If it isn't Euler it's Gauss. Remember folks, if a murderer comes into your house and asks who proved a theorem you never heard of the Best survival strategy is to say "Euler or Gauss"
I don't know if this story ever reached Finland, but in the US, there was a mathematical genius who had Euler-like talent, but became a Ludditic terrorist. If anybody would have done something that deranged, it would have been this guy: en.wikipedia.org/wiki/Ted_Kaczynski
OTOH it shows up on about page 20 of the first number theory book I ever saw, and the previous 20 pages largely deals with basic number theory (like unique prime factorization, greatest common divisor, etc.) and items interesting unto themselves whether or not quadratic reciprocity had ever been discovered, and each item typically takes under a paragraph to prove. Other books have briefer versions of the same proof than the 4 pages in that one, but that book covers the case for 2 and -1 all at the same time, and actually has two proofs of the more arduous section, the second one of which is a diagram that sums up what just took a couple of pages of algebra (provided you don't cafe about 2 or -1.) I didn't want people to think this was going to be one of the hard things if they wanted to pursue number theory.
(28 March) Really bizarre, this video was basically invisible for almost two weeks with hardly any recommendations going out to fans. Only now RUclips has decided to actually show it to people. Who knows, maybe it was a mistake to mention cat videos in previous videos and the RUclips AI is now under the impression that the target audience for these videos has changed :) The longest Mathologer video ever, just shy of an hour (eventually it's going to happen :) One video I've been meaning to make for a long, long time. A Mathologerization of the Law of Quadratic Reciprocity. This is another one of my MASTERCLASS videos. The slide show consists of 550 slides and the whole thing took forever to make. Just to give you an idea of the work involved in producing a video like this, preparing the subtitles for this video took me almost 4 hours. Why do anything as crazy as this? Well, just like many other mathematicians I consider the law of quadratic reciprocity as one of the most beautiful and surprising facts about prime numbers. While other mathematicians were inspired to come up with ingenious proofs of this theorem, over 200 different proofs so far and counting, I thought I contribute to it's illustrious history by actually trying me very best of getting one of those crazily complicated proofs within reach of non-mathematicians, to make the unaccessible accessible. Now let's see how many people are actually prepared to watch a (close to) one hour long math(s) video :). Have a look at the description for relevant links and more background info. The first teaching semester at the university where I teach just started last week and all my teaching and lots of other stuff will happen this semester. This means I won't have much time for any more crazily time-consuming projects like this. Galois theory will definitely has to wait until the second half of this year :( Still, quite a bit of beautiful doable stuff coming up. So stay tuned.
I can confirm that I've just watched the video for the first time today, even though I exclusively use subscriptions feed page and watch all of it thoroughly. It's very likely that it wasn't in the subscriptions feed at all at the moment of posting.
-He says something about rings -I think it will be another Mathologer analogy or something -He is actually talking about damn rings from abstract Algebra -I get excited
@@camerontankersley3184 if you mean field in the common way, I love topology, measure theory, functional analysis and differential geometry. If you mean THE field, then I'll go with the reals xd
@@jksmusicstudio1439 Omg su. But the reals do be poppin doe. Anyway, I'm in college and my parents freak out whenever I wanna take any math courses with big numbers in them. But they kinda be paying the bills doe, so I have to listen to them. So Anyway, since I can't take the juicy stuff in school I pretty much have to self-study. So my goal is to go through a textbook on Linear Alg this break. Since ur a math expert, do you think I'd be ready for Abby Alg after that? I took a foundations course this semester so I have a touch of set theory and I can prove some super basic stuff. I'm trying to sneak in a combinatorics course next semester with calculus three, so should I wait for that to finish? Like what's the verdict and what should I take after Linear and Abstract Alg?
I think this is it for me with one hour long videos for a while :) Just think about it. If a video like this takes one hour to watch how long does it take to make?
@@Mathologer Rest assured that your efforts are greatly appreciated. This is one of the (if not the) best mathematics channels on youtube and the reason is that people recognize quality and hard work when they see it. Happy π-day!
Was gauss not in the thumbnail at the time of this comment? Maybe he's only recognized that easily by the countrymen who paid with bills with his face on it for years?
Please add a COMMA: "Take that, Mathologer!". Here are some examples with/without a comma: Take that Mathologer and keep him locked away. Take that job and... shove it. Take that, Job, said God. "Ouch", said Job. Let's eat grandma [said the cannibal to his brothers]. Let's eat, grandma! Etc.
I watched this video 5 times and I have to watch 10 times more in order to really get it, but I just wanted to tell you how much I love your videos and the way you teach. Your are exceptional!
I can't believe it's actually finally here. I've been waiting for this since the joke at the end of pi is transcendental proof. Thank you, Mathologer. You've somehow outdone yourself yet again!
RUclips's recommendation is, those days more than ever, under authority in order to broadcast only OMS approved informations about Coronavirus (or to censor any criticizing information on the situation or the behaviour of governmemts and their links to pharmaceutical lobbies or finance : it depends of your consideration of free speech)
There is actually a ring for each integer, but the negatives just copy their positive counterparts, 0 gives the integers itself, and 1 gives a very boring very mini ring that only has one element. :)
Three Rings for the Math-kings under the sky, Seven for the Physics-lords in their halls of stone, Nine for Computer Programmers doomed to die, One for The Professor on his dark throne In the Land of Ideas where the Colossus lies. One Ring to rule them all, One Ring to find them, One Ring to bring them all, and in the darkness bind them, In the World of Imagination where the Shadows lie
@Ron Maimon Great summary! I only know a little bit about abstract algebra, but you just summarized multiple hours of video lectures into one YT comment! 😄👍 Just one thing. You said, "There's only one integer divisible by 0, and that's zero." Wouldn't it be more accurate to say that, "There's only one integer that's a *multiple* of 0, and that's 0."? Even 0 is not divisible by 0, right?
@@robharwood3538 I think graphically you can see in the curve for 1/x there is often a vertical line at x=0 so you could ask, "if x=0, what is y?" (to which the answer is "it could be anything" or as others put, "undefined").
4 года назад+36
"I'm not about to propose" You just broke my (math) heart!
I understand every word you say, but your sentences are mostly beyond my understanding! I am amazed at how you can explain these issues (I last studied Math over 50 years ago, but remain fascinated).
What makes Quadratic Reciprocity so special, and in particular, why was it so important to Gauss? Why is squaring integers so important in Number Theory? To understand the answers to these questions, you need to appreciate the work Gauss did in the theory of quadratic forms and their application to differential geometry. Gauss's "fundamental theorem" of differential geometry is about how the 2nd order partial derivates of a function that defines a "surface", e.g. `f(x,y)=height`, depends on a specified quadratic form called the "metric tensor" of the surface (and its first derivates), which gives a rule for how to measure distances "on the surface". This shows a profound and deep insight into the nature of the relationship between a large number of otherwise seemingly unrelated abstract concepts. Squaring numbers plays a fundamental role in both Number Theory and Geometry. After watching this video twice, going on a few wild goose chases, and not being able to stop thinking about it: I finally think I understand why Gauss placed such a high degree of importance on this particular theorem, and feel like I have a lot of reading to do now!
I think I read (wiki?) that Fermat gave specific examples of the "law". So lots of mathematicians were motivated in this direction to prove/generalize another example of something Fermat stated without proof. Very interesting video, far from anything I've ever been exposed to. (And a welcome distraction in these confusing times.)
I cant believe I hadnt heard of Quadratic Reciprocity considering my honours dissertation was on finite fields. Granted, it was efficient computation of matrix opperations under GF2 on GPUs, but still, I cant believe I'd never come across any of this.
i love your channel and all of your videos, but this must be the most mind blowing one in a positive way. your passion for math and didactics is just so fun to watch. in german there is the expression of a spark jumping “over” when someone successfully communicated something. at least in my case you made many sparks jump over/through this medium and ignited interest and sparked fires but in a forest clearing sense, burning ignorance/not-knowing. the best thing watching and rewatching the videos is the feeling of being part of an experience you and your team obviously planned to be captivating and entertaining but not compromising (maybe impossible to compromise) complexity. i love that! whilst making it seem light and easy to lay out al kinds of layered/interconnected topics and parallel to that showing the struggle with the task/empathizing with an audience and our common attention spans/rivaling media fun/game et cetera. self-referential in a postmodern sense but by god not so heady and dead serious! Vielen Dank und Grüße aus Deutschland.
20:33 “Where did that come from?” The big “whoa” moment for me! Thanks for making it so much easier to understand why this equation is significant. I could never understand why when I was taking a course in number theory.
