Mathematicians Use Numbers Differently From The Rest of Us
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- Опубликовано: 5 июн 2023
- 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:
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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
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
@@leeroyjenkins0 revanced youtube moment, ads blocked, sponsored segments skpped automatically
@@geniuz4093 yessir
It's all thanks to Patreons and sponsors
@@leeroyjenkins0 I'm not paying time with time. Ads are not for everyone just like adblockers.
@@leeroyjenkins0 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.
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!
absolutely
Waiting for 3B1B to pop up somewhere
Like half of his videos are math related lol
Next-up I want him to look at parker square.
Yeah...not math RUclipsr to me, but a sleep helper RUclipsr. 😂
0:24
"so, does this pattern continue?"
me immediatelly:
"patterns fool ya, paterns fool ya, ..."
3b1b reference
can confirm
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
Its interesting to think of lack of depth as an insult. Why would this be so?
@@SOC- 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.
@@andy07070 yea I enjoy the depth as well, especially now as it is so easy to use A.I to make content.
what a mature comment section
Yo Andy, remember me?
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.
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.
@@bryan200023 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
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-.
How can you learn more from a video on RUclips than a graduate course?
@@josephvanname3377 "...can you learn more from a video on RUclips than a graduate course?"
Yes. There is a difference between studying and learning.
I can’t focus for 5 mins at school but can watch a full 30 minutes video from you no problem
Lol
This is something you choose to do, school isn't
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 ;-;
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
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 .
Bravo! You covered in 30 minutes what took me semesters to master in my youth. I am totally inspired.
You mastered this in your youth? You're a genius!
I'd heard of p-adic numbers and was vaguely familiar with their definition, but didn't know much about their motivation or applications. After watching your excellent video, I'm motivated to learn more about them.
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.
It makes you feel like your learning but it's basically entertainment because you will forget anything about it the next day.
@@trout3685 well with my adhd, I essentially forget the previous sentence because I’m having such a hard time following.
@@CyclingGeo Do you remember leaving either of these comments several hours later? I'm here to remind your brain
@@CyclingGeo 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.
yeah because you probably finished 12 years of school and didn't get shot in the process
Thanks for bringing this amazing topic to us
übernyusiiiiii
Nice video
👌👌👌
Very nice
@@neelamopm 15:50
The quality of these videos is insanely high. Thank you very much!
Derek, you have literally been the person teaching me the most since I found RUclips. 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.
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.
I don't know what income 2.2m views on youtube achieves... do you?
the video editing has been done by AI tho...?
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
@@gwynsea8162 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.
its done by python software named "manim". i also made these type of videos to teach my students basic operations.
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.
I see the connection, but they are finite in length... So how does it work?
@@pranavps851 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.
@@griffinkimberly7695 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.
@@griffinkimberly7695 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?
@@griffinkimberly7695 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..
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.)
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.
My Granddad used to play P-adics numbers game with me. He started by asking me to write any random numbers before decimals and he used to write his random numbers below them, And sum of them always comes Zero. His techniques and methodology amazed me and fascinated to learn More Math. Miss you Granddad ! And Thank you Veritasium for this Video
That's a brilliant way to inspire an early fascination! I'll have to remember this one
May he rest in peace
honestly W grandad
Your grandparents may be cool, but they will never be teaching-p-adic-numbers-as-a-game cool 🤯
he was reincarnated as veritasium
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.
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.
Subtract the subject matter from my attention span and you get a p-atic number
@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.
Wait 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?
@@leif1075 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.
A better trick for converting decimal to any other base is by doing this
Divide 17 by 3, the remainder is 2 and the divisor is 5, now divide the divisor by 3, the remainder is 2 and the new divisor is 1. The process should be repeated till the newest divisor is less than the new base. Now arrange the remainders in the order of the first remainder in one's place, the second in the tens place and so on. Here the number becomes 122.
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
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.
if you want maths topics with graphics and engaging commentary, I'd reccomend 3blue1brown
Yes I have viewed 3blue 1brown on several topics. So wish RUclips 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.
The most amazing thing is that there was no film involved ;)
@@drewendly89 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.
