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 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 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-.
@@josephvanname3377 the graduate course instructor and his textbook didn’t provide motivation and the geometric insight so I disliked the material back then. That’s why I didn’t pursue it anymore. It would have been better if it was presented like this.
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.
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.
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.
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
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!
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.
@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.
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.
@@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..
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.
Yes I have viewed 3blue 1brown on several topics. So wish UA-cam 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.
@@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.
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!
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
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.
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.
@@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.
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!
@@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
@@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.
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.
So much of college is about time and financial constraints. You only have so much time to teach/take a class when everyone needs to be in the same room, and the instructor/assistants only have so much time to grade. Students need relatively quick feedback on their work, which isn't always easy to give when dealing with complicated problems. Life, however, almost always comes with time constraints attached, so imposing _some_ structure is a must. Financially, it costs more than ever to go to college as income inequality rises. I think community colleges and a hybrid online/in-person model will be the future. The UK has had the Open University for decades, but the USA has been very slow to catch on. My personal preference would be to have standardized tests for various subjects, then study for them in groups at my own pace: this is how professional certification works, for example. You have to show mastery of all of the material at test time, so you have to study quickly enough to keep most of it fresh in your mind.
3:30 Nope, I can’t let my concept of reality break anymore. I’m sure this is a great video and maybe I’ll eventually watch it, but today I just want to pretend the world is real.
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.
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.
@@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 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.
@@_-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?
@@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.
This video blew my mind. As a middle-school French student, I haven't seen this type of math before, and I'm not sure to totally understand what is going on with that... But this video is amazing. The explanations feels simple, but are keys to comprehend much more complex problem. Great work, Veritasium ! 😀
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!
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.🤯
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.
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.
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).
I not only finally understand p-adics a little bit but I understand much more now the astonishing leaps of thought that it took to come up with this stuff. Yet another absolutely fantastic video, Derek!
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.
@@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.
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.
@@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
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.
_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._
@@sethmobit type in Terrence Howard math Neil degrasse Tyson. It's a thing. This video will only confuse him more. (Hopefully not, but it would be funny to see him react to this video lol)
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.
@@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.
@@nicbajito 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.
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.
I love P-adic numbers, it’s such a crazy concept, bends reality, or maybe just bends our minds, how does adding infinite digits to infinite digits equal zero, crazy!
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.
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.
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.
This is basically whats found in binary, when you add 1 to the biggest number represented by U2 system, you get the smallest negative number represented by U2
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. ❤
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.
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 :) ∆
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
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!!
Thank you Derek and everyone that contributes to Veritasium. A shining beacon of light in the form of rationality, science education and the history that comes with it. I hope this channel and it's videos remain as fun and interesting to make for you all as they are fun for us to watch!
Yes, in the standard decimal numbers, 0.3333... = 1/3. Don't forget the three dots at the right, because 0.33333 is certainly _not equal_ to 1/3. However, in the 10-adic numbers, ...33333 is equal to -1/3, and 1/3 is equal to ...66667.
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!
@@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.
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.
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).
Veritasium explains mathematics even better than channels that are dedicated to maths and exam tips. He introduces us with the history, proof, experts, pleasant background score, great visualization and definitely Derek's great skill of educating. Keep going on, sir.
Channels dedicated to maths don't need to motivate people with history, presentation or a score because they know their viewers are inherently interested in the mathematics. This isn't the case with Veritasium
@@epicmarschmallow5049 Well that makes him even better than them in a way. Doing extra factors good make the overall explanation amazing. This way, people less interested in maths, get interested towards it. In the end, I think the golden gift of maths should be enjoyed by everyone. And Derek's presentation succeeds in it.
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’ve only completed up to 12th grade math classes, but I’ve always been extremely fascinated with math theorems and rules that I’ve never heard of. It literally fuels my brain to learn about the ever-growing world of numbers.
Take classes at a community college if your interested, higher level maths are more theoretical if you apply them. Discreet math, and linear algebra include mostly simple concepts yet are used across computer science
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.
@@KrisTheGreatest Which hasn't been physically proven or has any concrete proof or evidence. The simulation theory is precisely a theory and nothing more.
Wow, fantastic video! Although I have a PhD in number theory, I never really understood p-adic numbers that well, but this video clarified them for me tremendously! They're very counterintuitive, yet very useful!
