Q Series via Box Arithmetic
Q Series via Box Arithmetic | Math Foundations 239 | N J Wildberger
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We have a look at the interesting topic of q-series from algebra / analysis / combinatorics / number theory from the point of view of our new "box arithmetic" which prominently utilizes anti-boxes along with boxes. This is a chance to get some more familiarity with this curious new arithmetic in which the role of "nothing" is different from what we are used to.
To interpret Euler's pentagonal formula we introduce an unbounded extension of the current Box arithmetic.
This new way of thinking appears to have obvious computational advantages, in that the notation is capturing some of the arithmetic more efficiently. It also finesses the need for philosophical discussions about "variables".
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[Music] Good day everyone I'm Norman Wildberger we're looking at box arithmetic today I want to
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give some applications of relatively simple box arithmetic at the poly number level to an
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interesting topic in modern algebra / analysis / number Theory / combinatorics: namely q-series.
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Now if you're interested in this topic and want to know more about it I have an entire series on
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this topic in the Wild Egg mathematics courses channel so the story of q-series perhaps starts
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with Euler's pentagonal number theorem which is a formula for obtaining a certain sequence of
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numbers called pentagonal numbers which are sort of five-fold analogs of triangular numbers, and
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the formula is is given by this infinite product 1 plus Alpha to the N and ongoing from 1 to Infinity
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okay (1 - Alpha) * (1 - Alpha^ 2) *( 1 - Alpha3) Etc and as I've explained also in my Wild Egg Mass
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courses and there the Advice for amateur research mathematicians there talk about processes and a
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kind of convergence of processes that can allow us to think about what's happening with the identity
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in something like this without having to resort to you know infinite completed processes so what
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we're doing here is we're trying to expand this much as it as if it was a polunomial
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and and looking at various powers and happily any particular power like Alpha to the 5 only occurs
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a finite number of times it's the critical thing that allows you sort of to have a well-defined
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result of this thing and it's a very interesting sequence with plus and minuses and uh and these
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um these rather interesting numbers and it's particularly important because of a remarkable
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connection that Euler realized with the partition function so if P of n is the number of partitions
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of n number of ways of writing a natural number as a sum of other natural numbers like P of five
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is seven because you can write five as five or 4 + 1 or 3 + 1 + 1 or 3 + 2 or 2 + 2 + 1 or uh 2 +
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1 + 1 + 1 or 1 + 1 + 1 + 1 + one so this actually gives a recursive way of computing this P of n
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function and you see that these these terms here correspond exactly to the to the the exponents and
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the signs that are appearing up here. Okay so the questionwe want to address today is can we express
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this kind of ongoing identity in the context of box arithmetic so a good place to start is with a
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finite version and we'll just have a quick review of box arithmetic so if we just look at the first
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two terms (1 - Alpha )* (1 - Alph^ 2) that's 1 - Alpha - Alpha2 + Alpha cubed so how might
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we express that in box arithmetic? fFrst of all a quick review in box arithmetic: these days we
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have two zeroes, the black zero and the red zero this is zero and this is anti-zero and one of my
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viewers made the interesting suggestion that I should think of something that makes the the red
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a little more visible re the black for color blind people. So I'm going to try to make this red box
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typically more oval-shaped so it will have rounded corners and perhaps be a little bit thicker uh so
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hopefully that will uh distinguish it more clearly from the black ones are going to be right angled
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sharp rectangles okay so those are the zero and then a natural number is a box of zeros and the
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boxes can be either black or red boxes so this thing here is the basic new object a box inside
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a box so that's like a box containing zero that's the definition of the number one while the number
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two is a box like this with two black boxes inside it and if we have a black box with a red box
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inside it then that's minus one or anti-1 and The crucial thing is that if you add this box to this
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box by that we mean we dump out the contents into another box okay uh then the black box and the
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red box being inside the same box they annihilate that's the crucial particle anti-particle property
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of these black and red objects okay so that's why we get minus one okay and then going down one
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level to the polinomial version or poly numbers as I prefer to call them if we have a box inside
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a box inside a box so that's a box containing one one that's a new object we'll call it Alpha
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and then minus Alpha you get by um making red the top innermost box okay if we make the inside box
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red then we get Alpha to the minus one here's what Alpha squar looks like and so on okay so now let's
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look at 1 - Alpha * 1 - Alpha squ okay so here is 1- Alpha it's a box containing an empty box there
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so that's like the number one if this thing wasn't here that would be the number one and if this
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thing wasn't here we would have minus Alpha this one right here so this is the combination of this
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one and this one the sum of this one and this one 1 + minus Alpha if we take 1 - Alpha squar then we
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are taking the sum of this swing then we're taking the sum of this one with minus Alpha squ so minus
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Alpha squ is obtained by taking this thing and making red that um biggest inner box okay namely