@@rtagaming7663 could you link/give the title of that very video?
I'm jealous that Derek gets a personal lecture from such an amazing mathematician.
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.
@@lonestarr1490 Now, he'll be jealous of you too:)
An operation that only works in base 10 is not Mathematics it's just Arithmetic.
we are too since he's credited as coauthor of the video :)
@@nitinsharma7947 That was the intention behind telling him, lol.
Me and the boys finding the last digit of Pi by calculating it backwards:
My brain’s gonna be sore tomorrow
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!
Me too!
Don't want to be negative but as a computer scientist it should all add up...
@@raylopez99 I _think_ that was a pun?... :)
@@raylopez99 nice
@@raylopez99 Intended pun?
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
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.
@@BradyPostma This is a better description of "escaping the matrix" than anything else I've heard.
Just what I have been thinking about for a long time; Thank you for sharing your opinion to the world ❤
The ChatGPT stuff will defiantly help in Pedagogy/Teaching even at high levels; it's like having a pocket TA.
Best explanation of p-adic numbers I've ever seen. I never got it until now. Thank you so much! :)
I am taking Math 105 for teaching Math to Elementary and Middle School students and this video touched and reinforced so many concepts I have learned these last few weeks. It was exciting seeing how they are implemented.
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!
I know that computers use negative numbers like that. But now I understand why it works.
that what I was thinking and I don't get any of this actually - happy to see my unconcious getting it
yes, that works the same way (although on computer you are generally limited to 32 bits or 64 bits)
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
Who would have thought that overflow errors exist in real life 😅
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!
This works whenever p is 1 mod 4
@@quantspazar6731 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
@@user-rx3ny9ji8i 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
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.
@@user-rx3ny9ji8ii would encourage you to look at fermat’s theorem on the sum of two squares.
I watched this video 2 times and I got to say that spending some time on Brillient really helped.
The 1st time I didn't really get the arithmetic rules and methods of the 10-adics and p-adics.
Thanks to courses like "number-bases" I understood much better the 2nd time.
This video is fantastic. I am very impressed with your math videos Derek, the topics are so well-presented.
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.🤯
Having a computer science background I immediately thought "the structure of universe has an integer overflow problem!"
@@cholten99 Bro me too
Unhandled Exception: "Mind" is missing
Also though this will eventual lead to 2’s complement arithmetic, but didn’t(
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
By the way, if you're wondering about the solution that comes from (2, 0), you end up getting ...111112, which is just our first solution, -1/2, plus 1, which is just 1/2. Since all of the xs in the equation are raised to an even power, this solution works in about the same way as the one shown in the video.
Oohhh I was wondering about the (2,0) one. Thanks!
Yeah, I sort of figured it's the complement of (1,0) given that we're working with squares and 1 and 2 are complements in base 3, but it's nice to see a confirmation of that.
So are those the only rational solutions to the equation in question? I assume not (or unknown).
@@lonestarr1490 I would assume that if you use other values for p (5, 7, 11, ...) you get other (possibly infinite) solutions.
OK, so upon expanding the full equation, the only part that doesn't get removed due to the modulus is 2mnx + n^8 + n^4 + n^2 = 0 mod 3m (where the original equation is rewritten in terms like (n+mx)^2,4,8). Unfortunately, it seems that the reason why this formula always works when x is equal to one is very complex, since it seems to depend on how the specific forms of n (like 1, 4, 13) and m (the powers of 3) interact with the modulus. Fortunately, I think that equation that we get for all of the digits (after the first) only has one solution (it's a linear equation, and the coefficient of 2mn means that it only "loops" through the modulus once x goes over 3 and becomes too large to make sense), so it's probable that these are the only solutions (besides 0).
A lot of loves for Veritasium. Thanks a lot for making such a complex and important concept very easy to digest.
Thanks for a reasonably accessible description of the p-adic numbers!
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.
That sounds so cool
@@_-FreePalestine-_ 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
@@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.
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
@@aaronfactor6838 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?
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.
You like it because you are not going to get tested.
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 Christ Bot
@Repent and believe in Jesus Christ no thanks. I'd prefer my children not being raped.