I love how there’s always more to learn in math, from p-adic numbers, to space time, to how imaginary numbers were created, thank you for always coming up with more amazing, fascinating, inspiring, interesting content❤❤❤
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" !
@@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.
@@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.
These have been a pet favorite thing of mine since they made a brief appearance on numberphile in 2015. Freshman year of college we got to do an open presentation topic, and I chose p-adics, so seeing new resources pop up to make it more approachable and common knowledge fills me with warmth. This is an interesting approach! Starting with the magic and then sort of letting the details come out of the mechanics. I started by explaining how Cauchy completeness is used to define real numbers, and then the bit about the p-adic norm, and then onto all those magical properties like fractions and negatives and diophantine equations and FLT and openness and so on in decreasing specificity, as you would expect from a math lecture. Hopefully this inspires follow-up, if not from you then from fans. They're really all introductory videos 😭
As a maths student, we don't learn about the p-adic numbers in any course, so I found this really interesting. Somehow you managed to explain this in a way that normal people can understand and yet is very interesting to me as a math student, Amazing!
P-adics are too exotic! It's only relatively recently we're actually using them to do mathematics. It's arguably the modern equivalent of calculus. Similar to what calculus was in the Renaissance
I’m not a genus and failed school because I never went still have a good job tho and for me I understood this very well and very interesting I wish I studied math
There's actually something very similar to p-adics in fractal geometry. If you define an iterated function system (IFS) of n functions then you can refer to anywhere in the fractal by either a finite or infinite base n number called the address. Then you can determine the distance between addresses by defining a metric on the codespace of addresses. There's quite a bit more to it but if your interested Michael Barnsley's Fractals Everywhere is fantastic book on fractal geometry and chaos theory.
hi I’m also still an undergrad maths student but I’m very lucky that one of my professors is actually researching padics so in the lecture about algebraic number theory we also talked about them, as said in the video they have some really interesting applications. I have to say that the video might be a better introduction into the topic than anything I heard from a professor yet
You don’t learn about them in any undergraduate course? I learned about then in the first algebra course (Algebra I), we didn’t learn a lot about them at that point, but at least we were made aware of their existence at the very beginning
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.
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!
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
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
06:07 - Its behavior is similar to computer programs limited by minimum and maximum values. For example, in an 8-bit signed number, the range is from -128 to 127. This is because -128 is represented by 11111111 and 127 by 01111111. This concept is similar to the idea of an infinite number of 9s.
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 just wanna say you've managed to rekindle a bit of interest in math in me that I didn't expect. I grew up really interested in a lot of nerdy things like this, but over time I lost touch with that. Maybe I'm just feeling a certain way right now, but I want to thank you for this video.
One of your best videos ever! I’ve tried to understand p-adics many times and hit a brick wall of complexity. Your video makes it accessible to anyone who remembers at least some high school algebra. Amazing stuff!
As someone who works with hyperreal numbers and baseless 10s, p-adic numbers feel like the opposite of that, and for me, trying to understand them is like trying to read an undiscovered language with no linguistics skills. Edit: i never got why the "p" in p-adics was specifically a prime until this video. Many thanks!
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.)
In recent years, there were more and more videos of p-adic numbers from 3b1b, Eric Rowland, Numberphiles and now Veritasium. Each videos give a unique perspective of p-adic numbers which are very mesmerizing. If you are interested in this subject, I recommend you to watch all of these videos.
I don't think I've ever been blown away by something so small ever. I mean, I remember learning about the scales of the universe and The Poincare (excuse the spelling) theorems and structures but this.... Goodness gracious! Thank you, Derek. I'm gonna be watching this a couple more times. You nailed it.
Derek, you have literally been the person teaching me the most since I found UA-cam. 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.
I’m 15 and currently sitting my GCSEs (SATs) if you are American, and I love learning maths and wish to pursue it further. Videos like this inspire me to do that and though they can be deeply confusing for someone as inexperienced as me, they make me want to learn more and kelp me understand simpler concepts easier. Thank you for challenging my understanding so much!
I always find the name SATs for GCSEs annoying, because the UK has SATs 5 years before GCSEs, and actually pronounced as a word (sats) rather than Americans pronouncing each one individually (S A Ts)
your animators are doing gods work, hope you pay them well :) tell them they did a great job on this video like usual. I have shared your videos with my less science inclined friends and they keep talking about how much they enjoy them and how their brain has been "hurting" keep up the great work, love what you do and am very happy you get to do it
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.