This approach is awesome! We covered the law of quadratic reciprocity in number theory class, but the proof was omitted, and it came down to uninspired manipulation and flipping of Legendre symbols. But now I'm 95% convinced that I understand why it all works :) I suppose something that could have been expanded upon for the benefit of other viewers is what the Legendre symbol is used for (which was briefly mentioned in the video), such as an example of solving a quadratic congruence over Z/pZ. Maybe that would make the whole LQR/Legendre business feel more motivated, but it's great as it is. And maybe another visualisation of the quadratic residues/non-residues mod a larger prime (like 17 maybe?) so we can get a feel for the numbers that appear along the diagonal, in particular what to expect (somehow it is easy to forget obvious things like (49/p) = 1 for any prime p, no need to find remainder first, as the symbol can lose attachment to its definition when purely evaluating by rule).
You always explain things so clearly that I know exactly where I stop understanding the maths. I often don't quite understand the last section or two of your long videos, but they're still fascinating.
I'm reading a proof-writing book and it's covering modular arithmetic right now so I'm going to give a proof that the diagonal placement covers the whole board (or try to). Claim: If we're placing the cards down diagonally on a grid with prime dimensions, we will cover the entire board Proof: Call the top left square (0,0) and the bottom right square (p-1,q-1) where p and q are distinct primes. Then the nth card we place down (starting indexing at 0) will have dimensions (n mod p, n mod q). Assume we are not covering the entire board, so there is a card we place down at some point, say, the kth card, that ends up in the same position as a previous card, the nth card, where k < pq and n
It essentially follows from the Chinese remainder theorem: If x = n mod p and y = n mod q, then there is only one solution for n mod pq for a given pair (x, y), provided that p and q are coprime.
Well done with the video. I had to watch it a few times to understand it all! I was OK to the halfway point then started to struggle, but stayed until the end. Thank goodness for the chapters. Keep making them.
It was crystal clear for me. But I must admit I'm a french PhD in math, however not a specialist in Number Theory (in fact in Complex Geometry, Conformal Field Theory with a little extra knowledge in Non-Commutative Geometry, Intuitionist Logic and Measure Theory...what a conceited & pompous mess !). You've remembered me of my undergraduate years at the University and the wonderful mathematicians I had the chance to meet there. A pure joy !!!
So glad I stumbled on some of your videos today. I remember when we both worked at The University of Adelaide. Your enthusiasm and humour have inspired many, many people for several decades. I am thrilled to see you are still actively exciting people about Mathematics. I will have to view all your videos when I can! Best Wishes, DB
@@ccarson I guess it was back in the ‘90s. Gosh, I feel really, really old now. Of course, in those days, it was still correctly referred to as “The University of Adelaide”, as it was formally named and incorporated in the 6th November, 1874 Act of South Australian Parliament. Yes, the name included “The” - with a capitol “T”. The age of the Internet saw variations informally introduced and eventually embraced, which can be evidenced on its own Web site. I am old enough to remember when the correct name was important … lol. Embracing the term Adelaide University was eventually accepted, if only because it elevated the university on alphabetically ordered lists.
@@Xubono For me, the usage of ”The” with a capital ”T”, in proper names seems quite natural. For example, the longer version of the name of my and my Best Friend’s micronation: ”The Forest”, includes ”The” - with a capital ”T”. 😅
Firstly, thanks. I'm looking forward to this. Before I get too far in and while I remember: Do you take requests? Can I suggest a Beginner's Guide to p-adic Numbers.
Cycling answer: if you have a permutation of cards, looking at the last one, we see that all the other numbers are behind it. In particular, the lower ones are behind. Putting the last card in the front means creating inversions and flipping the sign that amount of times. But also all the numbers bigger than it are already behind it, and so there are already inversions, but putting the card at the front undoes these inversions and flips the sign by the amount of cards. So we see that combining both cases means that we flip the sign by the amount of cards behind tge last card. So if there are n cards, we flip n-1. If n is odd, n-1 is even, so the sign doesn't change. If n is even, n-1 is odd, so the sign changes.
The story of how Euler got into formulating quadratic reciprocity is fascinating. It's contained in "Primes of the form x^2+n*y^2" by David A. Cox. HIGHLY RECOMMENDED because it's an historical recount.
This is the happiest classroom ever. Even though I don't practice mathematics (I'm Law and Philosophy student), I really appreciate your videos. You guys rock!
In C↑D↓, the reason it only mixes up the rows and not the columns is because both C↑ and D↓ use the same vertical sequence (i.e., top, middle, button, top middle, bottom...). Only the horizontal sequence changes.
Professor, I am looking forward to you aiming at presenting the Galois theory the Mathologer way in the future. I have always considered the inability to solve quintics by radicals mysterious, hopefully your video will shed some light on the concept to non-mathematicians like me. Thank you very much for all the stuff you create, it is unimaginable to comprehend the amount of thought, time and effort to get such things done, with the above video being an excellent example.
Meanwhile the following may interest you: Galois theory is not necessary for proving the existence of a polynomial of degree five whose roots cannot be expressed by radicals; the prior Abel-Rufini theorem already establishes that; But Galois theory is helpful in finding an example of such a polynomial.
I've never heard about quadratic reciprocity before … and I want to know more about this! Thank you for providing good-quality post-bachelor math popularization!
Damn, this was, I think, your toughest masterclass yet. Almost all of it worked for me, but I was really stuck at the point where you used the powers of two and got the fact that the new permutation will be obtained by adding 3 to the new natural permutation. I would have benifited from a little step-by-step at that point. This was a lot of fun, though. I'm looking forward to learning more about these fields and the inversions and other properties of permutations. Thanks for this!
1) RC can be proven to have the inversions on the positive diagonals. It should not be skipped. 2) Pi (xi-xj) for i=0, show that it is Abelian - commutative, i.e. a*b = b*a. The question was posed by professor David Chilag of blessed memory from the Technion, Haifa Israel. The problem does not require any deep knowledge about groups. It is easy to show that a^(k+1) * b^(k+2) = b^(k+2) *b^(k+1) and if k+1 and k+2 do not divide the number of elements in the group and the group is finite, the powers k+1 and k+2 are one-to-one maps and therefore the maps cover all the elements in the finite group and we are done (there are no elements of order k+1 or k+2). However, the original question does not have any condition on the order of the group and the group can be infinite. If k=0 then the second condition is a * a * b * b = a * b * a * b then multiplication by the inverse from right by b^-1 and left by a^-1 gets a * b = b * a and we are done. The question only requires to know the 4 axioms of a group.
37:25 Alternate proof from the one of eliya sne (for those who know more about permutations) : Let Sigma be a random permutation, and Tau the cycled one. We can get Tau by "multiplying" the permutation Sigma with a fixed cycle of length n, let us call it Epsilon = (1 n n-1 n-2 n-3 ... 3 2) So the sign of the cycled permutation Tau is equal to the sign of Sigma times the sign of Epsilon, so we want the sign of Epsilon. It is easy to see that Epsilon has n-1 inversions, so if n is odd, then sgn(Epsilon) = 1 and the sign of Sigma is unchanged, and if n is even, then sgn(Epsilon) = - 1 and the sign of Sigma is changed. End of proof. I know this is a little bit more technical, but I hope it's a good alternative :)
39:16 tile an (ab)x(ab) grid with (a)x(b) rectangles and draw the top-left to bottom-right diagonal. If it intersects the bottom-right corner of a rectangle then a square is formed by (n)x(m) lots of (a)x(b) so na=bm, n!=b, m!=a so a,b cannot be prime.
I liked your proof exercise near the beginning because it made me realize *why* 0 works the same in modular multiplication-it feels obvious to me now but it’s because multiplying by 0 here is like multiplying by the modulus in our familiar ring of integers, and when you multiply a number x by the modulus m, x * m is always going to be congruent to 0 (mod m).
Finally, one of my favorite theorems! I want to leave you an exercise about this theorem that made me appreciate it even more: Let p be a prime such that p is either equal to 2 or 3 modulo 5. Prove that the sequence n!+n^p-n+5 has at most a finite number of squares
I am teaching math my thirty-odd y-o daughter, who had forgotten much of what math she had learnt at school. And we just ended a long chapter on variations, permutations, and combinations (with or w/o repetition), their formulae, and their equivalences to certain functions. Now this video is coming to be the top cherry to that chapter! Many thanks, Mathologer.
The pink identity can also be demonstrated as follows. Looking at the position of the cards after the permutation, each card makes an inversion with all other cards that are either higher and to the right or lower and to the left of that card. Such cards form smaller rectangles of cards that go either up to the top right corner or down to the bottom left corner. Counting all cards in all these smaller rectangles, we would count every inversion twice, so we only consider say the top right rectangles. To get the total number of inversions we sum up the number of cards in all possible smaller top right rectangles, which is Sum{i=1..p-1; j=1..q-1} i.j = Sum{i=1..p-1} i.(q.(q-1)/2) = (p.(p-1)/2)(q.(q-1)/2).
