more traction.

Fun fact, some p-addic systems have really interesting properties. For example in 5-addic the number:

…04340423140223032431212

Multiplied by itself gives:

…4444444444

Which is a representation of -1 (add 1 to it and you get 0).

This means that 5-addic system has the sqrt(-1), the imaginary unit, in it!

QuantSpazar3 months ago^{+55}This works whenever p is 1 mod 4

_3 months ago^{+15}@QuantSpazar Ho so for 5, 13, 17 etc - addic number this can happen because those mod 4 = 1 ? Really great tidbit. I didn't even though it would have been generalized already

QuantSpazar3 months ago^{+24}@_ I proved that for fun, it's very simple actually, you can check that if you have an expansion...a2a1a0 that squares to -1, by computing the first digit of the square, that a0 squares to -1 mod p, so that -1 is a square mod p (which is exactly when p is 1 mod4). Then to prove that it always work you can build (an) by induction

Just another man3 months ago^{+15}You would be interested to know about Hensel's lemma then. In p-adic situation, it would imply that for most polynomials, a root in p-adics would exist if it exists mod p. In your example, you were finding the root of x^2+1 which is possible mod 5 so a 5-adic root exists.

Theodore Astor3 months ago^{+1}@_i would encourage you to look at fermat’s theorem on the sum of two squares.

SUPREETH3 months ago^{+1}This video as always is a masterpiece

Kapchin Muan2 months agoI see, this is what I've searching for

A few back, I was trying to find out how to view mathematics as a language and then try to find connections of numers, by adding, subtracting, multiplying, squaring... etc

I'm glad I found this video, thank you VERITASIUM

Sam LeviMonth agoThe random injection of a mildly overly enthusiastic mathematician is the icing on the cake of this video. 😂. I like him. Hope to see more of him.

Shishir Upadhyaye3 months agoIt's amazing to know about p adics. I have had vision or dream about the concept of p adics when I first started calculus, which I have assumed the opposite of Infinitesimal. Indeed I was right, it does exist!

Sven PohlMonth agoSo great content time and time again. Thank you so much!

Meep3 months ago^{+232}learning about math without the pressure of college is pretty nice. i still feel completely lost after a certain point but the crushing pressure of needing to pass the class and putting stress on myself doesn't exist

BradyPostma3 months ago^{+43}I think that financial pressures have changed the college experience from an exploration of the full wonder of truth into a race to the narrow, utilitarian set of truths prescribed by the heartless needs of the employer class.

I don't want to be a machine for some owner's wealth accumulation. I want to explore the beauty of truth for its own sake.

Hassassinator3 months ago^{+10}@BradyPostma This is a better description of "escaping the matrix" than anything else I've heard.

21PMA123_ Reginald Bosco.G3 months ago^{+1}Just what I have been thinking about for a long time; Thank you for sharing your opinion to the world ❤

アlex3 months ago^{+1}The ChatGPT stuff will defiantly help in Pedagogy/Teaching even at high levels; it's like having a pocket TA.

IroAppe2 months agoThis is how university courses should be taught, at least in the beginning. Now I'm interested in p-adic numbers and also want to learn its rigorous mathematical definition. I didn't have that in university, which is why I switched courses. When you begin with the dry definition, then there is nothing in the brain to hang on to. Our brains work in networks. Information is extremely likely to be stored and accessible indefinitely, if there is a connection to other concepts that are already in the network. So introducing something completely new from the beginning, it is very unlikely to stick. University didactics should take that into account, and update their teaching networks to the "new" understandings of neuroscience. "New" in parentheses, because it has been known for many decades by now.

If you begin either with something students already know, or just show what the concept is able to do by quickly showing the process - even if students don't understand completely what is happening - you build interest. And then the brain's ready, in fact, thirsty for information input.

Lucas Castro3 months agohey Derek, could you do a video about Artificial Intelligence trying to uncover some internal pieces of language models? As i see it its something very related to your areas of knowledge and you would totally nail it!

Kevin Dadap2 months agoKudos to the animator! Very informative and satisfying :D

Tony Talks About MathMonth agoIt's hard to find good videos on p-adics. Thank you for this one!