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
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.
That proof at 5:00 somewhat misses to mention its axiomatic foundation. It makes some implicit assumptions concerning the characteristics of arithmetic operations of such infinite digit strings. We can define "real numbers" as infinite digit strings with only finitely many digits left of the decimal point. We can then define the arithmetic operations similar to the way in which it is done in the video, by first using only finitely many digits of the operands and showing that the digits of the result don't change any more at some point if more and more digits of the operands are involved. We would still have to prove that the laws of associativity and distributivity are really valid if we use this kind of definition. Which might be a bit cumbersome for this definition, and it is certainly not done in schools when they present this "proof". Actually, we find out that associativity is _not_ guaranteed if we don't identify some differing strings, like 0.999... and 1.000... Using the standard adding algorithm, we have 0.999... + 0.999... = 1.999... and 1.999... - 0.999... = 1 "Standard" meaning that we use the same number of digits of both operands for our intermediate results. And the law of associativity tells us: (0.999... + 0.999...) - 0.999... = 1.999... - 0.999... = 1 = 0.999... + (0.999... - 0.999...) = 0.999... + 0 = 0.999... Using the concept of "limits" (like you are proposing) is more straight forward in a way, as the identity 0.999... = 1 is conspicuous. Unfortunately, very many people who had quit mathematics after grade 10 don't have the slightest clue about this (and the other ones have forgotten). So popular-science articles and videos usually hesitate to talk about "limits".
I like this explanation: "In math, we define the sum of an infinite series equal to the number that it approaches, but never reaches. No matter how many 9s you write after that decimal point, you'll never reach 1. But since you are converging on the limit of 1, we say that the expression is equal to 1. That's just how limits are defined; deal with it. "
@@Strakester _"we define the sum of an infinite series equal to the number that it approaches, but never reaches"_ That's an unsuitable restriction, because we want to have series containing an end sequence of only zeroes to converge as well. Or series like 1 - 1 + 1/2 - 1/2 + 1/4 - 1/4 + 1/8 - 1/8.... This last one reaches the limit intermittently. Also, we have to define the concept of a limit and "convergence" thoroughly. Many students have difficulties understanding this in school. Apart from that, I think it's desirable to have a merely algebraic definition of "decimal numbers" and their operations.
This is the first time I had my mind absolutely blown by a math concept in quite a while. That's crazy that you can multiply a number of infinite length by a positive integer greater than one and not only get a rational result, but one that is less than the sum of its parts.
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.
This video blew my mind! I've seen other explanations of the p-adic numbers, but this is the one that really made the concept "click" for me. Keep up the great work!
If this video was made when I was 15 I would have started studying math a decade ago rather than a year ago like I did. Literally all of my largest brain-scratchiest math interests all rolled up into one beautiful video.
My theory for Fermat actually has always been that he had a notion of the necessary tools to prove his conjecture as a theorem, but, didn't know where to start as the tools that were needed hadn't been developed yet. So he put the comment in the margin as an 'get back to the latter' thing Fermat, like Gauss and Noether, was truly inspired to the degree that few could even imagine.
I believe for these people, pondering math such as this was their entertainment. If a person didn't like violent games (ie Colosseum), but did like learning, one had to imagine one's entertainment.
@Divergent_Integral I think whether or not it is satisfying is subjective. It was probably deeply satisfying to Andrew Wiles . Moreover, it gives extra value to things like the p-adic numbers when you hear that they were involved in that massive proof.
I think I heard from a podcast that Fermat had ideas for the proof for n=3 and n=4 and as the podcast conjectured, maybe he figured that the same methods could be applied again and again. Maybe he also thought that there was always some way to rewrite n where he could apply similar methods to components of the problem, maybe he had some intuitive feel for some complicated fractal proof structure or something.
Also naming new things would need an explanation with it and name convention wasnt that globalize. (Or even preventing that someone else came with the idea from another country at the same time as it happened 😂 could be anotjer layer)
Or he thought he had a proof and later realized it's not working out. I think many people overthink his comment and want to see the greater story arch when there probably isn't one.