Just a note - you don't actually need the existence of primitive roots in Z/pZ for the proof! If we let x\equiv q^{-1} mod p, then to find the sign of {x, 2x, \dots, (p-1)x} we can first consider the cycle (and subgroup!) {1, x, \dots, x^{ord_p(x)-1}} and note that all the cycles induced are just the cosets for our original subgroup, of which there are (p-1)/(ord_p(x)) of by Lagrange's Theorem. Since the sign is -1 to the power of the sum of one less than each of the cycle lengths, in this case we get (-1)^{(p-1)/(ord_p(x))*(ord_p(x)-1)}=(-1)^{(p-1)/(ord_p(x))}. If x is a QR, then x^{(p-1)/2}=1 mod p, so ord_p(x)|(p-1)/2 (hence the sign evaluates to 1), and if for contradiction it was also 1 for some non QR x, then ord_p(x)|(p-1)/2 which contradicts the well-known fact that x^{(p-1)/2}=-1 mod p in this case. Then the sign is (x/p)=(x/p)^{-1}=(x^{-1}/p)=(q/p) :).
Thank you for making this AND taking the time to caption it! It made it so much easier for me to follow, I really liked this one and can't wait for permutations!
What a insight, it reignite my spirit to study the quadratic reciprocity again which I did not get it in my number theory course in my university study in 15 years ago.
37:25 Proof: Let n be the number of cards, Let k be the number of cards bigger than the last one. By transferring the last card to the other side i eliminated k instances in which sometimes bigger than it is before it but also added (n-1)-k instances in which its bigger the other numbers (because the rest of the numbers are smaller then it). So eventually i just added n-1-2k . Since its a power of two i can ignore the -2k (its even). Now, i am left with just n-1 added to the power. If n is even then n-1 is odd and therefore the sine changes. If n is odd then n-1 is even and therefore the sine doesn't change. [|||]
Because this operation can be decomposed into (n-1) transpositions. And each transposition changes the sign. (-1)^(n-1) = 1 when n is odd and -1 when n is even.
Hi Burkard Polster. Thank you for this beautiful presentation of a very famous theorem. I'd like to point out that you look very similar of my all time favorite mathematician : Alexander Grothendieck. Grothendieck was a truly supreme mathematical genius of 20th century math whose contributions and influence are yet to be fully explored. I request you to bring out a video on any of his remarkable contributions.
Another gem 💎 of number theory! Only Mathologer could translate it into an accessible youtube video without compromising rigor! Gauss said of the proof, 'It tortured me'! Reading that I knew I wouldn't last one lemma before it, in my first encounter with it in Burton's Elementary Number Theory. Sure, the Preliminary Gauss's lemma and the main combinational argument of the proof was tough to understand, at the introductory level I first studied it an year ago😧. Now, most of that no longer dwells in my mind 😅 (forgotten! ), so imagine my ecstacy at this mathologer video😍! I'm sure it will help me comprehend it better than before, and I'll remember it longer (once I finish watching 😁)!
This is a most amazing presentation of the most amazing theorem in Mathematics! A must see for all budding mathematicians. As an aside, you gave tantalizing glimpse of finite geometry. Why don't you make a video on these topics? Particularly, the prime power conjecture for finite projective planes is a sadly neglected topic in RUclips. With its beautiful links with orthogonal Latin squares, this ought to be an eminently suitable candidate for a mathology video.
My proof for the little thing: Let’s look at the following order of cards: 4/5/3/1/2 lets say that the number of inversions here is る We do the cycling thing: 2/4/5/3/1 Consider this section: (4/5/3/1): notice how the number of inversions between these 4 numbers’ orders doesn’t change, let’s say that number is X So the only thing that changes after the cycling is the number of inversions for the cycled number that takes the first place after the cycling (2) in this case, the number of inversions it had in the initial order being logically る-X. Let’s look at the second order now: it’s number of inversions now becomes [( (5) -1 ) - (る-X)] with (5) being the number of cards we are playing with and (る-X) being it’s number of initial inversions Explanation: when it comes to ( (5) - 1 ): this is the max number of inversions a single member ( (2) in our case) can do with the other members because a single member cannot have an inversion with itself. When it comes to the subtraction i did: that’s because when we did the cycling we took out member from last place to first place cancelling out all the inversions it initially had with other members (because they are in growing order now) AND the it now has an inversion with members it did not have inversions with initially, basically the number of inversions it has now has become the compliment of the initial number of inversions ( their sum is equal to (5) - 1) Knowing all this, We can now move on to a more general case with (n) number of cards placed horizontally which have る inversions and the section of cards that did not change order after the cycling has X inversions ( we can do this in a more general case because all the explanations i did apply there too): the number of inversions after the cycling is therefore X + [n-1 - (る-X)] = 2X + n -1 - る ( lets call it N ) Notice how 2X is ever therefore it does not determine the parity of N so the parity of N in comparison to る is only determined by the parity of n-1: N and る have the same parity if n-1 is even aka if n is odd and the opposite is true as well. So the sign of the permutation doesn’t change if n is odd but it changes if n is even. Tell me if i have a mistake because im not experienced with this type of math
Thank you for all the hard work you put into that video!!! I will need to watch this one a few more times if I hope to understand it. Could you movtivate me with a really cool example of how quadratic reciprocity might be used to study pendulums or something?
16:26 the only squares mod 5 are 1 and 4. -1= 4(mod5) so the negative of the square 1 is also a square, and -4=1(mod5) so the negative of the square 4 is also a square, thus all the square on z/5z are squares. Q.E.D
This video is quite the Magnum Opus! It makes a result from deep in the heart (bowels?) of mathematics accessible to mere mortals, and introduces a bunch of mathematical constructs (squares in Zn, rings, sign of a permutation) along the way. For me, watching this video felt like being a tourist on an eloquent expertly-guided tour of a hidden room inside a massive museum. I am left in awe of the inventiveness of the mathematical minds of yore and supremely appreciative of Mathologer's efforts to spread his enthusiasm for mathematics. In response to Mathologer's query of what worked for me, with the first viewing I felt I was on top of the material until maybe the last 10 minutes. I expect that a second viewing will fix that, so... Congratulations Mathologer! I think you achieved your goal.
It's all about the last five minutes of the video. No, if you only watch them you will probably not understand anything at all, but these are really the time of the genius. You've done all the footwork and now you have to remember "why did I do this" and "how does all this - by now - easy stuff combine at all to help me in my original problem". More often than not also the question "what was the original question again?" pops up at this time. I remember how I in my thesis (in a completely different field, stochastics) was able to shorten a five page technical proof to a half page intuitive (and still correct) one and was immensely proud of myself. And then spent two weeks trying to remember how this achievement would help me at all.
The ordering property is not mysterious. Apply a reodering first, then your permutation, then the inverse of the reordering. The inverse of a reordering has the same parity as the reordering (inverse of a product of two-cycles is the same product but backwards), so the final result has the same parity as the original permutation.
33:00 That ”Positively Sloped = Inversion” -trick really makes sense, for our Row-Up -> Column-Down -permutation; because, in any positively sloped pair, the upper card is the earlier one, in the row-wise order, and the later one, in the column-wise order (and vice versa, for the lower card); because the row-wise order is: ”Left -> Right; Top -> Bottom”, whereas the column-wise order is: ”Top -> Bottom; Left -> Right”. So, any card: A that’s up and right, comes before any other card: B that’s down and left, in the row-wise order; and vice versa, in the column-wise order: The down-&-left card: B comes before the up-&-right card: A. Also; in this Row-Up -> Column-Down -permutation, we’re essentially switching rows and columns, which means that the internal order of any such pair gets flipped. Then, because we started with the cards in natural (row-wise) order (meaning: No Inversions.), the flipping of the internal order of any such pair amounts to an inversion. Whereas; in any horizontally or vertically aligned, or negatively sloped pair, the card that comes earlier, in the row-wise order, also comes earlier, in the column-wise order; thus, no inversions would emerge. 😌
39:15 Proof that the diagonalisation always works for _distinct odd primes_ p and q. First, imagine a coordinate system such that the card in the ith row and jth column has coordinates (i,j). (Origin at top left, numbers increase as you go right and down.) Also, p is the number of rows while q is the number of columns. If it works R⬆D⬇surely it works for C⬆D⬇as well. Because of the way the numbers wrap around, you can easily see that the row number of the card n is the smallest positive integer that is congruent to n modulo p (or the remainder that n gives when divided by p, except we now call 0 as p). Let this number be a. So for n=13, p=3, the row number is 1, while for n=12, p=3, it is 3. Similarly, the column number is the smallest positive integer that is congruent to n modulo q. Let this number be b. For n=11, q=5, we have b=1. Now we have the coordinates (a,b) for n. For two cards to overlap, they must have the same coordinates, i.e. leave the same remainders when divided by p and q. Let it be assumed, for the sake of contradiction that there exist 2 distinct integers m and n, smaller than or equal to pq, such that the corresponding cards overlap. Now, if two numbers have the same remainder after division by a third number, their difference must be divisible by the third number. (Use Euclid's division lemma, it is easy algebra.) Thus, |m-n| is divisible by both p and q. Since p and q are distinct odd primes, it must be divisible by pq. But since at least one of them is than smaller pq, the only option is that their difference is 0, i.e. m=n. This is a contradiction, since m and n were supposed to be distinct. Thus, no two cards can overlap. Q.E.D.