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that one there so how do we multiply boxes well we just take all possible sums of entries in this box
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and entries in this box so this one here this is 0 plus 0 is 0 0 plus this is just this this plus 0
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is just itself so the only other wise interesting one is this box with this box so we have a red box
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with a black one inside it another red box with two black boxes inside it and we're adding them
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well to add boxes we dump them all in so we dump them all in we're going to get three black boxes
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but what is the color of the bounding new box that's determined by the number of red boxes
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that are contributing to the sum there are two red boxes contributing to the sum so it's like
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a parody kind of thing so the two odds make an even and we get a black box as a result okay and
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then we confirm that that is indeed the same as 1us Alpha minus Alpha 2 plus Alpha cubed so this
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is a way of doing it with a very primitive sort of Assembly Language box arithmetic let's have a look
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at the same computation but just at a little bit higher level by looking at Poly numbers as boxes
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of natural numbers okay so 1 minus Alpha in terms of that would be a box with a zero and a one anti
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the one anti is a red box with a black box inside it a black box with a black box inside it is one a
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red box with a black box inside it is the anti of one that's a new kind of mathematical object that
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hasn't been in existence up till now okay but it's playing an important role because it's a
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constituent of this box box here to create this element for us in a box setting and similarly
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1 - Alpha squared is the Box consisting of zero and two anti so when we multiply these we have
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to add all possible ingredients so 0 + 0 is 0 0 + 2 anti is 2 anti 1 anti plus 0 is 1 anti
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and 1 anti plus 2 anti is like 1 + 2 it's three but is there an anti or a not anti well there's
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an anti and an anti there's two antis they sort of cancel out and we just have um a single three
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so again we get one- Alpha - Alpha S Plus Alpha cubed but this will be a little bit easier for
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for most people to recognize because it's very much just like doing integer or natural number
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arithmetic with the provisor that there are these anes floating around so we have to sort of take
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proper care of them but that's not too hard so here is another example just going up one more
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degree 1 - Alpha * 1 - Alp 2 * 1 - Alpha cubed so there is 1 - Alpha there's 1 - alpha s and there's
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1 - Alpha cubed and then when we form the product of these things we have to form all possible sums
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okay like 0 plus 0 plus 0 that's zero uh the ones involving two zeros are one anti plus 0 plus 0
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that's one anti or 0 + 2 anti plus 0 that's that one or 0 + 0 plus 3 anti that's three anti the
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one's just involving a single uh like 0+ one anti plus 2 anti that's three or 0 plus um 2 anti plus
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3 anti that's five or uh 0 + 1 anti plus 3 anti that's four and finally 1 anti plus 2 an plus 3
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anti is six anti because there's an odd number of anes okay so there are eight constituents
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Al together the eight different possible sums of of taking one element here one element here one
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element here now of course at this level here we see the crucial cancellation this three and this
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three anti are going to mutually annihilate each other crucially they do not give us a zero we're
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not getting zero we're getting nothing like a real nothing okay uh not just a symbol for nothing
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which is in fact something we're just actually getting nothing just completely disappears and
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we just get 0 1 anti 2 anti 4 5 6 anti which is the same as 1 - Alpha - Alpha 2 plus Alpha 4 plus
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Alpha 5 - Alpha 6 which is what you get when you expand this thing out okay so now let's make the
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big jump to try to understand Oiler pentagonal formula in this context of box arithmetic but
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definitely we have to step outside the bounds of what we've been doing up till now because
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now we're not just working with finite numbers of finite objects because we have this unbounded
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product and this unbounded sort of sum as the result of it but nevertheless we can understand
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this in the context of box arithmetic in a modest fashion in a modest sort of unbounded fashion so
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we don't pretend that we can go to infinity or any nonsense like that okay we recognize and
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resp ECT the ongoing nature of both this product and this sum but we have a precise meaning to the
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identity of this equaling this in the sense that you know of equating the number of times a given
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object appears on the right hand side and on the left hand side okay so how does that go so well
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here's the 1 - alpha 1 - alpha squ 1 - Alpha Cub 1 - Alp 4 Etc okay so we have all of these things
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and we have to take this unbounded product so what we mean by that is we're going to take sums okay
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but we're going to take sums that we can actually do and we really can only do finite sums so that
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means that we have to insist that the entries that we take are mostly zeros in other words we're only
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going to take a finite number of nonzero entries in this sum we're just doing addition okay but
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we're not allowed to say okay what is this 1 a plus this 2 A plus this 3 a plus this 4 a
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etc we're not taking that kind of sum we're only taking a finite number of any of them okay that's
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by agreement that's what we mean okay and so then what we can do is we can tabulate how many times
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we get various things okay so when we look at this thing here now uh there's only one way of