Very informative, packed with information that can be connected to High School Mathematics. Not only that, It also provide that many things are not crazy, we're just not used to it. Thanks!
I love when you make a pure math episode and I love that I knew the voice of your mathematician firend before we saw his face. He is so good in these videos. His enthusiasm is infectious.
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 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-.
@@josephvanname3377 "...can you learn more from a video on UA-cam than a graduate course?"
Yes. There is a difference between studying and learning.
@@josephvanname3377 the graduate course instructor and his textbook didn’t provide motivation and the geometric insight so I disliked the material back then. That’s why I didn’t pursue it anymore. It would have been better if it was presented like this.
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.
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.
Why are p-adics needed for a high school math teacher course? Are you studying of your own interest?
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
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 UA-camr to me, but a sleep helper UA-camr. 😂
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.
Thank you for helping me understand p-adic numbers. Ive come across them before, but never has it been anywhere near this clear and digestible.
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..
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 UA-cam 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?
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?
5:33 in coding we sometimes use -1 as infinity
Thanks for bringing this amazing topic to us
übernyusiiiiii
Nice video
👌👌👌
Very nice
@@neelamopm 15:50
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 .
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.
1:30 guessing p-adic
You guessed correctly good job
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
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
@@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
@@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.
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.
So much of college is about time and financial constraints. You only have so much time to teach/take a class when everyone needs to be in the same room, and the instructor/assistants only have so much time to grade. Students need relatively quick feedback on their work, which isn't always easy to give when dealing with complicated problems. Life, however, almost always comes with time constraints attached, so imposing _some_ structure is a must.
Financially, it costs more than ever to go to college as income inequality rises. I think community colleges and a hybrid online/in-person model will be the future. The UK has had the Open University for decades, but the USA has been very slow to catch on.
My personal preference would be to have standardized tests for various subjects, then study for them in groups at my own pace: this is how professional certification works, for example. You have to show mastery of all of the material at test time, so you have to study quickly enough to keep most of it fresh in your mind.
3:30 Nope, I can’t let my concept of reality break anymore. I’m sure this is a great video and maybe I’ll eventually watch it, but today I just want to pretend the world is real.
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 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?
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?
This video blew my mind. As a middle-school French student, I haven't seen this type of math before, and I'm not sure to totally understand what is going on with that...
But this video is amazing. The explanations feels simple, but are keys to comprehend much more complex problem. Great work, Veritasium ! 😀
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 😅
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).
I not only finally understand p-adics a little bit but I understand much more now the astonishing leaps of thought that it took to come up with this stuff. Yet another absolutely fantastic video, Derek!
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.
The style of explaining such a complex topic with amazing animation is mind blowing. Kudos to Derek and the Veretasium team
This is quite possibly the best explanation of p-adics that has ever been given. Amazingly done!
I am drunk and i am trying very hard to understand what is he saying
Im trying to sleep and same
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.
_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.
Whatever you do, DO NOT let Terrence Howard see this video.
Why? I didn't get it
@@sethmobit type in Terrence Howard math Neil degrasse Tyson. It's a thing. This video will only confuse him more. (Hopefully not, but it would be funny to see him react to this video lol)
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 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.
@@nicbajito 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
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.
I love P-adic numbers, it’s such a crazy concept, bends reality, or maybe just bends our minds, how does adding infinite digits to infinite digits equal zero, crazy!
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.
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'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.
3:15 Algo similar pasa cuando calculas la raíz cuadrada de 51. Tiene decimales muy parecidos a 50/7.
This is basically whats found in binary, when you add 1 to the biggest number represented by U2 system, you get the smallest negative number represented by U2
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!
These math + history videos continue to be your very best work. Thank you - I learned a lot!
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.
My brain’s gonna be sore tomorrow
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 :) ∆
Your comment is being stoled by bots
@@MZOfficial104 Yeah it's really insane...
@@artophile7777 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
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 Derek and everyone that contributes to Veritasium. A shining beacon of light in the form of rationality, science education and the history that comes with it. I hope this channel and it's videos remain as fun and interesting to make for you all as they are fun for us to watch!
I read bacon. Now i'm hungry. Thanks.
@@christiankrause1594 and i am thrilled
I love how the format has changed drastically since the beginning, yet it still feels the same.
So can we exceed the speed of light by using a different number system?