I figured out that if you think of the grid as being on a torus, you can easily see why modular arithmetic and the rings apply here. The rows and columns are nothing but the two fundamental loops on the torus.
Just: WOW! 💪👍 Thanks a lot for your motivation but much more for your dedication! Was tough for me to follow and had to stop to rethink few times but then I grasped it in the end! ... still searching for the applause button (; 👏👏👏
These mini rings are what music theorists call modifications. Basically, you think of a clock with n values and you count around the clock face until you find the remainder
1. I missed this episode somehow. 2. YT showed this as watched, clearly it was very late and inmust have fallen asleep. 3. Mathematically this is one of my fav (i should make my ranking), but from the fact of how much you had fun it must be your fav too.
Thank you so much for your effort in creating these enjoyable and accessible videos! As a technical writer who is entering the world of mathematics late, I find these really help me to internalize so much of what I need to go study. Cheers, Rick Lehtinen
Most fantastic! It reminds me of the mysterious connection of quadratic reciprocity with the linking number of two knots (in 3-space), which Gauss found a formula for (in terms of an integral). The diagonal dealing certainly looks like a (p,q)-torus knot, and maybe the R and C dealings are just circle running inside (and "outside") the torus. I'm sure others have thought it all through.
THANK YOU! Understanding QR has been a goal of mine. I didn't fully follow the argument on first pass, but now at least I have a Mathologized argument to study. Again, thanks for all your effort.
What An hour? *hand hovers indecisively over play button* and it's a masterclass? *hand drifts away from the play button* and I need to do my flute practice *hand retreats to lap* (oh very funny, plz don't go there) OTOH it's a video with a funny bald German in it *hand approaches keyboard* but there are probably no cats in it being cute *hand falls to side* *crickets* *more crickets* reciprocity prime what the? wat IS dat? *Mathologer smiles impishly and wiggles the hook* Command voice:" Lock all targets and fire at will! All ahead Warp 6! Take us in Mr Sulu!" *Punches the big red button...*
Je suis impressionné par la qualité des traductions (commentaire et sous-titres) bien que je sois parfaitement anglophone ! Content de voir que les communautés de vulgarisation et d'enseignement mathématiques se joignent !
7:22 Proof Commutative Law for any Z/n: Since Z/n is just Z mod n for any whole number, we just have to add a+b and divide by n, then take the remainder, and that's our answer. Let C be the sum of a and b. But no matter what order you add whole numbers a and b, you always get the same number C by the ordinary commutative law of addition. So you will divide the same number by n no matter what order you add a and b in. Distributive Law for Z/n: The argument is basically the same, except you use the ordinary Distributive Law. You get the same number, which we will now call D, on each side of that second equation, so you will always be doing the operation D/n. This completes my proof for any modular ring Z/n being valid for both equations. QED. Let me know of there is any error in this logic.
Or you could use the fact that a commutative ring quotient out an ideal is also a commutative ring. Z/n is the same as Z/(n), where (n) is the ideal generated by n.
This fascinating proof put me in mind of a very impressive trick that one can do (if one has the time) with a normal pack of 52 playing cards. Put the cards in suit order and then deal the pack into four hands of 13 cards. Assuming, in bridge notation, you are South. Pick the piles of cards clockwise starting with west - hence WNES. Now deal the cards a second time (D2) and pick them up in the same way. Continue until after D13. Now examine the cards and you will find that they are back in the original order. My low grade graduate maths is not up to explaining this, and, of course, 4 is not prime, even though 13 is. But it is another of the wonders of theories of ordering things.
ZED SEVEN Maybe you're confused by the U.K. versus U.S. pronunciation of letter 'Z': ZED vs. ZEE. Also, as is easily done, you swapped Z/S which are a voiced/unvoiced pair.
@@ReasonableForseeability At the timestamp in question, Burkard actually does say Sed Zeven. This is easy to see when you contrast it with how he says it at 15:54.
It's a simple spoonerism, made likelier than most by the fact that "s" and "z" sounds are produced the same way, except that the vocal cords are vibrating for "z" but not for "s".
This channel is what I needed in my life. I am an established (in the sense that I probably earn more than u, the hater) programmer, but I lack in basic math skills. I have watched a few more videos on this channel and they are absolutely treat! Thank you for doing this man, I wish you were my teacher (as in school days. You are, right now hehe). :D Subbed!
If it isn't Euler it's Gauss. Remember folks, if a murderer comes into your house and asks who proved a theorem you never heard of the Best survival strategy is to say "Euler or Gauss"
You are damn right
Kkkkkkkkj
Then why Euler is not a part of trinity which comprises Archimedes, Newton and Gauss?
Such a shame
I don't know if this story ever reached Finland, but in the US, there was a mathematical genius who had Euler-like talent, but became a Ludditic terrorist. If anybody would have done something that deranged, it would have been this guy:
en.wikipedia.org/wiki/Ted_Kaczynski
Pragadeesh J - You can declare any trinity you like. What difference does it make?
WOW IT TOOK ME MY ENTIRE DEGREE TO MAKE SENSE OF THIS THEOREM AND HERE MATHOLOGER COMES SHUFFLING CARDS TO PROVE IT CLEARLY
It makes more sense the more proofs you read
I feel like there was an important lesson to be learned from this video:
NEVER LET THE MATHEMATICIAN DEAL THE CARDS!
OTOH it shows up on about page 20 of the first number theory book I ever saw, and the previous 20 pages largely deals with basic number theory (like unique prime factorization, greatest common divisor, etc.) and items interesting unto themselves whether or not quadratic reciprocity had ever been discovered, and each item typically takes under a paragraph to prove. Other books have briefer versions of the same proof than the 4 pages in that one, but that book covers the case for 2 and -1 all at the same time, and actually has two proofs of the more arduous section, the second one of which is a diagram that sums up what just took a couple of pages of algebra (provided you don't cafe about 2 or -1.)
I didn't want people to think this was going to be one of the hard things if they wanted to pursue number theory.
I dunno if it's clearly to the un-degreed mind but yes i believe he proved it
i love your username
(28 March) Really bizarre, this video was basically invisible for almost two weeks with hardly any recommendations going out to fans. Only now RUclips has decided to actually show it to people. Who knows, maybe it was a mistake to mention cat videos in previous videos and the RUclips AI is now under the impression that the target audience for these videos has changed :)
The longest Mathologer video ever, just shy of an hour (eventually it's going to happen :) One video I've been meaning to make for a long, long time. A Mathologerization of the Law of Quadratic Reciprocity. This is another one of my MASTERCLASS videos. The slide show consists of 550 slides and the whole thing took forever to make. Just to give you an idea of the work involved in producing a video like this, preparing the subtitles for this video took me almost 4 hours. Why do anything as crazy as this? Well, just like many other mathematicians I consider the law of quadratic reciprocity as one of the most beautiful and surprising facts about prime numbers. While other mathematicians were inspired to come up with ingenious proofs of this theorem, over 200 different proofs so far and counting, I thought I contribute to it's illustrious history by actually trying me very best of getting one of those crazily complicated proofs within reach of non-mathematicians, to make the unaccessible accessible. Now let's see how many people are actually prepared to watch a (close to) one hour long math(s) video :). Have a look at the description for relevant links and more background info.
The first teaching semester at the university where I teach just started last week and all my teaching and lots of other stuff will happen this semester. This means I won't have much time for any more crazily time-consuming projects like this. Galois theory will definitely has to wait until the second half of this year :( Still, quite a bit of beautiful doable stuff coming up. So stay tuned.
I can confirm that I've just watched the video for the first time today, even though I exclusively use subscriptions feed page and watch all of it thoroughly. It's very likely that it wasn't in the subscriptions feed at all at the moment of posting.
Yes. Very strange. I confirm that this video was hidden from me too. Anyway I'm glad I watched it. A lot of hard work to explain rings.
Ok...now what do I do for the next 23 hours today?
Yeah it just got recommended even though I've been waiting for this to be done on a big math channel for a while now
I only just saw it today.
-He says something about rings
-I think it will be another Mathologer analogy or something
-He is actually talking about damn rings from abstract Algebra
-I get excited
Wooow, Exactly what I have felt :-)
They aren't just any rings, they're abelian rings!