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getting a zero 0 plus 0 plus 0 plus 0 to get a one or one anti those only only one way of doing that
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to take that one anti and the others are chosen to be zeros to get a two only the two anti there and
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zeros elsewhere to get a three well we could do that in one way by taking that three only
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that three so when I just say that three it means the rest zeros or we could take this one anti and
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that two anti if we add those we're also going to get a three because one anti plus 2 anti is three
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so we're going to get a three anti as well as a three and they're going to annihilate each other
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that's why there's no three here why is there no four well because there's a four anti that we get
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just by taking that one by itself or we could take this one anti and this three anti when we
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add them we get four that's going to cancel with this four anti so there's no four in there how
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about five well there's a five anti sitting there so we take that five anti or we could take a one
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anti and a 4 anti that would actually cancel with that one or we could take a 2 anti and a
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three anti so that's one five left over that's why there's a five there okay and um okay and then we
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can carry on to make this this this calculation uh but it's not you know a prior clear what the
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pattern is this pattern here is very regular but the pattern that it generates is um you
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know pretty interesting although Oiler figured out what the pattern was and was a very interesting
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and important pattern but it's not entirely obvious um at first so this is a calculation
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that should be done in quotes it's not covered by what we've talked about in box arithmetic so far
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it's an extension of what we've been doing and involves sort of unbounded or open boxes so I'm
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sort of leaving this one open you know suggesting that the thing carries on to me it would be some
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kind of intellectual dishonesty to put a bar there and to pretend that this is a closed thing that we
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can get the entire thing no it just keeps on going beyond our view that's the honest historical way
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of thinking about such sums now of the many hundreds of remarkable formulas in q-series
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um I'm going to just choose one more here to talk about today uh which is also due to Euler and also
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a very interesting uh and important uh discovery of his it's concerning a very similar kind of
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product product of one plus ala to the end now so a plus sign here so we have 1 + Alpha * 1 + alpha
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s * 1 + Alpha cubed and that turns out to be this series that starts this way with three dots of
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course 1 + Alpha plus alpha s + 2 Alpha Cub + 2 Alp 4+ 3 Alpha 5th + 4 Alpha the 6 and keeps
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going in some fashion which is not entirely obvious and remarkably Oiler realized first of
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all what the significance of this thing was that this was counting something pretty interesting in
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fact it was counting the number of partitions of an integer n into distinct parts so five can be
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subdivided or partitioned into distinct Parts either as five or S 4 + 1 or S 3 + 2 okay and
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the other ones don't count like 2 plus 2 plus 1 doesn't count because it has two repeated twos
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or or 3 + 1+ 1 doesn't count Etc so there's only three partitions of five into distinct parts so
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that's what that's counting and he by a little bit of clever algebraic manipulation was able to
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convince himself that this is actually the same as this product product of 1 over 1 - Alpha to 2
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nus one so we're taking the the fraction uh 1/ 1 - Alpha * 1 - Alpha cubed 1 - Alp 5 and um and in
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this context it turns out that this is then allows him to deduce that this number is also equal to
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the number of partitions of n into odd Parts into odd parts so for example five is equal to five and
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it's also equal to 3 + 1 + 1 all the parts are odd or it's equal to 1 plus 1 plus 1 plus 1 plus one
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all the parts are odd there's three of those so these two numbers are the same and that's not at
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all obvious but it's a result of this algebraic um identity here because these two series are
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encoding these two things so this remarkable work done in 1740 so let's have a look at this
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first thing how would we think about this from a box point of view so 1 plus Alpha now is the Box
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just containing the numbers zero and one that's what we think is the poly nomial or poly number
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1 plus Alpha sort of fundamentally it's just this box with a zero and a one in it here is 1 + alpha
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s here is 1 + Alpha cubed and you do the same thing you look at all possible and you do the same
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thing you look at all possible kinds of sums that you can get by taking finite numbers of nonzero
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entries here and you get a zero in one way you get a one you get a two you get a three however in two
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different ways and here I'm going to use that uh notation I introduced some time ago we'll put a
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little two as a left sub index here to denote a multiplicity so instead of having to write
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three three three different two different threes I'll just write a three with a two underneath so
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that means we have two threes floating around and what are those two threes well we could
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either take that three there or we could take one and two four there are two ways of getting
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that we choose the four here or we choose the 1 + three and then there's three fives four sixes
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5 sevens 68s 10 tens Etc so this is an alternate way of thinking about representing the information
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in this so let's stop and pause for a minute and think I know this is you know very unfamiliar to