@@andrewjenkinson7052 no
4:15 Interesting result! I always thought .33333 was equal to a third! 😅
Yes, in the standard decimal numbers, 0.3333... = 1/3. Don't forget the three dots at the right, because 0.33333 is certainly _not equal_ to 1/3.
However, in the 10-adic numbers, ...33333 is equal to -1/3, and 1/3 is equal to ...66667.
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.
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.
Veritasium explains mathematics even better than channels that are dedicated to maths and exam tips. He introduces us with the history, proof, experts, pleasant background score, great visualization and definitely Derek's great skill of educating. Keep going on, sir.
i mean he has a way higher budget and bigger production team and influence
Channels dedicated to maths don't need to motivate people with history, presentation or a score because they know their viewers are inherently interested in the mathematics. This isn't the case with Veritasium
@@armanigenes we have to admit that Derek has talent for those explanations, budget and animations aside.
@@epicmarschmallow5049 Well that makes him even better than them in a way. Doing extra factors good make the overall explanation amazing. This way, people less interested in maths, get interested towards it. In the end, I think the golden gift of maths should be enjoyed by everyone. And Derek's presentation succeeds in it.
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’ve only completed up to 12th grade math classes, but I’ve always been extremely fascinated with math theorems and rules that I’ve never heard of. It literally fuels my brain to learn about the ever-growing world of numbers.
Take classes at a community college if your interested, higher level maths are more theoretical if you apply them. Discreet math, and linear algebra include mostly simple concepts yet are used across computer science
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.
at aeound 1:40, if u look at the numbers on the background they appear to move slow, but if you look away, they move super fast
i think you’re seeing things
@@anything1456 No bro, try it for yourself, 1st look at the numbers, then over Derek's left shoulder...
@@tanmaypotdar575dawg u tripping
you're right
I see it
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!
Me and the boys finding the last digit of Pi by calculating it backwards:
this video feels like proof that we are in a simulation if i can find the last digit of any number
@@KrisTheGreatest Which hasn't been physically proven or has any concrete proof or evidence. The simulation theory is precisely a theory and nothing more.
Wow, fantastic video! Although I have a PhD in number theory, I never really understood p-adic numbers that well, but this video clarified them for me tremendously! They're very counterintuitive, yet very useful!
@Divergent Integral OK, I'll check that one out. Thanks for the recommendation!
I love how there’s always more to learn in math, from p-adic numbers, to space time, to how imaginary numbers were created, thank you for always coming up with more amazing, fascinating, inspiring, interesting content❤❤❤
"How it feels to invent Math" by 3Blue1Brown already blew my mind, and this went even further 🤯 Great video, Derek!
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" !
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.
Beautiful video! I need to say that for the first time I had problems visualising the p-adic numbers.
I didn’t expect to learn something so interesting today. Thank you for the exceptionally good lesson! Great job Derek!
dweeb
These have been a pet favorite thing of mine since they made a brief appearance on numberphile in 2015. Freshman year of college we got to do an open presentation topic, and I chose p-adics, so seeing new resources pop up to make it more approachable and common knowledge fills me with warmth. This is an interesting approach! Starting with the magic and then sort of letting the details come out of the mechanics. I started by explaining how Cauchy completeness is used to define real numbers, and then the bit about the p-adic norm, and then onto all those magical properties like fractions and negatives and diophantine equations and FLT and openness and so on in decreasing specificity, as you would expect from a math lecture. Hopefully this inspires follow-up, if not from you then from fans. They're really all introductory videos 😭
As a maths student, we don't learn about the p-adic numbers in any course, so I found this really interesting. Somehow you managed to explain this in a way that normal people can understand and yet is very interesting to me as a math student, Amazing!
P-adics are too exotic! It's only relatively recently we're actually using them to do mathematics. It's arguably the modern equivalent of calculus. Similar to what calculus was in the Renaissance
I’m not a genus and failed school because I never went still have a good job tho and for me I understood this very well and very interesting I wish I studied math
There's actually something very similar to p-adics in fractal geometry. If you define an iterated function system (IFS) of n functions then you can refer to anywhere in the fractal by either a finite or infinite base n number called the address. Then you can determine the distance between addresses by defining a metric on the codespace of addresses.