What's your favorite field of math
@@camerontankersley3184 if you mean field in the common way, I love topology, measure theory, functional analysis and differential geometry. If you mean THE field, then I'll go with the reals xd
@@jksmusicstudio1439 Omg su. But the reals do be poppin doe. Anyway, I'm in college and my parents freak out whenever I wanna take any math courses with big numbers in them. But they kinda be paying the bills doe, so I have to listen to them. So Anyway, since I can't take the juicy stuff in school I pretty much have to self-study. So my goal is to go through a textbook on Linear Alg this break. Since ur a math expert, do you think I'd be ready for Abby Alg after that? I took a foundations course this semester so I have a touch of set theory and I can prove some super basic stuff. I'm trying to sneak in a combinatorics course next semester with calculus three, so should I wait for that to finish? Like what's the verdict and what should I take after Linear and Abstract Alg?
I will never tire of 99999999 in a strong German accent
Isn't Mathologer's accent Austrian, or am I mistaken?
@@tracyh5751 ehh, German, Austrian, pretty close to the same account. Sie beide sprechen Deutsch.
Ja ja ja ja ja ja.....
*NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN, NEIN !!!!!*
* repeatedly slams desk angrily *
note to self, it's at 2:58
People will stay at home
Time to make 1hour long video
I think this is it for me with one hour long videos for a while :) Just think about it. If a video like this takes one hour to watch how long does it take to make?
Nooooo!! It's a nice way to spend your time indoors... and humanity needs you right now... keep it up!
@@Mathologer Rest assured that your efforts are greatly appreciated. This is one of the (if not the) best mathematics channels on youtube and the reason is that people recognize quality and hard work when they see it. Happy π-day!
@@mannyc6649 *the* best by far AFAIC
Or start a series on class field theory :-)
I actually guessed Gauss! Take that Mathologer!
Me: 1
Mathologer: 387ish
Never underestimate the power of trial, error and gausswork.
@@Tubluer :P
Was gauss not in the thumbnail at the time of this comment? Maybe he's only recognized that easily by the countrymen who paid with bills with his face on it for years?
Please add a COMMA: "Take that, Mathologer!".
Here are some examples with/without a comma:
Take that Mathologer and keep him locked away.
Take that job and... shove it.
Take that, Job, said God. "Ouch", said Job.
Let's eat grandma [said the cannibal to his brothers].
Let's eat, grandma!
Etc.
Antinatalist It’s a matter of style. I’ve seen several writers who omit that comma.
I watched this video 5 times and I have to watch 10 times more in order to really get it, but I just wanted to tell you how much I love your videos and the way you teach. Your are exceptional!
I can't believe it's actually finally here. I've been waiting for this since the joke at the end of pi is transcendental proof. Thank you, Mathologer. You've somehow outdone yourself yet again!
Why isn't this in my subscription feed... I had to search up the channel name to find it
Pankaj Chowdhury Partha same here. RUclips didn’t recommend it to me somehow.
Same for me... I only go to my subscription feed, but this video hasn't been shown there.
RUclips's recommendation is, those days more than ever, under authority in order to broadcast only OMS approved informations about Coronavirus (or to censor any criticizing information on the situation or the behaviour of governmemts and their links to pharmaceutical lobbies or finance : it depends of your consideration of free speech)
Pankaj Chowdhury Partha same
This just appeared in my Home tab. No way I could have missed this.
One ring for each integer above 1.
And one ring to rule them all.
That's called Z
There is actually a ring for each integer, but the negatives just copy their positive counterparts, 0 gives the integers itself, and 1 gives a very boring very mini ring that only has one element. :)
Three Rings for the Math-kings under the sky,
Seven for the Physics-lords in their halls of stone,
Nine for Computer Programmers doomed to die,
One for The Professor on his dark throne
In the Land of Ideas where the Colossus lies.
One Ring to rule them all, One Ring to find them,
One Ring to bring them all, and in the darkness bind them,
In the World of Imagination where the Shadows lie
@Ron Maimon Great summary! I only know a little bit about abstract algebra, but you just summarized multiple hours of video lectures into one YT comment! 😄👍
Just one thing. You said, "There's only one integer divisible by 0, and that's zero." Wouldn't it be more accurate to say that, "There's only one integer that's a *multiple* of 0, and that's 0."? Even 0 is not divisible by 0, right?
@@robharwood3538 I think graphically you can see in the curve for 1/x there is often a vertical line at x=0 so you could ask, "if x=0, what is y?" (to which the answer is "it could be anything" or as others put, "undefined").
"I'm not about to propose"
You just broke my (math) heart!
Same, I had a cardioid arrest
@@TheOnlyGeggleslegendre-y pun.
I understand every word you say, but your sentences are mostly beyond my understanding! I am amazed at how you can explain these issues (I last studied Math over 50 years ago, but remain fascinated).
That’s interesting that the addition and multiplication tables for 2 are the logic tables for XOR and AND gates 7:00
That's why it's taught on Master's classes in Electronics (or Electrical) Engineering Programs(or sub fields).
I noticed the exact same thing 👌🏻😌👍🏻.
@@HiteshAH That makes sense 💡.
What makes Quadratic Reciprocity so special, and in particular, why was it so important to Gauss? Why is squaring integers so important in Number Theory? To understand the answers to these questions, you need to appreciate the work Gauss did in the theory of quadratic forms and their application to differential geometry. Gauss's "fundamental theorem" of differential geometry is about how the 2nd order partial derivates of a function that defines a "surface", e.g. `f(x,y)=height`, depends on a specified quadratic form called the "metric tensor" of the surface (and its first derivates), which gives a rule for how to measure distances "on the surface". This shows a profound and deep insight into the nature of the relationship between a large number of otherwise seemingly unrelated abstract concepts.
Squaring numbers plays a fundamental role in both Number Theory and Geometry. After watching this video twice, going on a few wild goose chases, and not being able to stop thinking about it: I finally think I understand why Gauss placed such a high degree of importance on this particular theorem, and feel like I have a lot of reading to do now!
I think I read (wiki?) that Fermat gave specific examples of the "law". So lots of mathematicians were motivated in this direction to prove/generalize another example of something Fermat stated without proof.
Very interesting video, far from anything I've ever been exposed to. (And a welcome distraction in these confusing times.)
No that's a bit of a stretch. It's safe to say the metric tensor and quadratic reciprocity have nothing to do with each other directly.
I cant believe I hadnt heard of Quadratic Reciprocity considering my honours dissertation was on finite fields. Granted, it was efficient computation of matrix opperations under GF2 on GPUs, but still, I cant believe I'd never come across any of this.
Same. Except everything after "I can't."
i love your channel and all of your videos, but this must be the most mind blowing one in a positive way. your passion for math and didactics is just so fun to watch. in german there is the expression of a spark jumping “over” when someone successfully communicated something. at least in my case you made many sparks jump over/through this medium and ignited interest and sparked fires but in a forest clearing sense, burning ignorance/not-knowing. the best thing watching and rewatching the videos is the feeling of being part of an experience you and your team obviously planned to be captivating and entertaining but not compromising (maybe impossible to compromise) complexity. i love that! whilst making it seem light and easy to lay out al kinds of layered/interconnected topics and parallel to that showing the struggle with the task/empathizing with an audience and our common attention spans/rivaling media fun/game et cetera. self-referential in a postmodern sense but by god not so heady and dead serious! Vielen Dank und Grüße aus Deutschland.
20:33 “Where did that come from?”
The big “whoa” moment for me! Thanks for making it so much easier to understand why this equation is significant. I could never understand why when I was taking a course in number theory.
This approach is awesome! We covered the law of quadratic reciprocity in number theory class, but the proof was omitted, and it came down to uninspired manipulation and flipping of Legendre symbols. But now I'm 95% convinced that I understand why it all works :)
I suppose something that could have been expanded upon for the benefit of other viewers is what the Legendre symbol is used for (which was briefly mentioned in the video), such as an example of solving a quadratic congruence over Z/pZ. Maybe that would make the whole LQR/Legendre business feel more motivated, but it's great as it is.
And maybe another visualisation of the quadratic residues/non-residues mod a larger prime (like 17 maybe?) so we can get a feel for the numbers that appear along the diagonal, in particular what to expect (somehow it is easy to forget obvious things like (49/p) = 1 for any prime p, no need to find remainder first, as the symbol can lose attachment to its definition when purely evaluating by rule).
@Mathologer This is definitely 1 of the all-time gems. Thank You, Professor Burkard Polster. 😌
You always explain things so clearly that I know exactly where I stop understanding the maths. I often don't quite understand the last section or two of your long videos, but they're still fascinating.
Mathologer: " Avoid negativity "
Computer science: " A void negativity "
LMFAO
-|x|
kerning 💥
One ring to rule them all
that seems to ring true, but there's also wedding ring ... and suffering
My thoughts, exactly 🎯😅!
