More arithmetic with negative msets

3,312 views Mar 14, 2023 Math Foundations

Let's illustrate this new box arithmetic using anti msets to define negative numbers, with some more explicit examples. This involves using integers not just for coefficients of polynomumbers / polynumbers, but also to crucially extend the possibilities for exponents to negative numbers. This introduces a vital symmetry into the world of algebra with polynumbers that is almost entirely missing from usual algebra courses: where essentially Laurent polynomials appear not in a calculus / analysis context, but as pure objects of analysis. This will have major consequences for our understanding of algebra going forward. A big thank you to all my Patreons supporters and Members of the Wild Egg mathematics courses channel. Video Contents: More Arithmetic with negative msets Anti-objects Negative mset Examples of integer arithmetic Examples of integer polynumber arithmeti Multiplicative arithmetic with integer polynumbers Integral (Laurent) polynumber Arithmetic with integral polynumbers

More Arithmetic with negative msets

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foreign [Music] I'm Norman Wildberger and I'm going to be extending this novel arithmetic of ours based

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on multi-sets what we might call box arithmetic in the direction of negative numbers which we

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got at by introducing anti-m sets a rather novel arithmetical feature that's somehow mysteriously

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parallels the world of modern 20th century particle physics with the particle anti-particle

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duality or symmetry that seems to pervade the physical World perhaps something parallel to

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that actually sits at the heart of mathematics as well even though it hasn't really been explored

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very much So today we're going to have a closer look at all this interesting arithmetic with some

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examples okay so let me just remind you that the setup is that we're starting now with our zero

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World consisting of two objects not a single zero which is the empty amp set but also the

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anti-zero denoted with the same notation but with a superscript a and I'm also going to introduce

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some possible variants that we could use in some situations one being just to replace the

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black bounding brackets with red bounding brackets parallel to what an accountant would do in terms

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of black and red representing you know positive and negative assets liabilities and perhaps when

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we're using um sort of a typed situation with just on a line we could also perhaps introduce

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perhaps braces just a suggestion to give us an alternate way of representing anti-am sets

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Okay then we had these two very identical tables for the addition and multiplication

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zero plus zero zero zero plus anti-zero is anti-zero same thing here and add to

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zero plus anti-zero is zero and exactly the same actually for the multiplication

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and curiously especially here this gives us this unusual law that zero times anti-zero is anti-zero

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rather than zero which contradicts our experience that zero times anything is always zero so this

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makes us a little bit uneasy this is certainly a different kind of arithmetic than we're used to

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and as some viewers point out this also can be leading to a failure of the distributive law

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at this very Elementary level with just zeros and anti-zeros so for example if we look at zero plus

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zero times anti-zero Well zero plus zero is zero so that's going to be exactly this we're going to get anti-zero but on the other hand if we look at zero times anti-0 plus zero times anti-zero this

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one's an anti-zero this one's an anti-zero and according to this the sum is then going to be zero

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so the usual distributive law just at the level of these zero objects is not going to be valid

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now that does not mean that the actual arithmetic that we're primarily interested in having to do

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with numbers and poly numbers and multi numbers Etc is also necessarily inconsistent but it just

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means that we do have to be rather careful about about questions of The Logical structure of things

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and ultimately we want proofs and perhaps models of this that will ensure that what we're doing

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is is logically consistent at this stage here it's exploratory and in that Spirit we're going to give

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you some examples today of these these anti-m sets that are built up from this anti-zero and

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also then negative numbers and then intervals integral poly numbers and integer polynumbers

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extending the the negativity into the world of poly numbers so in some sense this is like a an

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Assembly Language approach to arithmetic you know we're we're going really right down to a sort of a

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fundamental level here and uncovering structure that's a little bit unfamiliar even though it

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does lead to pretty familiar arithmetic but we get to look at that arithmetic in a new light

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okay so last time we saw that as a consequence of having anti-zero we should probably extend

Anti-objects!

