KF Math Thread - Discuss Math

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General Tug Boat said:
I am personally glad there is a math thread on here finally and I am just going to pin this thread to see how it is progressively updated. I guess to also share one of my favourite math textbooks that actually helped me understand the application of math in engineering better is this textbook in particular Basic Technical Mathematics with Calculus, SI Version, Ninth Edition, 9th Edition . It's a bit of an older math book and I am sure that there can probably be some torrents that exist of it out there currently. This book doesn't get too crazy in depth in some of the more involved elements of calculus, or multivariable, but is a pretty good summarization. Along with a good start before delving into some of the more in depth books/material out there. I didn't like that Leibenz notation was extensively used before Larange notation, because to be honest. Larange notation is far more readable and just quantifies things and saves space a lot better. Personal preference in respect to feedback regarding that autistic little nuisance on my end.
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Here is a link to the pdf for those that share in my burning hatred for Pearson. And here is a link to the solution manual pdf.

General Tug Boat said:
I am almost tempted to begin getting back into math again, but just in a practical way now that I am a little more fluent in programming and to be honest I would rather have a practical approach with some of these concepts. Writing little programs in R probably might be my best bet, considering that I am currently in-between opportunities. So that might kill some time while I am job hunting at the moment.
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You might like functional programming languages like Racket and the more mathematical mindset needed when approaching different problems or tasks in such languages.
 
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I can't reply directly, but super based yats. The Pearson book does have its limits unequivocally, but that was the book that they ended up using for my college courses back in the day. Excellent you provided an archive link to the material. I am also going to explore Racket, I have never heard of that specific language before. Though, indeed would be an interesting language to investigate. Personally I have actually been going through all of my old notes just to play with LaTeX, and I will have to say that it has been a tremendous help in translating all of that stuff. It actually works 6 million times better than intended and even though the libraries took quite a bit of time to install. It has been incredibly useful in that specific endeavor.

I always found the notion of writing notes and especially finding ways of translating into some level of comprehension of explaining mathematical concepts has been vastly beneficial. A lot of that I personally put on the back shelf because of the time that I spent dedicated to a career in culinary. Now that I am being forced to upgrade because of economic factors.

As well, Canada has devolved into third world status as of late. Maybe exploring an opportunity in mathematics and computer science might be something that might fit me in this part of my age. After I am done translating the hundreds of pages of notes I have and re-exploring the fundamentals with a set of fresher eyes. All of that aside though to keep some level of discussion going when sharing some bodies of work that I admire.

I love the works of Gilbert Strang, he has a remarkable body of work. I embedded one of my favorite lectures of his below on complex numbers. The way he ties together the fundamentals so robustly gives me a great amount of respect.

I always liked the MIT opencourse related content. Especially when delving into some of the works of these professors. Gilbert Strangs series of calculus lectures are some of the best lectures I have watched to date in regards to the subject.


After I re-explore some of my old content and remove some of the cobwebs, I am defiantly reading his Linear Algebra texts.
 
Lurker Child said:
Of course, in the digits of pi, there is a pattern, we just don't know it.
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What even is a pattern? We have formulas to compute any of its digits, is that not enough of a pattern to you?
 
N Space said:
Then again, the probability that was the case is literally 1.
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Depends how you look at it. You could also say that having evolved to live in a universe where the laws of motion are low-order polynomials (macroscopically anyway), the probability that a number we're interested in is transcendental is surely less than 1.

I wonder what the actual proportion of transcendentals is among "numbers someone has thought of", how many have a pi or e (or whatever else) baked into them somewhere and how many don't.
 
Kosher Dill said:
Depends how you look at it. You could also say that having evolved to live in a universe where the laws of motion are low-order polynomials (macroscopically anyway), the probability that a number we're interested in is transcendental is surely less than 1.

I wonder what the actual proportion of transcendentals is among "numbers someone has thought of", how many have a pi or e (or whatever else) baked into them somewhere and how many don't.
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In that regard yes of course transcendentals are rare, it doesn't help that proving certain numbers are transcendental is pretty hard and thus not many have been proven to be so (π + e and the Euler-Mascheroni constant being examples), AFAIK the only useful theorems in this regard are the Gelfond-Schneider theorem and the Lindermann-Weirstrass theorem. The reason why I said it's probability 1 is because the set of non-transcendental real numbers is countable and thus has Lebesgue measure 0.
 


Just want to let you guys know that Alison, the popular and engaging online educational platform, actually offers diplomas in mathematics and it is accredited by the CPD.