I'm reading a proof-writing book and it's covering modular arithmetic right now so I'm going to give a proof that the diagonal placement covers the whole board (or try to).
Claim: If we're placing the cards down diagonally on a grid with prime dimensions, we will cover the entire board
Proof: Call the top left square (0,0) and the bottom right square (p-1,q-1) where p and q are distinct primes. Then the nth card we place down (starting indexing at 0) will have dimensions (n mod p, n mod q). Assume we are not covering the entire board, so there is a card we place down at some point, say, the kth card, that ends up in the same position as a previous card, the nth card, where k < pq and n
It essentially follows from the Chinese remainder theorem:
If
x = n mod p
and
y = n mod q,
then there is only one solution for n mod pq for a given pair (x, y), provided that p and q are coprime.
Well done with the video. I had to watch it a few times to understand it all! I was OK to the halfway point then started to struggle, but stayed until the end. Thank goodness for the chapters. Keep making them.
It was crystal clear for me. But I must admit I'm a french PhD in math, however not a specialist in Number Theory (in fact in Complex Geometry, Conformal Field Theory with a little extra knowledge in Non-Commutative Geometry, Intuitionist Logic and Measure Theory...what a conceited & pompous mess !). You've remembered me of my undergraduate years at the University and the wonderful mathematicians I had the chance to meet there. A pure joy !!!
*You've REMINDED me of my ...
se rapeller = remember
rapeller = remind
So glad I stumbled on some of your videos today. I remember when we both worked at The University of Adelaide. Your enthusiasm and humour have inspired many, many people for several decades. I am thrilled to see you are still actively exciting people about Mathematics. I will have to view all your videos when I can!
Best Wishes, DB
What years did the Mathologer work at Adelaide Uni?
@@ccarson I guess it was back in the ‘90s. Gosh, I feel really, really old now.
Of course, in those days, it was still correctly referred to as “The University of Adelaide”, as it was formally named and incorporated in the 6th November, 1874 Act of South Australian Parliament. Yes, the name included “The” - with a capitol “T”. The age of the Internet saw variations informally introduced and eventually embraced, which can be evidenced on its own Web site. I am old enough to remember when the correct name was important … lol. Embracing the term Adelaide University was eventually accepted, if only because it elevated the university on alphabetically ordered lists.
@@Xubono For me, the usage of ”The” with a capital ”T”, in proper names seems quite natural. For example, the longer version of the name of my and my Best Friend’s micronation: ”The Forest”, includes ”The” - with a capital ”T”. 😅
Firstly, thanks. I'm looking forward to this. Before I get too far in and while I remember: Do you take requests? Can I suggest a Beginner's Guide to p-adic Numbers.
I will second that request.
Me too.
@@chrislombardi3968 I will third that request 😁.
Cycling answer: if you have a permutation of cards, looking at the last one, we see that all the other numbers are behind it. In particular, the lower ones are behind. Putting the last card in the front means creating inversions and flipping the sign that amount of times. But also all the numbers bigger than it are already behind it, and so there are already inversions, but putting the card at the front undoes these inversions and flips the sign by the amount of cards. So we see that combining both cases means that we flip the sign by the amount of cards behind tge last card. So if there are n cards, we flip n-1. If n is odd, n-1 is even, so the sign doesn't change. If n is even, n-1 is odd, so the sign changes.
The story of how Euler got into formulating quadratic reciprocity is fascinating. It's contained in "Primes of the form x^2+n*y^2" by David A. Cox. HIGHLY RECOMMENDED because it's an historical recount.
This is the happiest classroom ever. Even though I don't practice mathematics (I'm Law and Philosophy student), I really appreciate your videos. You guys rock!
In C↑D↓, the reason it only mixes up the rows and not the columns is because both C↑ and D↓ use the same vertical sequence (i.e., top, middle, button, top middle, bottom...). Only the horizontal sequence changes.
Professor, I am looking forward to you aiming at presenting the Galois theory the Mathologer way in the future. I have always considered the inability to solve quintics by radicals mysterious, hopefully your video will shed some light on the concept to non-mathematicians like me. Thank you very much for all the stuff you create, it is unimaginable to comprehend the amount of thought, time and effort to get such things done, with the above video being an excellent example.
Meanwhile the following may interest you: Galois theory is not necessary for proving the existence of a polynomial of degree five whose roots cannot be expressed by radicals; the prior Abel-Rufini theorem already establishes that; But Galois theory is helpful in finding an example of such a polynomial.
@@loicetienne7570 Thank you.
I've never heard about quadratic reciprocity before … and I want to know more about this!
Thank you for providing good-quality post-bachelor math popularization!
Damn, this was, I think, your toughest masterclass yet. Almost all of it worked for me, but I was really stuck at the point where you used the powers of two and got the fact that the new permutation will be obtained by adding 3 to the new natural permutation. I would have benifited from a little step-by-step at that point.
This was a lot of fun, though. I'm looking forward to learning more about these fields and the inversions and other properties of permutations. Thanks for this!
1) RC can be proven to have the inversions on the positive diagonals. It should not be skipped. 2) Pi (xi-xj) for i=0, show that it is Abelian - commutative, i.e. a*b = b*a. The question was posed by professor David Chilag of blessed memory from the Technion, Haifa Israel. The problem does not require any deep knowledge about groups. It is easy to show that a^(k+1) * b^(k+2) = b^(k+2) *b^(k+1) and if k+1 and k+2 do not divide the number of elements in the group and the group is finite, the powers k+1 and k+2 are one-to-one maps and therefore the maps cover all the elements in the finite group and we are done (there are no elements of order k+1 or k+2). However, the original question does not have any condition on the order of the group and the group can be infinite. If k=0 then the second condition is a * a * b * b = a * b * a * b then multiplication by the inverse from right by b^-1 and left by a^-1 gets a * b = b * a and we are done. The question only requires to know the 4 axioms of a group.
"This video was so long, the Mathologer had hair when it started!"
. . . . .
"And so did I!!"
Fred
*Mathologer walks into a party*
"Hey wanna see a card trick?"
You sure wouldn't want to play poker with him!
This was a difficult video, but you made it much easier to understand. Thank you so much for your hard work.
37:25
Alternate proof from the one of eliya sne (for those who know more about permutations) :
Let Sigma be a random permutation, and Tau the cycled one.
We can get Tau by "multiplying" the permutation Sigma with a fixed cycle of length n, let us call it Epsilon = (1 n n-1 n-2 n-3 ... 3 2)
So the sign of the cycled permutation Tau is equal to the sign of Sigma times the sign of Epsilon, so we want the sign of Epsilon. It is easy to see that Epsilon has n-1 inversions, so if n is odd, then sgn(Epsilon) = 1 and the sign of Sigma is unchanged, and if n is even, then sgn(Epsilon) = - 1 and the sign of Sigma is changed.
End of proof.
I know this is a little bit more technical, but I hope it's a good alternative :)
So, basically invariance under conjugation ;)
I prefer Flammable Math's avoid positivity shirt -|x|
Excuse me what the frick
39:16 tile an (ab)x(ab) grid with (a)x(b) rectangles and draw the top-left to bottom-right diagonal. If it intersects the bottom-right corner of a rectangle then a square is formed by (n)x(m) lots of (a)x(b) so na=bm, n!=b, m!=a so a,b cannot be prime.
Thank you so much for continuing your amazing effort!!! You are a true gem in all this dimension.
I liked your proof exercise near the beginning because it made me realize *why* 0 works the same in modular multiplication-it feels obvious to me now but it’s because multiplying by 0 here is like multiplying by the modulus in our familiar ring of integers, and when you multiply a number x by the modulus m, x * m is always going to be congruent to 0 (mod m).
Finally, one of my favorite theorems! I want to leave you an exercise about this theorem that made me appreciate it even more:
Let p be a prime such that p is either equal to 2 or 3 modulo 5. Prove that the sequence n!+n^p-n+5 has at most a finite number of squares
I am teaching math my thirty-odd y-o daughter, who had forgotten much of what math she had learnt at school. And we just ended a long chapter on variations, permutations, and combinations (with or w/o repetition), their formulae, and their equivalences to certain functions. Now this video is coming to be the top cherry to that chapter! Many thanks, Mathologer.
@Mathologer I am happy that you are back, I love you..
The pink identity can also be demonstrated as follows. Looking at the position of the cards after the permutation, each card makes an inversion with all other cards that are either higher and to the right or lower and to the left of that card. Such cards form smaller rectangles of cards that go either up to the top right corner or down to the bottom left corner.
Counting all cards in all these smaller rectangles, we would count every inversion twice, so we only consider say the top right rectangles. To get the total number of inversions we sum up the number of cards in all possible smaller top right rectangles, which is
Sum{i=1..p-1; j=1..q-1} i.j = Sum{i=1..p-1} i.(q.(q-1)/2) = (p.(p-1)/2)(q.(q-1)/2).