@@BenjaminGoldberg1I best guess is that they use their money for wants and not needs. They don't compute their money so they just buy and buy till they discover that they're too late to pay their debt
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.
I 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
I'm a computer engineer. Along the duration of the video I started to relate this first to signed integer arithmetic (2's complement). After that, I heard "Fermat" and I immediately knew this was going to be about modular arithmetic and discrete mathematics (both very useful for cryptography). And finally, fractals (Sierpinski), which is also quite useful in CS.
I have used all that math, but I didn't realise it in the beginning until the video progressed. They taught me "just use this to calculate that", but I really had no idea what these tools really were.
Yeah, it's a bit shame though he didn't mention that every modern computer uses 2-addics to save any number...
Fermat, I assume you mean?
@@qj0n That's not true. P-adic numbers extend toward infinite. Computers use finite values. There is a similarity because p-adic numbers are mod p^inf, while computers store numbers mod 2^bitwidth
Yes, 2's complement is what I thought of, but this went into 3, 5, 7 - whatever complement, which is the p-adics.
@repentandbelieveinJesusChrist8 in electrical engineering, there is no god
I did my master thesis on the p-adic gamma function. I expect to see p-adics on 3Blue1Brown's channel but not here, so it was a pleasant surprise. Interesting how you postponed the metric, which is how we traditionally start, and first showed us some actual number theory problems that can be solved with p-adics.
The visualisations are top notch. The p-adic space is so counterintuitive that it dearly needs such representations to stick into one's mind. Well done!
I think it's a problem of mathematical education that we weigh people in definitions without motivation, but it's challenging because its so hard to motivate a problem when you don't already have a complete definition. Teaching someone is like trying to build a ship in a bottle-- you have to assemble the idea in their mind through a narrow opening. The best techniques to teach an idea are themselves breakthroughs, and we have far more things to teach than just those things that we know how to teach well.
Beautiful video! I need to say that for the first time I had problems visualising the p-adic numbers.
What an incredible introduction to p-adic numbers!!!! Thank you for your contribution!
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.
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?
Why even bother with intuition in maths ? 😀
@@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.
@@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.
@@leyasep5919 Intuition is very important in mathematics. They are needed for understanding old mathematics and creating new mathematics.
I was deeply fascinated with maths in my younger days, a subject I excelled in and genuinely loved. I went on to become an engineer and now I build software for a living. But every time I come across videos like these, there's this regret, making me wonder why I ever left the beauty of mathematics behind :')
Thanks man for making these videos
It's never too late! I'm in school right now so that once I have my degree I can take classes that interest me. Some of those are going to be mathematics classes.
As a software engineer, I agree with you. Sometimes I want to just whip out the pen and paper and start refreshing on calculus and go into these deeper concepts and ditch my depression generator machine.
@@rinzler_d_vicky eu queria exatamente o contrário. Deixar esses números de lado e ter um emprego como engenheira de software
capitalism is why
@@thewhitefalcon8539 at least in my country, math majors have one of the highest income expectations
I’ve never seen math explained this way. This is like beautiful art. Amazing and mind blowing! Thank you very much for sharing.
started 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
_I love the way Derek puts a decimal point to the right of the numbers he's showing. That tiny visual cue is worth its weight in gold when it comes to showing intent and aiding understanding._
It's more of a radix or fractional point since he uses more bases than just decimal.
@@Brauljo I think he stops using it once he moves away from 10-adic (base-10) numbers.
@@IhabFahmy Virtually every base is base 10, decimal is base ten.
@@Brauljo no only base 10 is base 10. what r u saying lol
@@Scotty-vs4lf All bases are base one zero.
I was introduced to the concept of p-adic through Greg Egan's science fiction novel "3-adica". He used the analogy of three nests packed together, each with three smaller nests in an infinite series, which gave me an intuitive feeling, but I did not understand the arithmetic meaning behind it. It wasn't until 25 minutes into this video that it hit me, and I suddenly made the connection in my head between the infinite nest picture and the divergence of the infinite series "converging" in a finite value. Then a few seconds later, the infinite cylinder appeared almost exactly as I had just imagined. Words cannot describe how I felt at that moment, it was wonderful.