I actually have not tried the second one, but the first one was a bit of a challenging breeze. The geometry and trigonometry one was a fun course to try, even with the videos helping you learn how to do it.

Either way, it doesn’t hurt to get a diploma in it if you don’t have one.
 
N Space said:
What even is a pattern? We have formulas to compute any of its digits, is that not enough of a pattern to you?
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That is a sufficient pattern, yes, for finding what the digits of pi are. But what about the behavior of those digits? If I gave you a sequence of digits, can you tell me whether they exist in that string and where? This is more in line with what I had in mind. Apparently that's an open question regarding some statistical "normality" in the digits of pi.

In maths, I'd say, being able to generate something is often just the beginning. The immediate questions are then on the generation itself. You have this "thing", what can you say about it?

Of course, with something like pi, usually you just need to know what it is. The primes are a better example of something whose existence only sets the stage for deeper qualitative questions such as existence of arithmetic progressions, distance between primes, etc.

Also, "pattern" is more of a personal slang than anything rigorous. If it's an observable regularity or consistency then I would call it a "pattern". It's how I understand problem solving, identifying and maneuvering on patterns. Don't think about it too much. It's supposed to be attitude, not a paradigm.

Edit: Saw your profile description. Made me lol. Non-measurable sets are abstract nonsense! Those sets turned up in measure theory to justify measurable sets not being all sets, and then never again lmao. Apparently there's an interesting discussion about it research-wise if the wiki page is anything to go off of.
 
Lurker Child said:
That is a sufficient pattern, yes, for finding what the digits of pi are. But what about the behavior of those digits? If I gave you a sequence of digits, can you tell me whether they exist in that string and where? This is more in line with what I had in mind. Apparently that's an open question regarding some statistical "normality" in the digits of pi.

In maths, I'd say, being able to generate something is often just the beginning. The immediate questions are then on the generation itself. You have this "thing", what can you say about it?

Of course, with something like pi, usually you just need to know what it is. The primes are a better example of something whose existence only sets the stage for deeper qualitative questions such as existence of arithmetic progressions, distance between primes, etc.

Also, "pattern" is more of a personal slang than anything rigorous. If it's an observable regularity or consistency then I would call it a "pattern". It's how I understand problem solving, identifying and maneuvering on patterns. Don't think about it too much. It's supposed to be attitude, not a paradigm.

Edit: Saw your profile description. Made me lol. Non-measurable sets are abstract nonsense! Those sets turned up in measure theory to justify measurable sets not being all sets, and then never again lmao. Apparently there's an interesting discussion about it research-wise if the wiki page is anything to go off of.
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It is true we know little about the digits of pi but if* it turns out to be a normal number then there'd be little point in analysing the matter much further. My point is that we already know how this digits come about so in a loose sense we know about a complicated pattern for them, sometimes that's just it and there's no simpler way to go about it.

Regarding non-mesurable sets, they are a royal PIA, so much so that proving the sum of random variables is another random variable is hard enough to be a tough homework problem.

*As far as conjectures go, this one is as close to a theorem as one may suspect to be given the fact that the set of anormal numbers has Lebesgue measure 0 and the statistical tests done to pi's digits agree, moreover, proving any one number is normal is absurdly difficult and pi is no different in this regard.
 
Lurker Child said:
Of course, with something like pi, usually you just need to know what it is. The primes are a better example of something whose existence only sets the stage for deeper qualitative questions such as existence of arithmetic progressions, distance between primes, etc.
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That is the thing math is actually. It is not about find the right answer it is about finding out that the right answer is the right answer and describing why. There are a bunch of strange connections that occur in Math that don't occur anywhere else. Some philosophers pointed out that Math doesn't actually prove any truths new since all it does is express that one thing is equivalent to another thing. For example 9 is the same as 3^2 only because 9 inherently contains 3^2 as an intrinsic part of 9. This is true but it ignores that the discovery of the link is valuable and is a proof of something true.

As for the pi randomness question I'd say this is all part of number theory.

If we could completely describe number theory we could answer so many questions.
 
Napoleon III said:
For example 9 is the same as 3^2 only because 9 inherently contains 3^2 as an intrinsic part of 9
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Well... that's been debated since pretty much forever. (Immanuel Kant famously used the example of whether 7+5=12 is an analytic proposition or not - he said no) But it seems like 9 would have to be an unimaginably "big" concept if we look at it that way. Are 1+8, 12-3, 2.5001+6.4999, sqrt(81), "the solution of 5x + 8 = 53", "the definite (Riemann) integral of 2x dx from 0 to 3", etc. all intrinsically part of 9?