Just a note - you don't actually need the existence of primitive roots in Z/pZ for the proof! If we let x\equiv q^{-1} mod p, then to find the sign of {x, 2x, \dots, (p-1)x} we can first consider the cycle (and subgroup!) {1, x, \dots, x^{ord_p(x)-1}} and note that all the cycles induced are just the cosets for our original subgroup, of which there are (p-1)/(ord_p(x)) of by Lagrange's Theorem. Since the sign is -1 to the power of the sum of one less than each of the cycle lengths, in this case we get (-1)^{(p-1)/(ord_p(x))*(ord_p(x)-1)}=(-1)^{(p-1)/(ord_p(x))}. If x is a QR, then x^{(p-1)/2}=1 mod p, so ord_p(x)|(p-1)/2 (hence the sign evaluates to 1), and if for contradiction it was also 1 for some non QR x, then ord_p(x)|(p-1)/2 which contradicts the well-known fact that x^{(p-1)/2}=-1 mod p in this case. Then the sign is (x/p)=(x/p)^{-1}=(x^{-1}/p)=(q/p) :).
Thank you for making this AND taking the time to caption it! It made it so much easier for me to follow, I really liked this one and can't wait for permutations!
Took me almost four hours just to make the captions ! :)
@@Mathologer perfect timing too
I already love math, but the passion that you have is contagious
Same. I used to solve Maths problems, as a pastime, in kindergarten 🙂.
This is no doubt one of the very best Mathloger videos. Thanks for creating this, I thoroughly enjoyed watching it.
Legendre: >:[
Wherever he is now, I'm sure he is much happier knowing that there is such a great video discussing his work!
or he is just dead
@@toniokettner4821 Wow! Blunt atheist is blunt. 😅
What a insight, it reignite my spirit to study the quadratic reciprocity again which I did not get it in my number theory course in my university study in 15 years ago.
37:25
Proof:
Let n be the number of cards,
Let k be the number of cards bigger than the last one.
By transferring the last card to the other side i eliminated k instances in which sometimes bigger than it is before it but also added (n-1)-k instances in which its bigger the other numbers (because the rest of the numbers are smaller then it).
So eventually i just added n-1-2k .
Since its a power of two i can ignore the -2k (its even).
Now, i am left with just n-1 added to the power.
If n is even then n-1 is odd and therefore the sine changes.
If n is odd then n-1 is even and therefore the sine doesn't change.
[|||]
Great! You beat me to it.
Because this operation can be decomposed into (n-1) transpositions. And each transposition changes the sign.
(-1)^(n-1) = 1 when n is odd and -1 when n is even.
Hi Burkard Polster. Thank you for this beautiful presentation of a very famous theorem. I'd like to point out that you look very similar of my all time favorite mathematician : Alexander Grothendieck.
Grothendieck was a truly supreme mathematical genius of 20th century math whose contributions and influence are yet to be fully explored. I request you to bring out a video on any of his remarkable contributions.
and Alexander Grothendieck was such a genius that he thought 57 is a prime number
"Oh, I'll just watch one quick Mathologer video before sleep." I don't know why I don't look at duration before making these decisions....
You should Mind Your Decisions.
Once again you @Mathologer nailed it...gotta love mathematics more than before
Another gem 💎 of number theory! Only Mathologer could translate it into an accessible youtube video without compromising rigor!
Gauss said of the proof, 'It tortured me'! Reading that I knew I wouldn't last one lemma before it, in my first encounter with it in Burton's Elementary Number Theory. Sure, the Preliminary Gauss's lemma and the main combinational argument of the proof was tough to understand, at the introductory level I first studied it an year ago😧. Now, most of that no longer dwells in my mind 😅 (forgotten! ), so imagine my ecstacy at this mathologer video😍! I'm sure it will help me comprehend it better than before, and I'll remember it longer (once I finish watching 😁)!
I am so excited to digest this video. I Love how you invite us to try using our Intuition and grasp these Concepts.
I'm a simple math lover. Mathloger uploads. I click.
This is a most amazing presentation of the most amazing theorem in Mathematics! A must see for all budding mathematicians.
As an aside, you gave tantalizing glimpse of finite geometry. Why don't you make a video on these topics? Particularly, the prime power conjecture for finite projective planes is a sadly neglected topic in RUclips. With its beautiful links with orthogonal Latin squares, this ought to be an eminently suitable candidate for a mathology video.
My proof for the little thing:
Let’s look at the following order of cards: 4/5/3/1/2 lets say that the number of inversions here is る
We do the cycling thing: 2/4/5/3/1
Consider this section: (4/5/3/1): notice how the number of inversions between these 4 numbers’ orders doesn’t change, let’s say that number is X
So the only thing that changes after the cycling is the number of inversions for the cycled number that takes the first place after the cycling (2) in this case, the number of inversions it had in the initial order being logically る-X. Let’s look at the second order now: it’s number of inversions now becomes [( (5) -1 ) - (る-X)] with (5) being the number of cards we are playing with and (る-X) being it’s number of initial inversions
Explanation: when it comes to ( (5) - 1 ): this is the max number of inversions a single member ( (2) in our case) can do with the other members because a single member cannot have an inversion with itself. When it comes to the subtraction i did: that’s because when we did the cycling we took out member from last place to first place cancelling out all the inversions it initially had with other members (because they are in growing order now) AND the it now has an inversion with members it did not have inversions with initially, basically the number of inversions it has now has become the compliment of the initial number of inversions ( their sum is equal to (5) - 1)
Knowing all this, We can now move on to a more general case with (n) number of cards placed horizontally which have る inversions and the section of cards that did not change order after the cycling has X inversions ( we can do this in a more general case because all the explanations i did apply there too): the number of inversions after the cycling is therefore X + [n-1 - (る-X)] = 2X + n -1 - る ( lets call it N )
Notice how 2X is ever therefore it does not determine the parity of N so the parity of N in comparison to る is only determined by the parity of n-1: N and る have the same parity if n-1 is even aka if n is odd and the opposite is true as well. So the sign of the permutation doesn’t change if n is odd but it changes if n is even.
Tell me if i have a mistake because im not experienced with this type of math
Excelent video Mathologer!
This didn't show up for me. Thought you might want to know that youtube seema to do some weird stuff again.
Yes, very strange, pretty much did not get recommended for two week :(
Thank you for all the hard work you put into that video!!! I will need to watch this one a few more times if I hope to understand it. Could you movtivate me with a really cool example of how quadratic reciprocity might be used to study pendulums or something?
46:28 okay, now that is the sneakiest conjugation I think I've ever seen
Isn't it soo nice :) ? You are the first one to remark on this.
@@Mathologer I didn't get it. Pls explain
I loved this. The law of quadratic reciprocity is one of the theorems that excited me when I was young.
16:26 the only squares mod 5 are 1 and 4. -1= 4(mod5) so the negative of the square 1 is also a square, and -4=1(mod5) so the negative of the square 4 is also a square, thus all the square on z/5z are squares.
Q.E.D
This video is quite the Magnum Opus! It makes a result from deep in the heart (bowels?) of mathematics accessible to mere mortals, and introduces a bunch of mathematical constructs (squares in Zn, rings, sign of a permutation) along the way. For me, watching this video felt like being a tourist on an eloquent expertly-guided tour of a hidden room inside a massive museum. I am left in awe of the inventiveness of the mathematical minds of yore and supremely appreciative of Mathologer's efforts to spread his enthusiasm for mathematics.
In response to Mathologer's query of what worked for me, with the first viewing I felt I was on top of the material until maybe the last 10 minutes. I expect that a second viewing will fix that, so... Congratulations Mathologer! I think you achieved your goal.
8:28 "actually, my first BOOK!" ... He says BOOK so loudly, I jumped.
Thanks for warning me, I checked the time in the video when I saw this comment, it was 3 seconds before, which was enough time for me to prepare.
R.I.P., people with headphones 😔.
It's all about the last five minutes of the video. No, if you only watch them you will probably not understand anything at all, but these are really the time of the genius. You've done all the footwork and now you have to remember "why did I do this" and "how does all this - by now - easy stuff combine at all to help me in my original problem". More often than not also the question "what was the original question again?" pops up at this time.
I remember how I in my thesis (in a completely different field, stochastics) was able to shorten a five page technical proof to a half page intuitive (and still correct) one and was immensely proud of myself. And then spent two weeks trying to remember how this achievement would help me at all.
Mind blown at the reordering property. I felt my brain was reordering too...
The ordering property is not mysterious. Apply a reodering first, then your permutation, then the inverse of the reordering. The inverse of a reordering has the same parity as the reordering (inverse of a product of two-cycles is the same product but backwards), so the final result has the same parity as the original permutation.