Eureka moment?
same lol
Yeah, when I read that novel I also didn't understand what's the author talking about.
Egan usually has that effect on me. Finally understanding the weird science that foregrounds one of his stories is a rare treat.
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
Type of roller coaster is this video. Bro had literally had me going like I have no idea whats going on, and then BOOM the computer science nerd in me remembered the counting systems like binary and then it clicked and broke my brain. I love this sort of dopamine hit!
I have a BS in math and computer science, and in one video you found a point of intersection for the last 20+ years of my working life. Awesome.
isn't cs a subset of math
@@play005517😂😂
As fields yes, as origins yes off course(this video with 2 adic and 10adic), as careers not necessarily correct but the deeper u go the more math u need.
@@nicbajitogaming8947 Is it pure math or does CS cover some aspects of the physics also? When it comes to building a computer math is the goal and physics is the method so surely the science of computation is the interface of these fields. Hell these days even classical computers need to factor quantum mechanics into their design because we are constructing systems at a near quantum scale and trying to engineer a product that behaves consistently with classical mechanics.
@@leeroyjenkins0 Ye, developers would be in maths for you. Still exists exception on computer science that dont need math or more math than school.
@@play005517 Yes
The style of explaining such a complex topic with amazing animation is mind blowing. Kudos to Derek and the Veretasium team
I've seen about 10-15 videos on p-adics on youtube but this is the only one that built an intuition for them by starting with a problem
2:37 this is actually similar to the decimal expansion of 1/7 just with a difference of the last digit which is 3.
This is quite possibly the best explanation of p-adics that has ever been given. Amazingly done!
At around 6:13 I thought
'....wait i know this, this is how negative integers get represented in binary!'
it feels awesome actually seeing the buildup and knowing how this particular math gets used in real life beforehand
Ah I thought it sounded familiar
Except that’s not how negative numbers are represented in binary. Computers read the first bit of a binary number as the sign. A first bit of 0 is a positive number and a first bit of 1 is a negative number. Also it would be a bit difficult to store infinitely long numbers in a computer.
Edit: I’m wrong don’t mind me, apart from the first bit computers indeed use a similar fashion to store negative numbers.
@@dagmarski4133 Negative integers are usually represented in a very similar fashion to what is shown in the video, look up "Two's complement". Of course you can't store infinitely many digits, but the rough idea is the same.
@@dagmarski4133 it is how signed binary digits are stored, it's known as 2s complement, and is specifically used for that thing of the fact subtracting is the same as adding the negative value.
Negative one in binary is 11111111 for however many bits you store a number in, using the fact that you can ignore the digits higher than your highest bit to avoid the fact it's not infinitely long.
@@dagmarski4133It's not exactly how, yes, but it's very similar which is what they meant. Also who doesn't love the fact that you can add 'negative numbers' to get the answer
Absolutely fascinating! I enjoyed watching this very much. Thank you for this great content.
I've watched this now 4 times I think, and I might almost be starting to understand it. It's not for any inadequacy in the explanation - far from it - it's that the concept is so foreign it's taking a very long time to sink in. Love it!
I knew p-adics ever I watched a few other videos on the subject. If you are quite intelligent in mathematics (as am I), learning things related to math might be easier than other subjects.
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.
Apart from the math, I LOVE your visual style! Simplistic but smooth animations at a low frame rate are eye candy. Amazing video Derek, you managed to blow my mind once again. ❤
It's amazing how smooth animation makes the ideas feel less intimidating.
@@BradyPostma that's very true
The low framerate bugs me off lol!
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.
Something that fascinates me about math is how there's such a tight connection between people from thousands of years ago and people from today. We are still working on the same exact problems. Math is universal across culture, space and time.
If you’ve ever read The Themis Files by Sylvia Day, this premise is a big chunk of the first book. That every species, including aliens, have to use math.
In my first couple of years of high school I thought prime numbers were the stuff of ancient Greek mathematics and too frivolous for modern research. I was so happy to find out I was wrong!
@@ronald3836 They kinda are a bit frivolous, though. But that makes them even more intriguing.
