Alternating

3,170 views Jun 15, 2023

We introduce subtraction into the world of arithmetic with integral polynumbers. This presupposes prior familiarity with negative numbers, which in earlier videos we introduced via the basic duality between msets and anti msets, or just amsets. The fundamental reflection symmetry denoted by sigma between natural numbers and their negatives allows us to define symmetric and anti symmetric or alternating polynumbers. This gives us an arena for very fundamental yet elementary investigations into Algebra which have largely been missed by our educational system focused, perhaps too much, on functions due to their role in analysis. We introduce the basis B_n of symmetric polynumbers, as well as closely related basis A_n of alternating polynumbers. The algebraic relations between these become important when we investigate a third class C_n of central polynumbers which play a big role in many areas of mathematics, including q-series, representations of SU(2), hypergroups, quantum groups, and in physics also. Video Contents The subtraction operatio Fundamental reflection symmetry Symmetric Binomials B Product formula for symmetric binomials Alternating binomials A Products of Alternating & Symmetri Central polynumbers C_n The Central Importance of central polynumbers

The subtraction operation

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foreign

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[Music] I'm Norman Wildberger we're talking about box arithmetic and a big thank you to

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all my Patreon supporters and members of the Wild Egg Mathematics Channel today

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we're going to be extending our algebra that we're developing in a quite novel direction

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almost a missing chapter of say high school algebra what we get when we enlarge our view

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of polynomials to include the sort of Laurent type polynomials with negative exponents this is

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going to allow us to analyze more carefully this notion of symmetry and anti-symmetry or altered

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any aspect of poly numbers that I introduced last time so at the end of the video I'm actually going

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to connect the the main objects that we're going to be creating today which are going to be called

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a central poly numbers with actually a whole range of different mathematical topics some of

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them are quite important in fact sort of really central importance in 20th century mathematics

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and physics so I'll just touch base very quickly with those things but I want you to appreciate

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that even though what we're doing very elementary things here okay we are arriving at objects which

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are a really very very fundamental for mathematics and physics okay so before

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we get onto that though let's just enlarge our arithmetic a little bit and introduce

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subtraction to our operations that we're doing with these msets so it's simple enough because

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we have a minus one now so we can Define if A and B are msets or anti-msets or maybe if we want to

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include both of them at the same time I might say a slash m sets okay to include the families

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of msets and anti-msets and and combinations of them so then we'll define A minus B to be

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simply A plus one bar B or in other words A plus minus one times B so here's an example here's an

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mset perhaps A and here's another mset perhaps B and we're taking the difference A minus B

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so this thing here okay um minus this thing here the definition is that we should take the same

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thing here and instead we should add minus one there's minus one times this second one

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now in terms of a more standard notation this thing here you see here it has a 2 inside it has

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an empty M set that represents zero and this thing here is an anti one okay so that's what this thing

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is and this thing here well there's a two there's another two so two two there's a one bar okay it's

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the negation of one and here is just an anti-zero okay so when we multiply the second one with the

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minus one what do we get okay well multiplying by minus one what it does is it takes all of the

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entries and it takes their antes so that means that all the bounding boxes have to change color

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from black to red or from red to black so this one here is going to change into a red box with two

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black ones same with this this one here is going to change into a red box with a red box inside

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this one here just going to change into from an anti-zero to a zero okay and once we have this

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then the addition is of course just the usual combining them so we just put everything in in

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an mbox but when we do that this thing here this 2 and this anti 2 are going to annihilate each other

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because these these things are antes they're aunties because they're the bounding box the

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containers are of different colors one's black and one's red and otherwise inside they're identical

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so they're identical except that one's the anti of the other so they're going to annihilate we'll

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still have one more of them left over we'll have two empty boxes and we'll have one of these and

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one of these and then when we list what we have we have uh one bar ante so that's that thing there

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okay and if it was black on the outside that would be a one bar but it's red on the outside

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so it's a one bar ante there's two zeros there's our anti-2 and this is an anti one okay so that's

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how we do subtraction in this mset world okay so when we went from poly numbers to integral poly

Fundamental reflection symmetry

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numbers the big step was that we went from msets of natural numbers to msets of integers and that

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means we have an interpretation now of polynomials with also negative integer exponents not just

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natural number exponents these are integral polynumbers and in this world there's this

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fundamental symmetry the sigma sort of reflection that takes alpha to the n to alpha to the minus

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n and then with respect to that we defined an I Poly to be symmetric if Sigma of p equals p and

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anti-symmetric or alternating if Sigma p equals minus p or p bar and I claim that the study of

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symmetric and alternating integral polynumbers is really something of a missing chapter a very

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elementary algebra something that you could have been taught back in early high school but because

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the emphasis there is strictly towards the positive exponent side of things polynomials