There's quite a bit more to it but if your interested Michael Barnsley's Fractals Everywhere is fantastic book on fractal geometry and chaos theory.
hi I’m also still an undergrad maths student but I’m very lucky that one of my professors is actually researching padics so in the lecture about algebraic number theory we also talked about them, as said in the video they have some really interesting applications. I have to say that the video might be a better introduction into the topic than anything I heard from a professor yet
You don’t learn about them in any undergraduate course? I learned about then in the first algebra course (Algebra I), we didn’t learn a lot about them at that point, but at least we were made aware of their existence at the very beginning
Accessing digits we didn't know we could is pretty wild!!
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.
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.
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
06:07 - Its behavior is similar to computer programs limited by minimum and maximum values. For example, in an 8-bit signed number, the range is from -128 to 127. This is because -128 is represented by 11111111 and 127 by 01111111. This concept is similar to the idea of an infinite number of 9s.
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.
I just wanna say you've managed to rekindle a bit of interest in math in me that I didn't expect. I grew up really interested in a lot of nerdy things like this, but over time I lost touch with that. Maybe I'm just feeling a certain way right now, but I want to thank you for this video.
One of your best videos ever! I’ve tried to understand p-adics many times and hit a brick wall of complexity. Your video makes it accessible to anyone who remembers at least some high school algebra. Amazing stuff!
As someone who works with hyperreal numbers and baseless 10s, p-adic numbers feel like the opposite of that, and for me, trying to understand them is like trying to read an undiscovered language with no linguistics skills.
Edit: i never got why the "p" in p-adics was specifically a prime until this video. Many thanks!
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.)
In recent years, there were more and more videos of p-adic numbers from 3b1b, Eric Rowland, Numberphiles and now Veritasium. Each videos give a unique perspective of p-adic numbers which are very mesmerizing. If you are interested in this subject, I recommend you to watch all of these videos.
And read some books on p-adic numbers, number theory, and algebraic geometry!
link plz
@@theflaggeddragon9472 then do a major in pure mathematics, then a masters and eventually a PHD
which is what I'm doing right now
This is what I was thinking myself. I posted a comment requesting more videos from the CC's you mentioned. We need more!
Then multiply everything by 5
I don't think I've ever been blown away by something so small ever. I mean, I remember learning about the scales of the universe and The Poincare (excuse the spelling) theorems and structures but this.... Goodness gracious! Thank you, Derek. I'm gonna be watching this a couple more times. You nailed it.
Awesome video, you really are one of the top engineers!
Derek, you have literally been the person teaching me the most since I found UA-cam. 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.
I’m 15 and currently sitting my GCSEs (SATs) if you are American, and I love learning maths and wish to pursue it further. Videos like this inspire me to do that and though they can be deeply confusing for someone as inexperienced as me, they make me want to learn more and kelp me understand simpler concepts easier. Thank you for challenging my understanding so much!
Good luck on your GCSEs
Good luck on your exam mate, not long now till they’re all over and you can enjoy your long summer!
I always find the name SATs for GCSEs annoying, because the UK has SATs 5 years before GCSEs, and actually pronounced as a word (sats) rather than Americans pronouncing each one individually (S A Ts)
@@squeaksquawk4255think about what you just said...
Thanks to everyone wishing me luck! It means a lot, and I’m glad to be part of suck a great community.
your animators are doing gods work, hope you pay them well :) tell them they did a great job on this video like usual. I have shared your videos with my less science inclined friends and they keep talking about how much they enjoy them and how their brain has been "hurting" keep up the great work, love what you do and am very happy you get to do it
6:18 so is the number line a circle?
thats interesting
I've thought about this a lot an year ago. I've have still got the journal entries explaining why this might be true. We can talk about it if you want
It's a sphere, because the numbers circle positive and negative positions infinitely.
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.
Derek: That number is its own square.
0 and 1: Look what they have to do to mimic a fraction of our power
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.
5:00 I never liked that proof for why 0.999999999... = 1. Imo the better proof is proving that there is no number B such that 0.99999999....
That proof at 5:00 somewhat misses to mention its axiomatic foundation. It makes some implicit assumptions concerning the characteristics of arithmetic operations of such infinite digit strings.
We can define "real numbers" as infinite digit strings with only finitely many digits left of the decimal point. We can then define the arithmetic operations similar to the way in which it is done in the video, by first using only finitely many digits of the operands and showing that the digits of the result don't change any more at some point if more and more digits of the operands are involved.