33:00 That ”Positively Sloped = Inversion” -trick really makes sense, for our Row-Up -> Column-Down -permutation; because, in any positively sloped pair, the upper card is the earlier one, in the row-wise order, and the later one, in the column-wise order (and vice versa, for the lower card); because the row-wise order is: ”Left -> Right; Top -> Bottom”, whereas the column-wise order is: ”Top -> Bottom; Left -> Right”. So, any card: A that’s up and right, comes before any other card: B that’s down and left, in the row-wise order; and vice versa, in the column-wise order: The down-&-left card: B comes before the up-&-right card: A. Also; in this Row-Up
-> Column-Down -permutation, we’re essentially switching rows and columns, which means that the internal order of any such pair gets flipped. Then, because we started with the cards in natural (row-wise) order (meaning: No Inversions.), the flipping of the internal order of any such pair amounts to an inversion. Whereas; in any horizontally or vertically aligned, or negatively sloped pair, the card that comes earlier, in the row-wise order, also comes earlier, in the column-wise order; thus, no inversions would emerge. 😌
Gauss and euler and reimann A JACKPOT!!!!!!!!!
It's like the best spin on the mathematician slot machine 😄
39:15 Proof that the diagonalisation always works for _distinct odd primes_ p and q.
First, imagine a coordinate system such that the card in the ith row and jth column has coordinates (i,j). (Origin at top left, numbers increase as you go right and down.) Also, p is the number of rows while q is the number of columns.
If it works R⬆D⬇surely it works for C⬆D⬇as well.
Because of the way the numbers wrap around, you can easily see that the row number of the card n is the smallest positive integer that is congruent to n modulo p (or the remainder that n gives when divided by p, except we now call 0 as p). Let this number be a. So for n=13, p=3, the row number is 1, while for n=12, p=3, it is 3. Similarly, the column number is the smallest positive integer that is congruent to n modulo q. Let this number be b. For n=11, q=5, we have b=1.
Now we have the coordinates (a,b) for n. For two cards to overlap, they must have the same coordinates, i.e. leave the same remainders when divided by p and q. Let it be assumed, for the sake of contradiction that there exist 2 distinct integers m and n, smaller than or equal to pq, such that the corresponding cards overlap. Now, if two numbers have the same remainder after division by a third number, their difference must be divisible by the third number. (Use Euclid's division lemma, it is easy algebra.) Thus, |m-n| is divisible by both p and q. Since p and q are distinct odd primes, it must be divisible by pq. But since at least one of them is than smaller pq, the only option is that their difference is 0, i.e. m=n.
This is a contradiction, since m and n were supposed to be distinct. Thus, no two cards can overlap.
Q.E.D.
I figured out that if you think of the grid as being on a torus, you can easily see why modular arithmetic and the rings apply here. The rows and columns are nothing but the two fundamental loops on the torus.
My 10yr boy watched this with me
- trying to inspire him- teach the language of math. 99.99999 he liked that part about who knows knows
99.999999...
So, 100? 😆
I used the "9.9999... really is equal to 10" to inspire my 11 yr old. Mathologer is great for all ages!
Just: WOW! 💪👍
Thanks a lot for your motivation but much more for your dedication! Was tough for me to follow and had to stop to rethink few times but then I grasped it in the end!
... still searching for the applause button (; 👏👏👏
These mini rings are what music theorists call modifications. Basically, you think of a clock with n values and you count around the clock face until you find the remainder
1. I missed this episode somehow.
2. YT showed this as watched, clearly it was very late and inmust have fallen asleep.
3. Mathematically this is one of my fav (i should make my ranking), but from the fact of how much you had fun it must be your fav too.
"I'll never do this again, promise" Actually, this was my favorite video of yours, so, please do it again?
Thank you so much for your effort in creating these enjoyable and accessible videos! As a technical writer who is entering the world of mathematics late, I find these really help me to internalize so much of what I need to go study. Cheers, Rick Lehtinen
Finally. Amazing.
Most fantastic! It reminds me of the mysterious connection of quadratic reciprocity with the linking number of two knots (in 3-space), which Gauss found a formula for (in terms of an integral). The diagonal dealing certainly looks like a (p,q)-torus knot, and maybe the R and C dealings are just circle running inside (and "outside") the torus. I'm sure others have thought it all through.
this is the only if not one of channels that i watch all its content. amazing as always!!
RUclips didn't show me this video until now and I'm annoyed so I'm commenting to try to convince the Almighty Algorithm to show it to everyone else.
Same
THANK YOU! Understanding QR has been a goal of mine. I didn't fully follow the argument on first pass, but now at least I have a Mathologized argument to study. Again, thanks for all your effort.
25:29 Your whimsy is boundless ^_^
This is the first time I know what happen about quadratic reciprocity, since many years ago I first time heard of it. Thank you very much.
What
An hour?
*hand hovers indecisively over play button*
and it's a masterclass?
*hand drifts away from the play button*
and I need to do my flute practice
*hand retreats to lap* (oh very funny, plz don't go there)
OTOH it's a video with a funny bald German in it
*hand approaches keyboard*
but there are probably no cats in it being cute
*hand falls to side*
*crickets*
*more crickets*
reciprocity prime what the? wat IS dat?
*Mathologer smiles impishly and wiggles the hook*
Command voice:" Lock all targets and fire at will! All ahead Warp 6! Take us in Mr Sulu!"
*Punches the big red button...*
I felt that
That was simply amazing. Such a nice proof, such clean presentation.
Nice shirt, did you know that Flammable Maths has an "avoid positivity" shirt?
Ah, yes; Flammable Maths, the ”Language Simp” of Mathematics! Such parodying (in this case, of Mathologer) definitely feels in-character, for him. 😅
Je suis impressionné par la qualité des traductions (commentaire et sous-titres) bien que je sois parfaitement anglophone ! Content de voir que les communautés de vulgarisation et d'enseignement mathématiques se joignent !
Yet, you opted to write your comment, entirely, in French 😅.
7:22 Proof
Commutative Law for any Z/n:
Since Z/n is just Z mod n for any whole number, we just have to add a+b and divide by n, then take the remainder, and that's our answer. Let C be the sum of a and b. But no matter what order you add whole numbers a and b, you always get the same number C by the ordinary commutative law of addition. So you will divide the same number by n no matter what order you add a and b in.
Distributive Law for Z/n:
The argument is basically the same, except you use the ordinary Distributive Law. You get the same number, which we will now call D, on each side of that second equation, so you will always be doing the operation D/n.
This completes my proof for any modular ring Z/n being valid for both equations.
QED.
Let me know of there is any error in this logic.
Or you could use the fact that a commutative ring quotient out an ideal is also a commutative ring. Z/n is the same as Z/(n), where (n) is the ideal generated by n.
@xavierstanton8146 Exactly 🎯!
Great video. This made me go read abstract algebra and group theory yet again to remember the properties of cyclic groups and permutations.
This fascinating proof put me in mind of a very impressive trick that one can do (if one has the time) with a normal pack of 52 playing cards.
Put the cards in suit order and then deal the pack into four hands of 13 cards. Assuming, in bridge notation, you are South. Pick the piles of cards clockwise starting with west - hence WNES. Now deal the cards a second time (D2) and pick them up in the same way. Continue until after D13. Now examine the cards and you will find that they are back in the original order.
My low grade graduate maths is not up to explaining this, and, of course, 4 is not prime, even though 13 is.
But it is another of the wonders of theories of ordering things.
15:59 What is sed zeven? :)
At least he doesn't pronounce it the way that people do north of The Wash.
"IZZARD seven."
:)
The ring (and field) of zhalen (integers) over the prime p=7.
ZED SEVEN
Maybe you're confused by the U.K. versus U.S. pronunciation of letter 'Z':
ZED vs. ZEE.
Also, as is easily done, you swapped Z/S which are a voiced/unvoiced pair.
@@ReasonableForseeability At the timestamp in question, Burkard actually does say Sed Zeven. This is easy to see when you contrast it with how he says it at 15:54.
It's a simple spoonerism, made likelier than most by the fact that "s" and "z" sounds are produced the same way, except that the vocal cords are vibrating for "z" but not for "s".
This channel is what I needed in my life. I am an established (in the sense that I probably earn more than u, the hater) programmer, but I lack in basic math skills. I have watched a few more videos on this channel and they are absolutely treat! Thank you for doing this man, I wish you were my teacher (as in school days. You are, right now hehe). :D
Subbed!
200 hundred different proofs. Me, an intellectual: Sorry this margin is too small to contain a proof 🤷🏻♂️
Fermat didn't have a proof, or if he did have one, it had mistakes since it was never checked by others
*prove me wrong*
'200 hundred'
That is a weird way of saying 20,000.
@gianlucamassari9431 You have the wrong Mathematician, in your pfp 😅.
Thank you, this sparks so much interest into looking into maths more again!