Braden Best2 months ago^{+1}6:34 this sent me on a whole train of thought about two's and one's complement in binary math (I'm a programmer) so I realized pretty quickly that what you are describing here is a ten's complement.

one's complement: NOT n (not meaning to invert all the digits, i.e. subtract them from 1)

two's complement: NOT n + 1

Say you have a number with an unknown number of bits, but you know that the last 4 bits are 1011 and every bit to the left is a 1. Whether the number is 4, 8, 16, 32, 64-bit, or even an odd configuration like 11-bit, the number, so long as it falls in this pattern, could be interpreted as a signed integer, and its negative representation is found via two's complement. So you invert the digits, turning it into a number with all 0's ending in 0100 (4 base 10), and then adding 1 to get 5. Therefore, the number, no matter how many bits it has, is representing -5. there is also one's complement, which omits the last step, but it is wasteful because you end up with two zeros.

Take a 4-bit number, for example. In a 4-bit number, you can represent the values [0, 15] unsigned, or [-8, +7] signed (2's complement). You can test this out yourself by writing a column with all 16 permutations of 4 bits (0000, 0001, ... 1111, you can figure it out) and their signed and unsigned decimal representations in columns to make a table. Use what I told you above, like 1011 being -5, and extend the pattern until you have the positives and negatives meet in the middle. You will find that once you hit 1000, the sign bit has been flipped and you get your first negative number, -8 (which happens to be -(2^(n-1))), and that -1 is the number with all 1's right before it wraps back around to 0 (all 0's). Using 1's complement, that all 1's number is "negative zero", which is a useless value, and you lose your -8 instead getting a range [-7, +7]. This pattern extends to 8, 16, 32, 64, etc. signed 2's complement numbers also have cool properties, like the fact that you can just add them together without having to do anything special and integer overflow will ensure that you get the correct answer, meaning you can implement subtraction by just 2's complementing the number being subtracted (invert all its digits and add 1) and then adding them together. 5 - 1 can be implemented as 5 + (-1) or 5 + NOT 1 + 1, or in our 4-bit representation, 0101 + 1111 which equals 10100 (20 unsigned), but since we only have 4 bits the leftmost bit is truncated (integer overflow) and we end up with 0100, which is 4, which is indeed the correct answer to 5 - 1. Signed integers are not so cool when you're working in a programming language like C where signed overflow is undefined and breaks your programs because you can't guarantee what the compiler will do, but that's a different matter dealing with human standards and technology compatibility.

But yeah, I have some experience with p-adic numbers. 2-adic mostly.

kaisoep3 months ago^{+347}I used to hate math class in school because I didn't understand it and there was a lot of pressure from teachers to perform well. Now I'm done with school and willfully watch videos about complicated math and enjoy it so much. It is genuinely so interesting to watch these videos, even if I don't understand every single thing. My mind was blown like 20 times throughout this video and my view on math has been turned completely upside down.

SAMBHAV MIRAJGAONKAR3 months ago^{+19}You like it because you are not going to get tested.

Benjamin Goldberg3 months ago^{+10}Fun fact, children who were poorly educated in math have a tendency to become adults who make bad financial decisions. It's almost as if money is made out of numbers or something.

Repent and believe in Jesus Christ3 months ago^{+2}Repent to Jesus Christ ““Honor your father and mother”-which is the first commandment with a promise- “so that it may go well with you and that you may enjoy long life on the earth.””

Ephesians 6:2-3 NIV

K

Logan Vollmin3 months ago^{+13}@Repent and believe in Jesus Christ Bot

Art Vandelay2 months agoI understood very little but it was super interesting nonetheless. Quite fascinating. Also: I wonder how fermat actually solved the problem, since he didn't use p-adic numbers (or did i misunderstand?).

Entropie -2 months agoIt seems likely that he made a mistake and the argument he envisioned would not have worked.

Kre CV3 months agoKeep the great job Tom!!

Oskar von Heideken2 months agoThis has a lot of similarities to binary math (2 is a prime so..). But this has a lot of parallels in e.g. fixed point notation for binary numbers, which is kind of cool. Only difference I guess is that we don't have an infinite number of bits

LostButMakingGoodTime3 months agoDerek, I always enjoy your content and your presentation skills, no matter where the subject takes us. In this case I understood every single word you said, and the illustrations and examples seemed straightforward and logical. But for the first time, I don’t have the slightest clue what you’re talking about or what it all means. But thanks! I think. 🤣🤣🤣🤣