The curious world of integral polynumbers | Math Foundations 233 | N J Wildberger
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- čas přidán 27. 07. 2024
- Arithmetic with multisets, or msets, incorporates negative numbers and integral arithmetic by employing the particle / anti-particle duality that has played such a big role in 20th century physics, pioneered by the great British physicist Paul Dirac.
Once we have integers, we can expand our arithmetic of polynumbers or polynomials in two different ways: one is to introduce integers as possible coefficients, and the other is to introduce them as possible powers, or exponents. This gives us a very different, perhaps even surprising, approach to this larger integral polynomial arithmetical world. We want to think about these objects here not in a calculus view, where a polynomial is considered as a special kind of "function", but rather in a more purely algebraic way involving again ideas from physics in the form of discrete mass distributions or measures.
In this way multiplication corresponds to traditional convolution of measures. And the entire story supports an important new kind of symmetry, that will play a major role in the further development.
NOTE: There's a typo with the last number of the first poly (a 4 instead of a 3) in the first slide: it should be 2+x+3x^4=
__
| 2
| 1
| 0
| 0
| 3 not 4 (thanks Federico Rocca!)
Video Contents:
00:00 Introduction
01:33 Integral Polynumbers and Ipoly
04:08 Integers as coefficients
07:37 BIG difference between Integral Polynumbers and functions
14:31 In algebra: avoid the "function " concept!
19:11 IPoly as measures/ densities
25:14 RPoly : Rational Polynumbers
29:40 The reflection symmetry
34:21 Symmetric /alternating integral polynumbers
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I actually got really interested in the function x+1/x at some point and did some investigations into it. (with some quite interesting results). and I've really had a feeling for a while now that this functions is quite central. (probably much more so than we are taught). When you started talking about the symmetry, that really resonates for me. this is amazing.
I think your legacy will be greater than you can possibly imagine, and I feel thankful that I got to watch some of that progress manifest online over the years. I hope your year is a satisfying one.
However this is already developed. Here is nothing new.
en.wikipedia.org/wiki/Laurent_polynomial
This is beautiful! It's all coming together now. Wow!! Thanks!
A real treat to watch. The first time I've seen measure/density treatment of poly numbers -- much like a wave or bar chart. Thank you for the math descriptions you have shared.
I love polynumbers at this point. XD
Defs have been diving in depth into them recently. Thanks for sharing this beautiful creation to the world. 🖤
I get goosebumps every time I see you developing these more in-depth. I cannot wait for the next video!
Incredibly important.
Thank you Norman. It takes a while to master the mapping between m-sets and polynumbers, especially with the anti objects.
Thank you for all of your work! I am excited to see how this develops.
In multiset notation, I've started using an underbar _ for negative coefficients & an overbar for negative exponents.
It's very interesting to see some of the mathematical identities follow along well like power expansion.
I don't think this is a coincidence, specially when some of your rules are different from the standard algebra rules.
It makes me wonder, if any system that defines the multiplication rules and the addition rules to satisfy the basic identities we have in our number system, means it should be consistent with the already existing Theorems.
That would mean you can use many of the already proven Theorem in your new mset approach.
This is fantastic to watch. I love it!
The weight approach stands also at the crux of some inequalities' proofs involving rational powers, where AM-GM, Jensen (etc.) fail. It's the correct approach.
Hi Catalin, That’s an interesting comment. Can you please elaborate a bit further?
@@njwildberger The generalized Arithmetic-Geometric Mean (artofproblemsolving.com/wiki/index.php/Power_Mean_Inequality en.wikipedia.org/wiki/Generalized_mean) is known to not have an algebraic proof yet. Currently is „proved” by using analytic properties of log and transcendental power functions. Seems to be provable only as an inequality of rational polynumbers, rather than numbers, by using rearrangements (sketch) and convolutions on rational polynumbers, with some basic theory of polynumber inequalities, that is, adding the concept of domain and comparison of polynumbers; per example, after the definitions of these two concepts, and a quick setup of the theory, the rational polynumber /alpha^{1/7} is considered "bigger" than /alpha on the domain [0,1], without computing any roots (at most, using algebraic differentiation on rational polynumbers). Off topic: be prepared to read soon from me a theory of numbers closed arithmetically under summation, substraction, multiplication, division, EXPONENTIATION, ROOTING and LOGARITHMATION. Basically what you want: the expansion of the rational number realm that is finitely-defined on all the 7 operations mentioned above. Not joking. I'm there but not quite yet. You'll be one of the first to be announced if succesful, since your various series started this idea.