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this anti-notion to msets very generally so that for any mset or object M in this arithmetic this

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is going to be an anti-object which we denote by m super a okay which has the property that

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if M and anti-m are together in an mset then they cancel out just like an electron and a positron

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and we just get an empty mset as a result and we also saw that this m a this anti-m can be

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evaluated by essentially adding the anti-0 to m in either order and in particular this shows

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that the ante of anti-0 is is zero and extending that sort of black red approach to notation let me

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suggest that we could use the following sort of two-dimensional version of that so if here's an

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amp set M okay it's a box with a couple of boxes in it this is an empty box this box here has uh

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two empty boxes in it okay so this would be what we would usually call this would be like one plus

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Alpha squared in polynumber language if we wrote it out in terms of you know linear

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structure there's a big mset with an empty M set there and here's an M set with head with two M sets inside it so if we were looking at this mset and trying to understand its anti-object so

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M super a then I propose that we could denote that with a red bounding box so we take this

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thing here and we we copy it essentially with the distinction that the outside box the containing

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box is now red instead of black but the interior things are just the same as they were over here

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and in the linear case that would involve replacing these outside brackets with red

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brackets instead of black brackets but crucially what's inside stays the same now if we didn't have colors at our disposal okay then we could replace these red brackets with say

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braces or whatever else we wanted to use I I like this two-dimensional uh representation because our

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minds are very much oriented to two-dimensional visualization and understanding and so this makes

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uh demonstrating things a little bit easier for us if we have too complicated something

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like this it's fine for an AI or a computer but our physiology prefers to see things in this form

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here so this is actually quite a useful way of illustrating some things okay so the first use

Negative msets

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of these anti-objects is to Define minus one or I like to write one bar so one bar or minus one is

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the mset consisting of a single anti-zero and it has the property then that if you add one bar with

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one then you're going to get a zero or the empty set and then the multiplication of one and one bar

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follows this sort of familiar pattern and then we can extend this negation to natural numbers

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so if N is a natural number then we can Define n bar or minus n to be one bar times n and that

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then introduces an extension of our fundamental objects from natural numbers to integers so so

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we create integers by multiplying natural numbers by minus one so now we have a wider framework and

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it's really this wider framework that is a very pleasant environment with lots of interesting

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applications and furthermore if we have a more General mset a pure mset M then we can Define

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its negative also M bar or minus M exactly in the same way we just multiply by one bar

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so I remind you that in this world we can always add and we can always multiply M sets Okay it's

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there's no requirement that these things necessarily be numbers those are just the top level things that there's a whole range of different more complicated kinds of objects and

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these operations extend to all of them and then we also saw that another way of thinking about this M Bar was that it consists of the mset whose elements are the antes of all the elements of

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capital M so it's the mset of m a as M ranges through capital M so let's do some examples of

Examples of integer arithmetic

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some simple integer arithmetic two plus one bar plus four bar well two is the m side consisting

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of zero zero one bar is the m set consisting of just the single anti-zero and four bar is

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anti zero anti-zero anti-zero anti-zero when we add them we just dump everything out into

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a single M set so we get zero zero anti-zero here and then a bunch of anti-zeros further

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and then once we have them all in an M set then the zeros and the anti-zeros will mutually

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annihilate each other okay so say that pair will disappear and that pair will disappear and we're

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left just with anti-zero anti-zero anti-zero which we recognize as three bar or minus three

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good okay what about three times two bar well three is a zero zero zero and two bar is anti-zero

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anti-zero how do we multiply these well we have to form the Big M set consisting of all possible

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sums of elements here with elements here so we have to be able to evaluate zero plus anti zero

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which we know is anti-zero and there's going to be six possible such things so we're going to get

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that thing which we recognize as six bar okay let's illustrate uh the sort of the fact that multiplication and in fact addition also

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are not essentially binary operations in this world they're in principle uh things that can

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input a whole bunch of things at the same time so let's multiply two Bar times four bar times

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one bar so that's anti-zero anti-zero there's two bar multiplied by anti-zero

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anti-zero anti-zero anti-zero there's four bar multiplied by just anti-zero that's one bar

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so what are we gonna have to do we're going to have to take a multi-set which consists of all

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possible sums of elements here F elements here with elements here okay so I'm going to take one

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of these anti-zeros one of these anti-zeros and one of these anti-zeros we're going to add them

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so over here is on the side let's just compute this thing anti-zero plus anti-zero plus anti-zero

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well anti-0 plus anti-0 is zero and zero plus anti zero is anti-zero so this just gives us

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anti-zero in fact more generally if we had K anti-zeros all added together

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then whether we get zero or anti-0 as the final result depends on whether K is even