Math is discovered, not invented. The universal language.
@@lonestarr1490 They are essential to modern cryptography, upon which all secure communication and secure electronic transactions rely. So all global spycraft, military secrecy and internet commerce depend on the primes. Kind of takes the frivolity right out of it.
A perfect teacher can understand the pain points in the learning process and then patiently clears them out to build the intuition of learner. Great work Derek Muller.
Couldn't agree more
I just successfully followed along with a video of a mind blowing type of math I’ve never heard of before. That is an amazing job
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.
Your math videos really are my most favorite ones.
Maybe it's just something about the aesthetic of your explanation and your way of breaking down the topic bit by bit and explaining it patiently at a pace good enough to make it really enjoyable and interesting :) ∆
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@@Tmharbesoujj Yeah it's really insane...
@@artlover7770 4 big guys
He explains it in the same style that you would if you were majoring in this stuff. He doesn't dumb things down just to give you an illusion of understanding.
This is why I enjoy his videos even though I am already familiar with the topics due to my major being physics and I also took a ton of math courses that usually only math majors take
yep hes so good at making maths digestible for people who arent even that fluent
did lose my attention when the other guy started talking/drawing like this is khan academy, veritasium quality is just better idk why he went down that route
These math + history videos continue to be your very best work. Thank you - I learned a lot!
This video is an excellent insight into how my mind works. Please continue making such beautiful videos about numbers.
Derek keep going, your videos are very interesting and useful! :)
As a designer and animator, I am incredibly impressed with your use of animation to enhance your storytelling. It’s only gotten better over the years. Well done!!
I agree! I have to say that the little specks are very distracting, however.
Thank you for not shying away from the nitty gritty calculations. Watching many of the pop science channels feels like eating a nothing burger. I don't know what I've learnt by the end of it. This video was not like that. It introduced me to a whole new concept in enough detail that I feel confident going in and researching further.
I love how the far end of infinity wraps back around to zero and negative numbers. Thanks for the mind-blowing video!
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.
Derek: That number is its own square.
0 and 1: Look what they have to do to mimic a fraction of our power
I love that this cutting edge mathematical concept can be entirely explained using high-school level algebra. Utterly fascinating how far you can get with seemingly "easy" and "limited" tools.
That's how math and nearly every field of science works. You need basics and simple aspects first.
@@itsgonnabeanaurfrommeThats completely true ! 💯
It isn't really cutting edge, you learn this in early number theory classes. However it is one of many pillars modern mathematics stands on. A portal into a very different but familiar universe
So "cutting edge" that it's been around for over a 100 years and is taught to undergraduates
Nothing you wrote in this blurb is accurate.
It's hard to find good videos on p-adics. Thank you for this one!
I couldn't follow it all. But it was still fascinating, and I can still appreciate how well the concept was taught.
The reference with the stars and the sun is actually a very nice analogy to Ostrowski's theorem! One way we define real numbers is by completing the rationals under the distance norm we are already used to. The theorem roughly says that the only other ways we can complete the rationals is by completing under the p-adic norms. So in the analogy, each star represents a prime number and the sun represents what mathematicians call "the prime at infinity" !
"How it feels to invent Math" by 3Blue1Brown already blew my mind, and this went even further 🤯 Great video, Derek!
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.
So great content time and time again. Thank you so much!
In computer science we have 2's complent.
There all ones is equal to minus one.
The calculation is similar.
To convert from a positive number to a negative number:
Invert all the bit and add "one" to th result.
I'm kind of amazed that I've never seen this sort of thing before.
I've heard of p-adic numbers and some of these related ideas in discussions around Fermat's last theorem, but I never saw an explanation for how they work.
This was explained in a way that made it very approachable, and helped open my mind to a whole other way of representing and working with numbers.
Something amazing that wasn't covered is that you can write sqrt(-1) as an actual written number in these number systems. I studied a simple version of this in a second year university paper which looked at topics normally studied in later years, and I've often thought the topics would be interesting to more than just math nerds.