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this is not visible okay so we want to remedy that today and we'll do that by starting with the two

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important families of integral polynomials the A_n's which are going to be alternating,

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and the B_n's which are going to be symmetric and we're going to define them for arbitrary natural

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numbers n so for arbitrary integers and so a_n for any integer n and also b_n for any integer

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and we could also restrict this to natural numbers but I think it's more convenient if we do this in

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this sort of more natural generality okay so let's start with the symmetric ones we might call them

Symmetric Binomials B_n

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symmetric binomials B_n is an integer here's the definition B_n is the m set consisting of n and n

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bar so that's an integer and that's its negation so the N corresponds to Alpha to the n and the n

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Bar corresponds to what we might call Alpha to the minus n so we're taking this obviously symmetric

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integral poly number okay what do they look like well here's sort of a list near zero when n is

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zero then both of these things are zero so we're getting zero zero in the that's the number two

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it's important to to make that a note of that okay B sub zero is two not one as you might

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otherwise think and then B sub 1 is going to be Alpha to the minus one plus Alpha B sub 2

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is Alpha to the minus two plus Alpha squared Etc and B to the minus 1 is Alpha to the minus one

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plus Alpha to the minus minus one so it's Alpha to the minus one plus

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Alpha it's the same as B sub 1 okay and B sub minus two I should put minus 2 there

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B sub minus 2 is out for the minus two plus Alpha squared that's again the same as B sub 2.

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okay so these are something of a basis for the symmetric integral polynumbers

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and what's particularly interesting is what happens when you multiply two of

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them so let's look at B_3 times B_5 there's B_3 Alpha the minus 3 plus Alpha cubed B5

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Alpha minus five plus Alpha the five and when we multiply them we get out to the minus eight uh

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this combination here is Alpha to the minus two this combination here is Alpha to the two

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and this combination is Alpha the H and we see that what we're picking up is a a B2 and a B8

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okay and note that the sum of three plus five is eight and the difference here is two

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okay if we squared once let's say take B sub 4 and we Square it there it is there we Square it well

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we get the square of the first term which is Alpha to the minus eight square of the

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second term which is Alpha the eight and twice the product we get two so that is then B zero

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there's b0 and Alpha the eight plus Alpha minus eight is B8 and again this um the zero is like

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the difference between the four and the four and this eight is like the sum of the four and four

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so as we've already noted B sub n is the mset n Bar N clearly B sub n bar is equal to B sub

Product formula for symmetric binomials

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n because these two things just interchange places and the formula that we've just looked

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at a special case of is that b sub M times B sub n equals B sub M minus n plus b sub m

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plus n and there's no restriction here on the indices they can be arbitrary integers

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a little bit more sophisticated is what happens when you multiply three of them together say B

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sub m b sub n b sub p then it's an exercise for you to check that you're going to get four terms

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you're going to get a b m minus n minus p b m plus n minus p b m minus n plus p and B M plus n plus p

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and remember that any one of these things is unchanged if we negate

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the index so for example B9 times B4 times B3 9 minus 4 minus 3 is 2 9 plus 4 minus three is

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ten nine minus four plus three is eight and nine plus four plus three is sixteen

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and also works when zero is involved so B7 B4 B3 7 minus four minus three is zero

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uh b seven plus four seven plus four minus three is the eight seven minus four plus three is six

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and seven plus four plus three is Fourteen and do note that this B 0 being 2 is you know consistent

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sort of essential for these formulas to work a little exercise for you try to figure out what

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uh the binomial expansion looks like in this case just a variant of the usual one but it's

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worthwhile um clarifying so if we take B_m for an arbitrary integer M and we raise it to the

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k where k is a natural number like zero one two three Etc what do we get when we expand

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all of that out so the B_n polynumbers were pretty well the simplest symmetric

Alternating binomials A_n

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integral poly numbers here are pretty well the simplest alternating integral poly numbers the A_n

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so they're going to be defined also for n an integer and they're defined in a very similar

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way A sub n is the mset consisting of n with anti-n bar okay and the ante has the effect of

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introducing a coefficient of -1 so we'll write minus Alpha to the minus n plus Alpha to the n

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and then we see that now if we negate the index n a sub n Bar we get the negation of a sub n

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and here are some examples so a Sub 0 is importantly zero because when n is zero there's

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a zero here and a zero anti and the zero and the zero anti annihilate and we just get an empty set

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or we just get zero a sub 1 is minus Alpha to the minus 1 plus Alpha a sub 2 is minus Alpha to the

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minus 2 plus Alpha squared and a sub minus 1 is the negative of this one so Alpha to the minus 1

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minus Alpha a sub minus 2 is the negative of this one so it's Alpha to the minus 2 minus

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Alpha squared and so on so let's do a general multiplication let's multiply a sub M times a