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us but logically and uh you know objectively if we can be um you know which is really more essential
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you see when you're taking this these products here okay we have these exponentials um first
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of all there's the question if I ask an ordinary mathematician you know what what do you mean by
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Alpha here or in the usual story Q okay what is Q well Q sometimes means this and sometimes
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it means that and sometimes it means something different okay and it's not so easy to say what
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a variable is in mathematics even a a computer scientist has some you know it's all kinds of
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possible ambiguity as to what that term actually means but for us this Alpha is just you know being
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represented by this box with a one in it so it's completely cut and dried and furthermore
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when we do this multiplication like Alpha squ time Alpha cubed okay that's Alpha to the 5th
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but it's really an addition isn't it I mean yes we're we're saying we're doing multiplication
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but computationally it's really an addition you know so what's happening here is closer
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to the computational heart of what we're really doing and moreover you can see that it's more
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efficient to represent what's going on because these Alphas in some way are extraneous right
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they're not really essential part of the story there sort of placeholders or or ways of pinning
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uh these exponents uh to something but here there are no alphas and also there are no plus signs
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okay so just have a look at um you know the the compression here right I mean in in this piece
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of information I've compressed I think the data a lot more efficiently so you know my guess is
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that uh you know these little uh Tic Tacs that are flying around in our neighborhood all these little
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aliens perhaps that are living under the water or whatever you know that we're soon probably
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going to be um chatting with um you know in the next few decades my guess is that you know this
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might actually be more in keeping with what they do we'll have to ask them when we start having
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serious chats with them won't we and what about that expression that happened in the denominator
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of Oiler formula the 1 - Alpha * 1 - Alpha Cub * 1 - Alpha 5th Etc product of 1us alpha is to
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odd Powers so that turns out to be at least the first few terms one - Alpha - Alpha Cub plus Alpha
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4th - Alpha the 5th plus Alpha the 6 and so on in some fashion which is not at all obvious so from
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a box point of view this is 0 one anti box this is a box with 03 anti there's a box with 0 five anti
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and again we just do the all possible sums where we're just taking finite numbers of um of entries
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so never or more than a finite number of nonzero entries so there's a single way of getting a zero
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and there's a one anti there's a three anti there is a four that's some one anti plus three anti is
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four there's a five anti there it is there's a six because the six is appearing uh not over here but
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it's appearing as 1 anti plus 5 anti okay and that's anti plus anti so that's just a regular
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six and then there's a seven anti and then there's 28s and then two 9 anes 2 10 2 11 an Etc you can
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kind of get the idea that the odd numbers are all appearing with ones so that should also have an
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ante and then the the identity that the the the first product was equal to one over the second
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product really amounts to uh the expression that that first big box that we got times
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this second big box that we got okay when we multiply these two we should just get one so
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that's a a remarkable thing okay so there's some interesting but non-trivial pattern here there's
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another interesting but non-trivial pattern here okay so what happens when we multiply
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these two things so 0+ 0 is 0 uh 0 + 1 uh and 0 + 1 anti so we get a one and a one anti they cancel
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so there's no ones um twos there's a 0 plus two that's two and one plus one anti is two anti so
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that two and that two anti cancel threes uh well there's two threes here so 0 + three there's two
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of them so we get two threes here um but we also get a a three anti here okay but we also get a one
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anti plus a two that's another three an so all the three anes cancel and the same thing happens with
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every single natural number go through it every single number is cancelled with something else
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and at the end of the day the only thing that's in this big product is the single zero that we
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got by adding that zero and that zero so the Box containing zero is one and that's the content of
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Oilers identity so it's a an interesting alternate way which one can argue Cuts closer to the chase
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of what we're really doing arithmetically which circumvent the need for elaborate discussion
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about the roles of variables and which uh is simpler notationally at least I think one can
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argue that and um and it's fascinating that these anti objects play an important role in
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this arithmetic nevertheless you can say yes okay so you're redoing something that's well known in
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in a different way with different terminology and that's right but in my next video I'm going
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to give you another application of box arithmetic to another aspect of number Theory actually also
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a very important formula of Eulers having to do with the Riemann zeta function and we're going
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to see in this case a a highly non-trivial and novel understanding of what's going on with a
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very important number theoretical situation by taking this box arithmetic point of view
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I hope you'll join me for that I'm Norman Wildberger thanks for [Music] listening