We would still have to prove that the laws of associativity and distributivity are really valid if we use this kind of definition. Which might be a bit cumbersome for this definition, and it is certainly not done in schools when they present this "proof". Actually, we find out that associativity is _not_ guaranteed if we don't identify some differing strings, like 0.999... and 1.000...
Using the standard adding algorithm, we have
0.999... + 0.999... = 1.999... and
1.999... - 0.999... = 1
"Standard" meaning that we use the same number of digits of both operands for our intermediate results.
And the law of associativity tells us:
(0.999... + 0.999...) - 0.999... = 1.999... - 0.999... = 1
= 0.999... + (0.999... - 0.999...) = 0.999... + 0 = 0.999...
Using the concept of "limits" (like you are proposing) is more straight forward in a way, as the identity 0.999... = 1 is conspicuous. Unfortunately, very many people who had quit mathematics after grade 10 don't have the slightest clue about this (and the other ones have forgotten). So popular-science articles and videos usually hesitate to talk about "limits".
I like this explanation: "In math, we define the sum of an infinite series equal to the number that it approaches, but never reaches. No matter how many 9s you write after that decimal point, you'll never reach 1. But since you are converging on the limit of 1, we say that the expression is equal to 1. That's just how limits are defined; deal with it. "
@@Strakester
_"we define the sum of an infinite series equal to the number that it approaches, but never reaches"_
That's an unsuitable restriction, because we want to have series containing an end sequence of only zeroes to converge as well. Or series like 1 - 1 + 1/2 - 1/2 + 1/4 - 1/4 + 1/8 - 1/8.... This last one reaches the limit intermittently.
Also, we have to define the concept of a limit and "convergence" thoroughly. Many students have difficulties understanding this in school.
Apart from that, I think it's desirable to have a merely algebraic definition of "decimal numbers" and their operations.
So in other words, you prefer analysis over algebra.
This is the first time I had my mind absolutely blown by a math concept in quite a while. That's crazy that you can multiply a number of infinite length by a positive integer greater than one and not only get a rational result, but one that is less than the sum of its parts.
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.
This video blew my mind! I've seen other explanations of the p-adic numbers, but this is the one that really made the concept "click" for me. Keep up the great work!
What an incredible introduction to p-adic numbers!!!! Thank you for your contribution!
If this video was made when I was 15 I would have started studying math a decade ago rather than a year ago like I did. Literally all of my largest brain-scratchiest math interests all rolled up into one beautiful video.
My theory for Fermat actually has always been that he had a notion of the necessary tools to prove his conjecture as a theorem, but, didn't know where to start as the tools that were needed hadn't been developed yet.
So he put the comment in the margin as an 'get back to the latter' thing
Fermat, like Gauss and Noether, was truly inspired to the degree that few could even imagine.
I believe for these people, pondering math such as this was their entertainment. If a person didn't like violent games (ie Colosseum), but did like learning, one had to imagine one's entertainment.
@Divergent_Integral I think whether or not it is satisfying is subjective. It was probably deeply satisfying to Andrew Wiles . Moreover, it gives extra value to things like the p-adic numbers when you hear that they were involved in that massive proof.
I think I heard from a podcast that Fermat had ideas for the proof for n=3 and n=4 and as the podcast conjectured, maybe he figured that the same methods could be applied again and again. Maybe he also thought that there was always some way to rewrite n where he could apply similar methods to components of the problem, maybe he had some intuitive feel for some complicated fractal proof structure or something.
Also naming new things would need an explanation with it and name convention wasnt that globalize. (Or even preventing that someone else came with the idea from another country at the same time as it happened 😂 could be anotjer layer)
Or he thought he had a proof and later realized it's not working out. I think many people overthink his comment and want to see the greater story arch when there probably isn't one.
Very informative, packed with information that can be connected to High School Mathematics. Not only that, It also provide that many things are not crazy, we're just not used to it. Thanks!
Wow big donation haha. Good for you bro💯
@@solomon8273it's not dollars
@SilverVibes898no what
@@Cris_Formage it’s 3 usd
So in other words, basic arithmetic has an in-built overflow error.
I love when you make a pure math episode and I love that I knew the voice of your mathematician firend before we saw his face. He is so good in these videos. His enthusiasm is infectious.