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sub n so there's a sub M there's a sub n when we multiply we get four terms so the product of these

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two is Alpha to the minus M plus n the product of this one and this one or maybe this one and this

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one is minus Alpha to the m minus n the product of these two is minus Alpha to the N minus M and the

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product of these two is Alpha to the M plus n and from that we can see that the product is actually

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well this thing here these two terms okay are getting minus B sub M minus n

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so we have to take these two out and then we get two plus signs

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and we're getting B sub M minus n which is the same as B sub n minus m and these outside terms

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both also with positive one coefficients giving us B sub M plus n because here's Alpha to the M

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plus n and Alpha to the minus M plus n so the product of these two A's is a sum of two B's

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and you might like to write down the corresponding formula for the product of three of these a some

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n's and also how do you take the corresponding binomial power of given a sub M again K being

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a natural number so it's not too hard to see that if you have two symmetric

Products of Alternating & Symmetric

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integral polynumbers and you multiply them you're going to get another symmetric one if you have two

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alternating ones and you multiply them you're going to get a symmetric one however if you

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have a alternating one times a symmetric one in that case you're going to get an alternating one

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so that's the formula that we're going to look at now what happens when we multiply an alternating

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binomial with asymmetric binomial A sub M there it is there times B sub n there it is there well just

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four terms just as before we get a minus Alpha to the minus M plus n we get an alpha m alpha minus

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n Alpha to the m minus n and these terms Alpha to the N minus M with a minus sign and a plus

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sign with Alpha to the M plus n and then these two terms here these ones Alpha to the m minus n minus

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Alpha to the N minus M that's a sub M minus n and these outside ones are a sub M plus n now the A's

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remember are are alternating so if we negate the index there which we also could then we'd

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have to put a minus sign so we could also write it as minus a sub n minus M plus a sub n plus m

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so this is a a nice formula and a very interesting and important special case is when m is just one

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m is just one in that case you get a 1 times b n equals minus a n minus 1 plus a n Plus 1.

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and that then has the consequence which this little formula here which we might call on Lemma

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okay that if you have a chain of B's B sub k b sub K plus 2 B sub K plus four so going up by two each

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step up to some B sub l so these are separated by an even number of steps then the result of

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multiplying by A1 is minus a sub K minus 1 so 1 less than that index plus a sub L plus one one

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bigger than that index and the reason is I'm sort of immediate from this because when you multiply

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the A1 times the BK you're getting a minus a sub K minus one that's that term there plus a plus

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a sub K plus one but that plus a sub K plus one that you've just got is going to be canceled

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with the the term that you're going to get by multiplying the A1 times the B K plus 2 the the

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left term it's going to actually exactly cancel that because you can get a sub k plus one and a

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sub K plus 3 okay so in the intermediate spaces between these things the two A's that you're

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getting from the left and from the right always have opposite signs and so they're going to cancel

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and all what's going to be left is the one to the left of the left one and the one to the right or

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the right one the name of this formula here now there's sort of two sort of special cases of this

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also when k equals 1 in this formula if we start with one and they go out by two so one three five

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up to some L then the first term is going to be minus a Sub 0 because 1 minus 1 is 0. and that's

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zero so there'll only be one term that's a sub L plus one on the other hand if k equals 2 and we

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look at sort of even ones if we start just with B2 B4 B6 all the way up to b m then the formula

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is going to tell us what we should get is minus a sub 1 plus a sub M plus 1. but a sub 1 okay is

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so there'll be that on on this side here is the same that's over here so we can bring it to the

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left hand side by putting a 1 there okay so if we rewrite the formula that you get by starting with

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this one with B sub 2 we rewrite it and bring the a sub 1 that we get over here then the result is

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just a single a sub M plus 1 on the right hand side so in both cases if we have a string of B's

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that are going up by two each time and we multiply by A1 we are getting a single a

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in fact the a is just has an index one bigger than the biggest B that appears in each of these cases

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that's a quite uh important kind of formula which motivates us to the definition of the objects that

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actually we're really interested in which are the central poly numbers denoted by C sub n

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okay so the central poly numbers C sub n and now we'll stick with natural number indices n equals 0

Central polynumbers C_n

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1 2 Etc so C Sub Zero is just going to be 1. C sub 1 is going to be Alpha to the minus 1 plus Alpha

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so it's symmetric that's the same as B1 so there's a picture of c0 in terms of a

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mass distribution or a density single mass of unit size sitting above zero here's a

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picture of C1 unit masses above one and minus one on our integral number line

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C2 is going to be Alpha to the minus 2 plus 1 plus Alpha squared that's 1 plus B2 has three

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equal masses equally separated on the number line symmetric about zero at minus two at zero at two

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c three similarly Alpha to the minus three plus Alpha to the minus one plus Alpha plus