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or odd if there's an odd number of anti-zeros we're going to get an anti-zero if there's an even number of anti-zeros we're going to get a zero and because the addition and the

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multiplication tables are the same that's the same statement if we replace the additions here with multiplications so anti-zero times anti-zero times anti-zero is also going to be anti-zero

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okay so it's clear what we're going to get here we are going to get a whole bunch of anti-zeros

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a whole bunch of them one two three four five six seven eight and so the final result is eight bar

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okay so now we can convince ourselves that the arithmetic that we've cooked up here really does obey the same rules as our usual integer arithmetic with respect to the operations addition

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and multiplication okay so let's go down one level to the poly numbers so I remind you that a poly

Examples of integer polynumber arithmetic

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number is an mset of natural numbers so let's look at an example here's an mset of natural numbers p

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is the mc00225 these are all natural numbers and this has an interpretation in terms of sort of

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more usual notation as 2 plus 3 Alpha squared plus Alpha to the five and if we connect with

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the my usual preferred notation for poly numbers like we use in the algebraic calculus course or

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that we've talked about extensively in the Math Foundation series we can think of a poly number

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it's like a polynomial as a list a vertical lists say typically of coefficients the actual variable

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itself is sort of secondary it's sort of implicit it's really the coefficients that are the crucial thing so here there's a two there's zero Alphas three Alpha squares zero Alpha Cube zero Alpha

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to the fourth one alpha to the fifth so let me just point out again the important distinction

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here between this more classical approach to what a poly number is namely as a list okay

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um and this this other this new point of view this mset point of view towards what a poly number is

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okay so these are closely connected in the sense that this thing here is really the multiplicity

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list for this m set this is keeping track of how many of each of these objects we have because

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the objects appearing here are naturally ordered they're natural numbers so 0 1 2 3 Etc so we can encode this information just sort of by counting how many zeros there

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are how many ones there are how many twos there are how many threes there are Etc

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but there's a sense in which this is actually a more basic or fundamental

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object okay because the ambiguity that's appearing over here that you're allowed

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to add additional zeros below wherever you are and doesn't change that is sort of removed at

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the multi-set level because we don't have to say that we're just talking about a multi-set

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so this is a a subtle but important shift that I do want you to take on board that

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perhaps a poly number is really sort of more fundamentally on mset of natural numbers

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okay so now we want to enlarge that view because now we have integers okay we have integers so

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let's see what happens if we extend this notion to involve integers so here's the negation of this p

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okay how do we negate an amp set of P bar or minus B well we just take the anti of each one of these

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so we have zero anti or anti-zero anti-zero anti-2 anti-2 anti-2 anti-five and that's also

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a poly number A Familiar kind of poly number but one involving negative coefficients so

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if we wrote it out in as a as a poly number in Alpha it'd be two bar plus

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three bar Alpha squared so there's really three um Alpha squares here but there's there's antes here

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so it's representing three bar Alpha squared and one bar Alpha to the five and as a vertical list

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we could write a minus two zero minus three zero zero minus one or I prefer 2 bar 0 3 bar 0 0 1 bar

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notice also that I'm putting the one bar here okay we're sort of forced to here because we

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don't have a subtraction operation so it would be wrong for us to say minus Alpha the five not

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wrong but not really consistent with what we're doing so far so far we really only have addition and multiplication so strictly speaking to be consistent we probably also should put a 1 here

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okay it's probably a little bit more consistent if we treat all these things sort of uniformly

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and you know treat the ones uh same as the twos and the threes I suppose we could even

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go one step further and write zeros zero Alpha et cetera but that would that's maybe going a bit too far but anyway so this is the negation of of this thing here in sort of an obvious way

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but so we're also allowed to mix and match these things so if we can have a situation where we have a poly number which has both positive and negative coefficients for example this Q thing here have a

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look at it it's uh M said 0 1 a or anti one two a and three so as a poly number this would be well

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there's a single one um uh there's one there's a one to the a here so that's like uh one bar

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times Alpha and there's a one bar times uh Alpha squared and there's a one bar times Alpha cubed

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so what would happen if we added say p and Q well we have to put them all together in a Big

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M set okay so um there's a zero here a zero here and a zero so it'll be three zeros there's a one

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a there's a two two two we'll put two two two in and we also have to put a 2 a in an anti-2

