Advice for Maths Exploration
A key challenge for amateur mathematicians is finding a fruitful and accessible and interesting area for investigation. This is not so easy: classical number theory is certainly very interesting but it is highly difficult, perhaps even unrealistic, to hope to make really new discoveries here at the amateur, elementary level. Not impossible, but do you want to spend a huge amount of time and effort for little probable gain?
A good alternative to classical Number theory is "Poly Number theory" which concerns itself with the rich algebra of polynomials / polynumbers, in which interesting families feature prominently -- like the Chebyshev polynomials of the first and second kinds, and their more modern variants -- the spread polynomials, which arise in Rational Trigonometry as a natural replacement to the Chebyshev polynomials of the first kind. My book "Divine Proportions: Rational Trigonometry to Universal Geometry" has an entire chapter on these (available at wildegg.com).
In this video we introduce these families, with particular emphasis on the spread polynumbers and their remarkable factorization, which sets them clearly apart from the Chebyshev polynomials and is a strong indication that they are really much more fundamental!!
There is a lot of number theory that emerges from this factorization, which was found by Shuxiang Goh about 15 years ago while an undergraduate at UNSW (he is now a practicing medical doctor).
Along the way, I mention why the famed "Fundamental Theorem of Algebra" is in fact a fiction: it is only approximately correct, not exactly correct, and it really is crucial for modern undergraduates to appreciate this. Factorization of polynomials / polynumbers is a MUCH more interesting topic than it is currently perceived to be.
And I also stress that identities involving "cos x" and "sin x" etc should NOT be viewed in terms of "equality of functions", but rather more concretely in terms of equality of coefficients in a power series / polyseries. This makes much more sense, and is actually doable! So there are advantages in thinking more clearly about foundational issues in mathematics --- we can connect better with modern computation, and new theoretical doors open.
This video is part of a series on Advice to Prospective research mathematicians at my sister channel Wild Egg Maths, available to Members there (see below).
A big thanks to all my Subscribers, Members of the Wild Egg Maths channel, and Patreon supporters!
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Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
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Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
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Solving Polynomial Equations: https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
Algebraic Calculus One: https://www.youtube.com/playlist?list...
Algebraic Calculus Two: https://www.youtube.com/playlist?list...
0:00
hello everyone i'm Norman Wildberger and thanks for joining me i'm going to give you some more
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advice today along the same direction as my last video where we were talking about
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extending number theory by thinking more broadly in terms of poly numbers not just regular numbers
0:17
and i sort of stressed to you that factorization in some sense is more interesting and um
0:25
more insightful really in the the bigger poly number world so factorization of ordinary numbers
0:33
as we mentioned last time is unpredictable but it's also a little bit tedious every every number
0:39
has a factorization it's almost uh random you know how the factorizations go and if you know
0:45
the factorization of a big number you still don't really know very much however the story with poly
0:52
numbers is much richer and uh factorization there is a is a very interesting phenomena so when you
1:00
have a poly number or polynomial which actually factors then it's telling you something important
1:08
about the underlying mathematics now i have to point out in a very strong way that
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this is a counter to the usual story that you're taught in undergraduate mathematics which is all
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about the fundamental theorem of algebra telling us that polynomials basically always factor
1:26
well let me tell you very clearly they do not always factor and uh there's a common statement
1:33
for example that that polynomials with say you know rational coefficients or over the real
1:37
numbers factor into linear and quadratic factors this is just absolutely plain wrong approximately
1:44
yes okay but as an exact statement it's wrong and it's really important for you to to get
1:50
on board with that thinking and it's in fact sort of demonstrably wrong you just write down any any
1:55
polynomial of you know a reasonable degree even something relatively simple like x to the seven
2:01
minus x plus three try to factor that into linear and quadratic factors and see how far you go okay
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so the point of view that i'm trying to explain to you is that factorization is actually an indicator
2:12
of structure and it's an indicator of interest okay and so this is really important especially
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when we're looking at these families of poly numbers that i was telling you about my last video
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which are you know playing really important roles in lots of areas in mathematics physics chemistry
2:30
etc so today we're going to have a look at the the arguably the most important family of poly numbers
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now if you ask a regular mathematician um the pure mathematician you know one of them the
2:42
most important families of polynomials they'll probably say that Chebyshev polynomials of either
2:48
the first or second kind but we're going to see that the spread poly numbers which are variants of