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Alpha cubed that's B1 plus B3 represented by four equally spaced unit masses symmetric around zero

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at minus three minus one one and three and then we carry on like this C4 is 1 plus B2 plus B4

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sort of sort of corresponding to the even powers of B except that's not b0 okay that's only half

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of B zero that's why I write it separately so 1 plus B2 plus B4 well the odd ones B1 plus B3

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plus B4 are like that so we just sort of Follow That pattern for even and for odd so C4 is Alpha

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to the minus four going up by steps of two all the way up to Alpha to the four C5 going from

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alpha to the minus five to Alpha 5 including all the the intermediate Powers which are separated

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from the Bottom by a multiple of two as we go up minus five minus three minus one one three five

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so these are the central polynomials they're all symmetric okay so you should think about them as

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being perhaps a somewhat more sophisticated or a little bit more complicated variant to the B's

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they also form a basis of the symmetric poly numbers so if we have an arbitrary symmetric

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polynomial we can write it uniquely in terms of the B's but it's not too hard to see we can also

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write it uniquely in terms of the C's and the seas are a little bit more sophisticated but

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they're also a little bit sort of more uniform and it turns out that they're connected with a huge

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amount of significant mathematics okay so very briefly now I just want to indicate very roughly

The Central Importance of central polynumbers

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all the remarkable or at least some of the remarkable aspects to these Central

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poly numbers c0c1 C2 C3 C4 Etc that we've defined going on in this direction okay

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they're connected with so many important aspects of 20th century mathematics and physics

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um let's start here okay they're intimately connected with Chebyshev

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polynomials which you will have run into actually Chebyshev of the second

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kind which you will have run into if you a member of my Wild Egg Mathematics Courses

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uh YouTube channel where I have a whole series on interesting families of orthogonal polynomials

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and how to look at them in a somewhat novel linear algebra slash number Theory point of view

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they're related to Gaussian binomial coefficients and q Series okay and there's also in fact over

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at Wild Egg there's also a series of videos that I'm doing on q Series and the Gaussian

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binomial coefficients figure very prominently there so these are some sense of variants of

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the gaussian binomial coefficients which are in some cases more natural or maybe more useful than

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the gaussian binomial coefficients but not in all cases there's a good reason for studying both of

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them and uh in fact that that other point of view is is closely related to Quantum groups also in

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physics these things are coming out very naturally in in angular momentum quantum mechanics they

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figure prominently in representation theory of Lie groups okay so in the world of Lie

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groups there are things called compact simple Lie groups and the simplest one is SU(2) the simplest

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non-abelian one is SU(2) okay and something called representation Theory and it turns out

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that these things here are are essentially the characters the irreducible characters of SU(2)

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when you view them in the right way um and as such they're also intimately connected with

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harmonic analysis on the three-dimensional Sphere not the two-dimensional sphere that we find in

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our space but the three-dimensional sphere which is another manifestation of SU(2) but you can do

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sort of spherical harmonics on that thing and it turns out that these things are intimately

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connected with spherical harmonics on S3 they're very close to the Weyl character formula and and

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arguably I've already shown you the Weyl character formula today in the simplest of all cases okay

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um I actually I also have to put in yeah I have to put in another Ade graphs okay Ade graphs

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um Ed so the vile character formula applies to all of these graphs but in the a n case sort of

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a special case this is actually the simplest a case it's the A1 case so this is like the

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the simplest version of some huge Story and there's also a series in my wild egg mask

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of course Channel all about ADE graphs and Dynamics on Graphs and uh you know and these

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things will figure there hypergroups well on this channel I have a series called Diffusion

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Symmetry where I'm introducing you to the world of hypergroups these are very close to being really

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central objects in in hypergroup Theory okay so I'm going to be meeting them in in that series

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of videos the closely connected with that they're intimately connected with random walks and going

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back to physics they're intimately connected with modern Theory the electro weak force in physics

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so there's a lot to this story but in fact I have this sort of aspiration to explain a lot of this

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stuff maybe maybe all of it at some point um but in any case since this series is only dealing

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with Elementary Algebra I want to alert you that this very naive point of view we're just playing

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around with msets and you know now we have these integral poly numbers Etc and what's a rising sort

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of like naturally is we're just sort of following our nose doing the simplest possible thing our

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our objects whose algebraic structure is actually completely key in a lot of really important areas

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and I want to talk more about what this the nature of that algebraic structure is it has to do with

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the multiplicative structure of these Central polynumbers and you might like to play around

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with with that and ask yourself what happens if I multiply say C3 times C7 okay symmetric

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polynumbers what's the product going to be that's an interesting question we'll talk about that in

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our next video I hope you'll join me for that I'm Norman Wildberger thanks for listening [Music]