Fantastic, it's something incredible that you have found, and it has allowed me to find a mathematical explanation
for a hypothesis of mine, namely that the "real" world operates entirely on Mod 2 and that Mod 1, adopted up to now,
is correct but not in all situations, Mod 2, on the other hand, could potentially explain a lot of things
27:00
"But if they differ in the 27th place, we say that they differ not by 27, but by 1 over 27."
Seems to be a mistake. They would differ by 1/3^27.
If they differ in the 3rd place, they would differ by 1/3^3 = 1/27.
interestingly, this actually gives a really useful perspective on how computers store numbers! almost every modern computer stores negative numbers as two's complement, which makes it relatively easy to change between negative and positive numbers, and allows you to use the same kinds of addition on them.
Indeed… yet I wish that computers today would make further usage of p-adic numbers, beyond the positive/negative complementarity.
I noticed that too.
Actually, one can argue the connection is even stronger - if you add two integers which are too big for the amount of memory you have, you get integer overflow, where the most significant digit is lost. For example, if you can only remember 2 binary digits and you try to do 11+11, you get 110, but you forget the leftmost 1 and get a 10.
In the reals, this sounds really bad, since you are losing the most important digit. But in the 2-adics, this makes perfect sense, since really 10 and 110 are pretty close.
So not only is 2's complement exactly how negatives work in the 2-adics, one could argue that computations with fixed precision integers are in general just fixed precision 2-adic computations, where you only keep some agreed upon number of digits before the decimal place (which in the p-adics has the same meaning as after the decimal point for the reals).
5 years of computer science, can confirm this was a trip down memory lane to the valley of the shadow of death (my first programming class was doing math in different bases for 3 months).
Yes! The 2-adics are ∞-bit integers.
I'm a master's student in mathematics and p-adic numbers is more or less my specialty. It's nice to look at it through fresh eyes because sometimes I forget how incredibly non-trivial these constructions are.
3:22 So, *_THAT’S,_* why 1/7 (and any a/7, where a ≠ 7k, where k is an integer) gives that particular decimal expansion 🤔.
Wow. Just when you thought you could count, along comes this and blows your brain. Full marks for actually managing to explain this.
I find it also interesting that our computers do negative numbers by exploiting a form of 2-adic numbers with a set maximum length.
In effect the modulo is built right into them. The term used in the field is 2's-complement signed numbers.
p-adic numbers are such an intriguing topic, thank you for covering them!
I came here to mention 2-s complement math as well
Nice I was going to mention 2s compliment as well
Some older BCD computers and calculators used 9's or 10's complement representations, like his 10-adic examples.
“This feels even crazier than negative numbers or square roots of negative number” “Thats cause they’re less familiar” I love this quote
This video enhanced my understanding of the two’s complement binary representation of negative numbers at a deeper level.
Mind-opening and inspiring, brilliantly articulated and demonstrated. Marvelous! 👍👏🤛
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.
I’m going to go into calculus for both of us, bro, wish me luck
@@tquasa07good luck👍👍😃, get good grades there.
As a number theorist myself, this was quite a good exposition. Excellent job!
That just sounds so cool as a profession
I admire your work Sir. I have seen your articles.
@@damondeleon5115 What do you mean, "at the front"? The numbers agree to arbitrarily many digits from right to left (simply do the standard multiplication step enough times). Since we can make these numbers arbitrarily close just by doing more operations, we say that they are the same.
You can think of the new number as the limit (in the calculus sense) of doing more and multiplication steps as the number of steps tends to infinity.
@@damondeleon5115 without going into all the details here, every "infinite" calculation done in the video can be made rigorous and precise using limits and some ideas from calculus.
Wow!
Thanks for bringing this very interesting topic to my attention and this in such a fascinating way!
Wow!
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…
If you're an embedded software developer, then either you avoid negative numbers like the plague, or you should already be intimately familiar with ...11111 = -1 in the 2 adic integers. Most programmers just call it "2's complement" instead.
What you might be less familiar with is that 0x...AAAAB, or in 8-bits 171 = -85, * 3 = 1. Most programmers would probably call this "overflow", but the 2-adic integers would genuinely call this "one third."
28:55 "Just cause it's less familiar"
Brilliant