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the Chebyshev polynomials are pretty clearly more important than the Chebyshev polynomial so i want
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to indicate that to you that should be a flag of just how big uh and and exciting a new area
3:08
we have here because these spread polynomials are are so interesting not just from a geometrical
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point of view where they're crucially important but also in terms of analysis
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and in terms of number theory and the highlight will be a remarkable factorization found by
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my honors student susheen go some years ago the goal factorization so let me start with
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the Chebyshev polynomials it's a very big subject and i'm going to ultimately make a whole series on
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on spread poly numbers and the Chebyshev polynomials will figure prominently so i will
3:46
at some point elaborate on these so very quickly okay the the simplest way of introducing the
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Chebyshev polynomials is through a recurrence relation okay so there are t sub n and x so n is a
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an index natural number starting with 0 and t 0 of x is defined to be 1 t 1 of x is defined to be x
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those are the initial conditions and then the recurrence which drives
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them is t sub n of x equals two times x times t sub n minus one of x minus t n minus two of x
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okay so we use this recurrence to successively create more and more of these Chebyshev
4:28
polynomials of the first kind and pretty clearly a t n of x is going to be a polynomial of degree
4:33
x because each time we're bumping up the degree by one okay so what do these things look like
4:40
well they're polynomials that can be graphed and it turns out that their values between -1
4:49
and 1 are particularly important okay because they bounce around from a lower value of minus 1 in
4:57
that interval to an upper bound of plus 1 in that interval and they do so very beautifully so they
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start so for example t sub 4 is this light blue one here okay so it starts up here at minus one
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one and comes down touches minus one goes up tangent two plus one goes down touches
5:19
minus one and comes back up right up here so they all do this of course outside this interval they
5:25
they go up you know typically you know towards positive infinity or or negative infinity
5:30
depending on their degrees and so on okay so what actual values are are we talking about here
5:37
well it turns out that there is a an interesting generating function which is a
5:41
sort of a remarkable way of uh sort of capturing all of them okay it's just given by this little uh
5:49
maybe i'll just highlight this that's a great little formula the generating function is let's
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display it t of x comma t which is one minus t x over one minus two t x plus t squared what is this
6:04
well this is uh like a a ratio of poly numbers but there are two variables floating around the x
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which is going to be the x that we're using to describe the chebyshev polynomials
6:16
and a further variable called t so if you look at this as a function of t it's a rational function
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okay so you have a linear polynomial in t in the numerator and a quadratic one in the denominator
6:30
so basically it's a division right this this formula is in some sense asking you to do a
6:35
division and if you do a division well you can do that in the context of power series or poly series
6:43
okay you're going to get a a series in t but each of the coefficients of that series is going
6:50
to be a polynomial in x and happily all of the Chebyshev polynomials um get marched along one
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by one as coefficients of the various powers of t in the expansion here so if i get my computer
7:08
to just expand this as a power series in t i get this expansion here one plus t tx plus t squared
7:17
2x squared minus 1. that's telling me that the second Chebyshev polynomial is 2x squared minus
7:22
- and then t cubed times 4x cubed minus 3x that's the third Chebyshev polynomial and so on okay and
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of course it carries on so it's a little bit more easily visible here in this matrix so this column
7:37
okay so we see all of the Chebyshev polynomials from t 0 up here to i guess t sub 10 down here
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now by the way uh the reason i'm writing it this way usually i like to write polynomials or pull
7:55
in numbers starting from the lower degree terms but my computer just automatically outputs this
8:00
and it's too much work for me to reverse the order so just keep that in mind okay so what um
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what do these things mean well one uh sort of standard interpretation of this is that it's
8:13
a way of relating cosines with cosines of multiples okay so for example you may know
8:21
that cosine of 2x okay equals well one way is to write it as cosine squared x minus sine squared x
8:34
perhaps you've seen that okay let me just see if my computer recognizes that as a true statement
8:40
okay yes that's true now there's another way of writing that because we can replace
8:45
the sine squared with a 1 minus cos squared so we could get rid of the sine squared and just express
8:53
it in terms of cosine and we get 2 cos squared x minus 1. that's an alternate expression for cos2x