Beautiful! Convolution is nicely intuitive, and I like the idea of weights/densities very much! Could we, perhaps, improve on that and consider weigths/densities 'inertial masses'?!
Before goind deeper into the inertial mass idea, a short discussion on notation. I prefer a sort of type theory where n< is for positive numbers and n> for negative numbers, as the symbol < intuitively 'increases' and the symbol > 'decreases'.
Let's first generate an analogue of the rational number line Stern-Brocot style, by a row by row structure instead of the usual tree structure; with also "antinumbers" included.
< >
< >
< >
< >
etc.
Let's note that because of the chiral notation chosen, the rows are wholly palindromic, not only symbol-wise, but also geometrically.
On the 2nd row, we get the concatenation , which we define as the denominator element, a new type of number n. That allows us to turn the weight idea around, and interprete < and > as integral accelerations of the numerator aspect, and as inertial mass. Thus we can interprete e.g. the expression
How about GRPoly, using Gaussian Rational elements on the i-1 number line?
When are two integral poly numbers multiplicative inverses? Expanding their product after writing each as linear combinations of alpha powers gives:
p = sum_{i} a_i (alpha)^i
q = sum_{j} b_j (alpha)^j
pq = 1 implies...
sum_{i+j = k} a_i*b_j = 0 whenever k != 0
sum_{r}={a_r * b_{-r}} = 1
(i, j, k, r all integers)
The matrix formed by the outer product of the two coefficient vectors (ab^T) sums to zero along each antidiagonal above and below the leading anti diagonal, but the (main) antidiagonal sums to 1. This must be a special kind of matrix. Does it have a name?
Bravo! I am not sure, but it looks to me that your alpha could very be some gauge in a gauge-invariant (physics) field theories. Anyway, I dream (meaning I hope to see in my life) a revolutionary paper back, American style, with Big journalistic citations on the back cover and, on the first cover, explosive things like "Brand New", "Marvelous"... and "Real Number Free, just like "cholesterol free", "zero calories" or "no GMO"! Thanks.
Ha Luc! I like it! “Real number free” and “no artificially added infinities “ and maybe even “purely organic thinking”. !
@@njwildberger 'purely rational thinking' :D
@@njwildberger Exactly! We are in the same gang. I like particularly the "purely organic thinking" which I never thought before, and which is already significant: the present text is GPT-free!
can we have multisets with rational multiplicities also, so we could assign rational weights too?
Yes certainly we can do this
I wrote a comment about an idea of numerator mark-antimark symmetry < or > as acceleration, and denominator concatenation as inertial mass.
Can that be expressed without relying on the multiplicity list representation though?
When implementing this in a program the multiplicity list would require rational numbers to be defined already as simpler msets. We can do that e.g. based on another inductive list-like data type, but then we'd have two different representations, which is not ideal.
@@BalazsEndresz Yes, there are some thorny details in that regard, but I think a fruitfull way to proceed is how Wildberger adviced, to start from 'expression' as compact information, and then interprete and give more analytical definitions in various ways. Isn't that the general mereological idea/practice of type theory, to start from generalities and proceed from there to distinguish particulars? Search for intuitively clear and communicable as well as efficient notation runs through the whole process, and finding an ideal notation is not an easy task.
For exemple, the string expression interpretations method.
I spoke too soon. Actually trying this out in Haskell, it is possible make it work with only one data type fairly easily. It just shouldn't be thought of as a Map type.
How about calling these things 'polyintegers'?
A Wild Berger Appears!
❤❤him. But I quit
@@brendawilliams8062
Infinity-chu, I choose you!
@@Nah_Bohdi 🤣
I do not understand what is actualy new here. What is the "meat".
Laurent series and Laurent polynomials is already developed theory with negative exponents.
en.wikipedia.org/wiki/Laurent_series