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but that 2A is going to cancel with one of the twos okay so they're going to go so we're just

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gonna have two two and then we'll get a three and also a five so the result would be three plus one

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bar Alpha plus two alpha squared plus one alpha cubed if we wanted to plus one alpha to the five

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and you know more colloquially we would write this or say it as 3 minus Alpha plus two alpha squared

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plus Alpha cubed plus Alpha Phi that would be the more familiar uh sort of high school way of saying

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it but that does assume that we have a subtraction operation which we're not assuming at this point

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okay so that's the way anti numbers generate negative coefficients so it's the antes inside

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the brackets that are generating negatives sort of outside the brackets maybe that's

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one way of thinking about it great so let's do a little bit of multiplicative arithmetic

Multiplicative arithmetic with integer polynumbers

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with these integer polynumbers say 2 plus Alpha times 3 minus Alpha squared first of

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all let's over here just draw a picture of this thing here okay so what is 2 plus Alpha

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so it's a box that has what in it well it's going to have two empty M sets so we just had that that

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would be the number two okay but it also has an alpha in it what's an alpha well an alpha is

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when you put a 1 inside this box here so 2 plus Alpha looks like this

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and now we're multiplying it by 3 minus Alpha squared what does that look like well that's also a box okay and it has a 3 in it okay okay so it's the three zeros in there that

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makes the three and then there's a minus Alpha squared so what does that look like well an alpha

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squared would be like when you had um a box here with two empty boxes inside okay but we have a

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minus Alpha squared so I actually need to change colors and introduce a red box because we have

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um we have a minus Alpha squared so I'm going to put a box here and a box here have a look at this

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okay so if we didn't have this thing here okay if we remove those three things and just looked at this big black box with a red box inside it and two black marks this would

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be the negation of the the thing that we got by replacing this red box with a black

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box okay if we replace the red box with a black box then we would have Alpha squared

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all right so negative Alpha squared you get that by by taking this thing here and and changing the

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color of all the ingredients in other words replacing all the ingredients with antes so

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the minus Alpha squared is represented by this big box with a red box inside it containing two black boxes okay so you know we could do the arithmetic at the Box level

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but uh so let's let's uh sort of do it in a more regular fashion so 2 plus Alpha

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so um what's this going to look like so this is going to be uh an amp set it's a poly number so

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it's an M set of natural numbers or these days integers so 2 is representative zero zero and the

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alpha is represented by the one okay and we have to multiply it here's another M set consisting

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of what well there are three uh zeros in it and then there's a minus Alpha squared what's that

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well that's like a 2 that's we're sort of representing Alpha squared but we have to put an anti 2 there because we have a minus one of them

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okay so let's see if we can do this calculation um so there's there's quite a lot of terms here

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there's three times times four so it'll be 12 things all together so what do we have to do in order to multiply these two things well we have to take all possible sums okay so zero plus

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zero is zero so I'm going to do zero times only so I get zero zero zero two and I'm just adding

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0 to all of these okay I have to do the same thing with this zero add zero to all of these

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now I have to do the same thing with this one add one to all of these 1 plus 0 is 1 1 and 1 plus 2 Alpha so that's an interesting here so

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we we need to calculate over here perhaps a different color so what is one plus two a

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so 2A um so another way of thinking about that is it's one plus two plus anti zero right so

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anti the anti of two okay one wave of saying what that is it's 2 plus anti zero so if we write it

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this way and sort of we use distributivity we would get three plus anti-0 which is

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anti-3 so according to that calculation we should have a three a there

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and in terms of uh an actual um you know poly number what poly number is this okay so we get

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uh six zero so that's six uh we have these three ones here so that's like three Alpha

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and then we have these two uh two A's or anti-2s okay so that's like Alpha squared but with a minus

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so minus two alpha squared and there's three a anti three corresponds to minus Alpha cubed

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does that look like what we'd expect two times three uh three times Alpha

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yes minus two alpha squared yes and minus Alpha cubed good

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let's do another example say one minus Alpha cubed okay so that's going to be what well one minus

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Alpha as an M set in the sort of more familiar way that's a zero representing the one and the

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minus Alpha it's like a one but with a an a here okay anti anti one okay so times zero anti one

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times 0 and T1 all right so we're going to get an M set consisting of all possible triple sums