9:03
in terms of powers or a polynomial just in cosine x okay and that's where this second
9:11
Chebyshev polynomial is coming from it's one explanation for it so the third one which will
9:17
be less familiar to you cosine 3x so the claim here that you can see by reading off the third
9:23
polynomial is that this is 4 cosine cubed x minus 3 cosine x let's see if my computer believes that
9:37
yes that's true too now whenever we talk about such transcendental
9:44
objects um we should clarify what we're talking about so in most modern analysis courses
9:51
the interpretation of such a statement like i've just written down is in terms of
9:56
functions okay that cosine 3x is a function that inputs x and outputs cosine 3x and this right hand
10:05
side is also function and these functions agree in the sense that if you plug in a real number x and
10:11
compute it then you get the same thing on the left and right hand sides that's not as in capital n ot
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the way i want you to think about the meaning of this statement okay the meaning of the statement
10:24
is much better the meaning is much better understood in terms of poly series or power series
10:37
and let me indicate what what i mean by that okay so let's take cosine 3x and let me get my computer
10:46
to write out a poly serious for it or a power series okay in x now because it's an x
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okay so there it is in x up to degree 10. and now let's take the right hand side which is another
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power series ultimately in in x and we can compute it or my computer can compute it i will do the
11:10
same thing in x please thank you and we see that these terms are the same at least up to degree 10.
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so the meaning of this statement as far as i'm concerned okay is that we have an equality of
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power series that's or poly series better so there's a power series or poly series on the left
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there's a power series or poly series on the right and these are equal and what does it mean to be
11:38
equal it means that their coefficients are equal one by one as we increase powers in this case of x
11:46
it does not mean that they are equal as functions because
11:49
there's no way of evaluating these things as functions as functions this is an
11:53
infinite task that we're not able to do when we can't pretend that we can evaluate such a thing
11:59
but what we can do is we can successively march through these things okay and notice that we
12:05
have equality of coefficients and then aspire to proving that the general coefficient of x to the n
12:12
for the first one is equal to the general coefficient of x to the n of the second one so
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i'm making a big deal of this because it's really important that you you start thinking this way
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it's crucial for the algebraic calculus 2 course which i hope some of you are taking okay and it's
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just a better way of thinking about what's going on with with analysis that's just the way it also
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fits much better with the way physicists think okay so those are the Chebyshev polynomials of
12:43
the first kind okay a little bit more quickly now the Chebyshev polynomials of the second
12:47
kind are very similar they also satisfy a recurrence relation with u 0 of x being 1
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u 1 of x equals 2 x and actually exactly the same recurrence relation for the higher degree terms so
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the only difference between the first kind and the second kind Chebyshev on almost is the u1
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with t's we had just x here now we have 2x and here are some graphs of some of the
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the first ones u0 u1 is just a line u2 is the quadratic one uh where is it here in green
13:26
you know etc okay and and what about the actual values well there's actually a generating
13:32
function too the generating function is actually even a little bit simpler
13:35
it's just 1 over 1 minus 2 t x plus t squared and you can just spew that out and write down
13:43
these various values so these are particular values for the Chebyshev polynomials
13:50
of the second kind now before we get to the spread polynomials which is really um the main interest
14:00
here i want to i want to have a look at a question of factorization so suppose i take one of these
14:08
these things okay say this is the Chebyshev polynomial of the first kind
14:12
something that you should routinely do when you when you see some of these polynomials
14:16
is ask do they factor as i mentioned at the beginning that's a kind of a crucial question so
14:23
let's actually see if our computer can factor that okay it factors it but not in a very spectacular
14:29
way but let's have a look at this one here the degree eight one how does that factor if at all
14:38
now we can't expect such a simple factorization and it doesn't factor okay how about number nine
14:55
hey that fact is in kind of an interesting way that's x times 4x squared minus
15:01
3 which is a little bit like this thing here this one here has such a factor also i guess
15:09
and then there's this other term which is 36x squared okay it's not in order minus 3 plus
15:14
36x squared minus 96x to the fourth plus 64x to the sixth so some factorization is going on here
15:28
occurs here and people have studied this but it is say let's say not so obvious what the pattern is
15:38
let's just say that at this point okay
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now let's look at the chebyshev polynomials of the second kind
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okay and oh by the way i should uh should show you something cool so i can do all this factorization
15:53
at once since i have all of these things in a matrix my program will allow me if i just
15:58
put factor to factor all of them at the same time isn't that nice so we see that there's
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some sort of non-trivial factorization for for this one i guess that was degree degree 6
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and degree 9 and also degree 10. okay there's some some number theory going on here right this is
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this is the algebraic equivalent of factoring numbers
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and yes it's also a little bit random but maybe not that random maybe we can
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understand the factorization let's have a look at the chip polynomials of the second kind here they
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are let's factor them all at once okay we get something over here oh my goodness look at this
16:48
see this is a much richer factorization and there's some you know interest here there's
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some factors which are re appearing uh at least a little bit um although there's new ones too i
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guess these ones here are more or less like like these ones yes uh have we seen this one before
17:08
i don't think so okay this so there's something uh to be investigated here uh with these uh
17:14
Chebyshev polynomial so the second kind so the factorization seems a little bit more pleasant
17:22
factorizations here uh appear
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to be say let's say richer but we're not really sure about that
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okay the spread poly numbers or polynomials were introduced in my book on rational trigonometry
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about 15 or 17 years ago and they are very closely connected with the chebyshev polynomials as we'll
17:49
see but offer important advantages especially from a number theoretical point of view so
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what i'm trying to do here is i'm not going to go into details about these things but i want to
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to get you thinking about broad new directions and the the spread polling numbers are just a
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perfect example of so many opportunities that we have of discovering interesting new things which
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are really available to amateur mathematicians and i do intend to do an explore research level
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mathematics series on spread poly numbers so you can think of this as a gentle introduction to that
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okay they are defined by a recurrence relation very similar to the Chebyshev polynomials s0 of s
18:38
equals 0 s1 of s equals s and then this recurrence which is a little bit different because there's a
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plus 2s at the end okay otherwise it's so it's 2 1 minus 2s times sn minus 1 minus sn minus 2 plus 2s
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here are some of the first values of the spread polynomials particularly interesting is the
19:04
second one that's worth noting so note that sub 2 of s equals this thing here which also factors
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maybe it's better to write it like this
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sometimes called the logistic map okay and there's further values so how do we get at these
19:32
things other than having to run the recurrence well there is a generating function just as
19:39
this was found by Mike Hirschhorn who's a colleague of mine at UNSW
19:44
and an expert in all kinds of interesting analytic formulas and combinatorics
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and he discovered that the generating function is say this function of s and variable t
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t is the same w variable that was appearing in the generating functions
20:01
for the Chebyshev polynomials so it's t times s times one plus t over one minus t times one
20:08
minus two t plus t squared plus four t s so a little bit more complicated okay a little bit
20:14
more complicated but you could still use this to generate some of these things now
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the connections with the Chebyshev polynomials are really important to mention so this one here is
20:26
is prominently the most important one it's telling us that the the spread polynomials
20:34
are are intimately um are really just an affine shift of the Chebyshev polynomials and maybe we
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can sort of illustrate that here with this little diagram here okay it's maybe a bit too far up
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so here's the relation okay it's saying that s n of s equals one minus t n of one minus two s over
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two and what that really effectively is doing is it's it's taking the chebyshev polynomials
21:00
and sort of compressing them so instead of in going from the interesting range minus one to
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one if we look at that that box plot that i showed you before uh x going from minus one to one y or
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yeah y going from minus one to one and sort of so translating that shrinking it by a factor of half
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and then translating it over then it occupies a box going from say x or s in this case from zero