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element here element here element here adding them up okay so um so uh okay so there'll be

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like a zero plus zero plus zero that'll be one and then there'll be a zero plus zero plus one

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a okay and then there'll be a zero plus one a plus zero it'll be a zero plus one a plus one a

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that's four of them and then with the 1A so one a plus zero plus zero one a plus

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zero plus one a one a plus one a plus zero and one a plus one a plus one a

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okay what are we gonna get when we do all this okay so the first thing here is zero plus zero

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plus zero that's going to be zero what's this thing here zero plus one a so that's gonna be one a and in fact uh so is that and so is so is so is that so we're gonna get one a

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and so is this one so it'll be one a one a one a so I dealt with this one here this one here

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this one here and this one here okay what about this thing here zero plus one a plus one a well

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what is 1A plus 1A anti one plus anti one well uh let's do a little sum over here

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okay so anti 1 plus anti one okay well one way of thinking about what an anti one is it's one

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plus anti zero so maybe we could write it like this okay that makes it clear that this is 2

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plus anti zero plus anti-zero is zero so we're getting two so anti one plus anti one is two

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it's the same kind of story with the parity there's there's an even number of antes and so we're getting something it doesn't involve aunties so this thing here is going to give us a two

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as is this one and this one oh sorry not Plus just a single two and also

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another two and another two and finally this last one here 31 plus 81 plus 81

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same kind of story but we're getting a three with an ante

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and if we wanted to write that out as a poly number let's squeeze it down here

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maybe in blue so this is what this is one there's three of these one a so that's minus three Alpha both things there there's three of

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these so plus three Alpha squared and there's one of these so minus a single Alpha cubed

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so we've done a little binomial expansion in this mset world okay but now we're able to

Integral (Laurent) polynumbers

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move in a different direction still staying in the world of poly numbers okay but we're going to use the fact that we have integers now in uh more subtle more expanding fashion

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and this leads us to what we're going to call integral polynumbers or perhaps one could say Laurent poly numbers but I'm not going to actually use that notation I'm

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going to prefer integral poly numbers so an integral poly number is an mset of integers

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poly number we said before was an mset of natural numbers like zeros ones twos Etc

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now an mset of integers

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so for example here P here's an M set of

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editors negative 2 negative one negative 1 negative one zero three three

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and we can now interpret this in an algebraic fashion by being sort of consistent and thinking

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of these negative sizes just representing powers of alpha which have negative numbers as exponents

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so this two bar here is giving us an alpha to the minus two these things here these One bars

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are alpha to the minus one there's three of them the single zero is a one and these two alpha cubed

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these two threes give us two alpha cubed so in terms of a box here's a box that corresponds to

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this so there's a box okay and it has a two bar in where's the two bar um well uh this thing here

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is essentially the two bar because a two would look like a box a black box with two empty boxes

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inside it if we want to negate that what we have to do is we have to take the the M side

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which consists of the antes of all the ingredients so we have to take the antes of the two involved

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here so this is the um what the the two bar looks like and similarly the One bars here that one that

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one and that one represented by a black box with a red box inside the zero is just represented with

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the single empty set and the threes well there's three and there's another three

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now what in terms you know in terms of our familiar uh notation for for amsets if we

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just look at the positive part of this maybe I should put it down a little bit

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let me let me write it like this okay so we just looked at the the positive part there's a single oh one okay there's a zero Alphas zero Alpha squared and there's two alpha cubes so

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that would be sort of the our usual poly number representation as a multiplicity list uh with

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just positive powers of alpha but now we have also three Alpha to the minus ones and an alpha to the

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minus two so we can represent that by essentially going in this other direction going up from this

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this bar here okay so there's three here in this position and one here in this position

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this is sort of the alpha to the zero position Alpha what to the one alpha to

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the two alpha to the three this is Alpha to the minus one here's Alpha to the minus two

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so this is sort of consistent with what we're doing previously but we're kind of going in a different direction so in my next video I'm going to have to say more about you know how

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to think about these integral polynomials like what what do they mean how do we represent them or how do we think about them but let's just do a bit of arithmetic with them to give us an idea of

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you know how this works okay so let's add two integral poly numbers here's an integral poly

Arithmetic with integral polynumbers

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number one bar one bar two here's another one three bar one two four to add them we just make

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a Big M set dump everything together so there's a one bar there's a one bar there's a two there's a

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three bar there's a one there's a two there's a four okay okay so we could rewrite this maybe a

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slightly more organized fashion like this one bar one bar uh one two two oh there's a three bars

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one we could put maybe the three bar there okay three bar one bar one bar uh one two two and four

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okay and in terms of you know interpretation in terms of poly numbers this would be here uh this

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would be two alpha to the minus one plus Alpha squared okay that thing there and this thing

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here would be an alpha to the minus three plus an alpha plus Alpha squared plus Alpha to the four

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okay round bracket and what are we getting out of this we're getting um an alpha to the minus three

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plus uh two alpha to the minus one plus Alpha plus two alpha squared plus Alpha to the four

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does that sound reasonable yes I think it does

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okay here's another example a little bit more sophisticated now because we're going to combine these two rolls of negative numbers in slightly different ways

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okay now have a look at this thing here three bar one bar two ante

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one bar anti zero two three first of all let me Express the meaning of this in more familiar

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sort of poly number language what is this thing representing it's representing the poly number Alpha to the minus 3 plus Alpha to the minus one and without the ant here would be an alpha square

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but there's an anti here so we have to think of it as being minus Alpha squared okay and what about

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this one over here well the one bar anti so one bar is representing Alpha to the minus one but

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we have a Nancy so it's minus Alpha to the minus one plus one plus Alpha squared plus Alpha cubed

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okay and then when we uh put them all together well let's do it over here so we just dump

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them all together three bar one bar two anti 1 bar anti zero two three okay and what is that

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well now there's some cancellation that happens because in the same asset we're seeing some

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objects and they're anti so there's a one bar here and there's an anti one bar they're going to cancel there's a a 2 and an anti-2 they're going to cancel the three bar and the three

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they are not aunties they're not going to cancel they stay where they are so we're going to get a three bar what do we say these guys cancel these guys cancel we're going to get a zero and a three

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so over here in on the polynomial side we should be getting Alpha to

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the minus 3 plus 1 plus Alpha cubed that I suppose that's reasonable

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okay so let's do a multiplication so two bar one times one bar one bar anti-3

36:24

okay so in terms of polynomials or poly numbers this would be this one here is like Alpha to the

36:32

minus two plus Alpha times 2 Alpha to the minus one plus three ante that's minus Alpha cubed

36:51

okay so at the level here what do we have to do we have to make a Big M set consisting of all possible sums of these things all possible sums okay so um two bar plus one bar what is that

37:05

well two bars the same as minus two uh one bar is the same as minus one when we

37:12

add them we're going to get minus three or three bar so we're going to get a three bar

37:18

that's that's sum this sum is another three bar we get two bar plus three anti let's think about that

37:25

um maybe I could do that over here so 2 bar plus three anti so the three ante you can think of as

37:36

three plus and t zero okay when we do it that way the two bar plus the

37:43

3 will give us a one so we get one plus anti-zero which is one anti

37:52

so uh which one was that that was this two bar plus three anti again sort of an odd number of

38:01

anti's gives us an anti but otherwise it's like two bar plus three so it's like one so we're

38:09

getting um one ante so we've done um three sums so far okay so we have uh still one plus one bar

38:19

well that's zero one plus one bar that's zero and one plus three

38:26

anti that's the same story as that the one plus three is four so we can get four ante

38:35

so what is this telling us we should be getting uh two alpha to the minus three it's that a one

38:44

plus plus two plus two a one ante so that's minus Alpha and a four anti so minus Alpha to the four

38:55

does that correspond to what we would expect two alpha to the minus three

39:02

um minus Alpha yeah two yes minus Alpha fourth I think that's that's right

39:14

so it seems as if we have an interesting arithmetic but now we've we've used these

39:20

negative numbers to enlarge our arithmetic or algebra of with polynomials or polynumbers in

39:26

in in in two ways not just in terms of adding negative coefficients the possibility of that

39:32

but also in terms of adding the possibility of negative exponents and this is a a departure from

39:40

the usual story of polynomials okay and I'm going to talk more about that in my my next video how

39:47

to think about that we're going to see some some sort of new insights into you know what the world

39:54

of polynomials and their algebra really looks like I'm Norman Wildberger thanks for listening [Music]