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to one on the horizontal axis and also from zero to one in the vertical axis okay so we're going
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from minus one one cross minus one one to zero one cross zero one and that's the effect of this
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this essential affine shift here that i showed you that's relating the values and here are
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well a graph of the first 16 spread polynomials or polynomials so there's uh s equals zero that
21:59
would just be the the axis here uh s equals s so that's that line that's going up there
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and then you can see this this parabola here right it's doing the same kind of thing that
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the Chebychev polynomials do they start you know down here and the end either down here or
22:13
up here and they bounce around up and down in this box tangent always to the to the bottom and
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the top of the of the box and you can see that there's all kinds of interesting things sort of
22:26
happening here there's sort of some concurrences and uh maybe you can even if you stare at it uh
22:33
in the right way you can maybe also see some sort of ghost curves it can be can you see a parabola
22:40
sort of in that general shape there that's sort of visible as of gap somehow formed by the the graphs
22:47
so pretty interesting well what else do we know about them there's also a connection
22:53
with the Chebyshev polynomials of the second kind given by this formula and um i might also mention
23:03
this is sort of interesting fact that if n is odd then sns has a factor of s and then the rest of
23:09
it is always a square while if n is even it has a factor s and other factor 1 minus s and then the
23:16
rest is also a square that's pretty interesting but there's lots of other things to say but the
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most remarkable thing that i know about them is uh this fact this is a conjecture which i'm going to
23:38
okay we're going to call this go factorization okay
23:41
so this is named after Susheng go who was an honor student with me at UNSW some
23:48
15 17 years ago and he wrote a very nice honors thesis on the spread polynomials
23:55
and he noticed this very elegant fact that the spread polynomials factor in a in a very nice way
24:07
so here's the exact statement for every k a natural number one two three there is a polynomial
24:15
phi sub k with integer coefficients and it happens to be of degree 5 k where that's the Euler
24:23
totient function sometimes called which counts the number of numbers which are less than a given
24:29
number and relatively prime to it okay so there's there are these poly numbers phi sub k and then
24:41
for any n equals one two three a natural number also the nth spread poly number okay
24:49
is a product of phi sub k's a product of which ones well all the phi sub k's were k divisor of n
25:01
it's exactly um a beautiful factorization so this is um now a list of these
25:10
these phi sub k's we might call them go poly numbers that occur in these
25:18
in these factorizations so there's now a secondary family of of pollen numbers which are acting as
25:26
some sense or like you know primes sort of like a prime factorization isn't it but we're not
25:32
factoring into numbers we're factoring into sort of simpler polynomials and these are almost all
25:38
squares um that sort of follows from the thing i just told you except for the first two okay but
25:44
all the other ones are squares and they are quite unpredictable i think safe to say okay and uh in
25:52
in my course on spread polynomials we're gonna have a very close look at this and there's
25:56
just you know so much investigation a really rich number theoretical investigation that uh that
26:03
emerges from studying this so let me illustrate this factorization okay so for example s12 12 has
26:10
factors or divisors 1 2 3 4 6 and 12. and so there's a factor of s 12 one for each one of
26:19
these go poly numbers of the associated counts so phi sub 1 phi sub 2 phi sub 3 phi sub 4
26:24
5 sub 5 6 division 12. okay phi sub 1 was just a s phi sub 2 2 will be this one here okay phi sub
26:35
3 is also appearing see it there phi sub 4 it's also appearing 5 sub 5 no but phi sub 6 is also
26:44
appearing there it is and phi sub 12 which is not on this list turns out to be i suppose this
26:51
thing here and there's a couple of factors of 4 merging from here and here to give us the 16.
26:59
so this is a quite remarkable and let me illustrate that by going up to our list of
27:07
poly numbers okay and i'm going to factor this so you can sort of see this in action factor
27:17
so here are the factorizations of this and if you compare these factorizations with the chebychev
27:24
polynomials factorizations you will see that these are much much nicer okay exactly because of this
27:31
this formula that was found by xiang that that the the factors that are appearing here are
27:38
are in some sense very predictable and they only depend on the divisors of the
27:43
the the number and that you're you're considering and these these polynomials
27:50
that are associated with divisors are fixed they do not depend on the n it's the same factors all
27:57
the time okay so this 4s minus 3 squared that's coming from here okay that's like a phi sub
28:06
- that will be occurring in in all of these places where we have something divisible by 3.
28:14
okay so it's a absolutely beautiful thing but of course it then opens up this question okay
28:19
how do we understand these five functions okay and that's a very rich question and i want to
28:25
encourage you to think about this as being the number theory of the future okay this is
28:31
this is like a really rich direction for a number theory to go and it's
28:36
almost a wide open direction okay so so much for amateurs who are interested in either
28:41
algebra or analysis polynomials special functions number theory to investigate.