There's also the side issue that your 9 probably contains some non-intrinsic parts from your mathematical foundations. Is 8 an element of 9? Yes, if you're constructing the natural numbers as Von Neumann ordinals, but I wouldn't say "containing 8 as an element" is really intrinsically part of "nineness".
 
Thank you for posting I actually genuinely appreciate it and enjoy and I'm glad you did. I say this because I'm going to be incredibly insulting and mean while I respond to your very interesting post.

Kosher Dill said:
unimaginably "big
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Yes any particular number is quite big in this sense. This should not be surprising when we consider due to base systems for counting binary base eight etc there are a infinite number of ways we could write 9 not even even using what we would consider as mathematical operations.

I would say that because that is true that the numbers exist not as a simple small thing but as an extremely large thing. Every time I do non-basic mathematics I get this feeling that I'm just seeing a small piece of the puzzle and that there is something incredibly large and complex hidden under the surface that I can't see. I am completely overwhelmed by this this feeling whenever I am doing any kind of Calculus. Is the chain rule so simple intrinsically? No there is a great machine underneath the surface with all kinds of moving parts we can't see that don't move in the specific scenario of the chain rule. Look at something like a half derivative and you might be able to see what I mean by machinery present underneath the surface.

Another example of the above that doesn't make me sound like a Schizo is Laplace Transforms. Why would this extremely strange function in which you do this strange half infinity integral be so useful for differential equations? And look at an inverse Laplace Transform. Look at how incredibly complex it is. Something more than a random useful convience is at work there


Kosher Dill said:
There's also the side issue that your 9 probably contains some non-intrinsic parts from your mathematical foundations. Is 8 an element of 9? Yes, if you're constructing the natural numbers as Von Neumann ordinals, but I wouldn't say "containing 8 as an element" is really intrinsically part of "nineness".
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WRONG very wrong. This is wrong because it breaks the aspects of what a number is into more pieces than is useful. This impulse is similar to the idea that the best way to map a coastline is to map it in smaller and smaller chunks until you figure it out perfectly. This of course is wrong you make a general rule to describe the coastline using a fractal. Mathematicians sometimes have a tendency to think like physicists and break things into neat little chunks when they should avoid that and find a general rule.
 
Napoleon III said:
WRONG very wrong. This is wrong because it breaks the aspects of what a number is into more pieces than is useful.
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I suppose what I'm driving at is that "what a number is" isn't the most productive way to look at things when you get down to brass tacks. Because...

Napoleon III said:
Mathematicians sometimes have a tendency to think like physicists and break things into neat little chunks when they should avoid that and find a general rule.
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Mathematicians do express things in terms of general rules - in fact so general that it doesn't matter precisely what a number is. Going back to the 9 = 3^2 example, and sticking just to the natural numbers, it would be more accurate to say that as, perhaps:
"In such-and-such a system of mathematics (ZFC usually), for any model of the Peano axioms, with the usual definition of multiplication and notational shorthands, '3 * 3 = 9' is a theorem." And proving this wouldn't be "discovering interesting facts about 9" so much as "discovering interesting theorems in our mathematical system".
That statement is true whether the object we call 9 is {0, 1, 2, 3, 4, 5, 6, 7, 8}, or {8}, or something else.
Whereas on the other hand "{3, 5, 6} is a subset of 9" may or may not be true, depending on your model. So really, the interesting parts of math are very often the structures rather than the particular objects we use. It doesn't matter if my nine is different from your nine, or that neither of them are identical with the eternal quintessence of nineness. All we care about is whether an object is "nine-like" in the specific sense and context we define, and as it turns out that's plenty.
 
Napoleon III said:
Another example of the above that doesn't make me sound like a Schizo is Laplace Transforms. Why would this extremely strange function in which you do this strange half infinity integral be so useful for differential equations? And look at an inverse Laplace Transform. Look at how incredibly complex it is. Something more than a random useful convience is at work there
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Maybe, but I wouldn't be so sure. The thing that makes the Laplace transform useful for solving some differential equations is that it turns derivatives into multiplication by a variable, so you can turn solving a linear ODE (with constant coefficients) into solving a polynomial equation. In other words, the utility comes purely from some fairly elementary algebraic properties it satisfies, along with the fact that it is actually possible to compute it in interesting, basic cases (e.g. rational functions and that kind of thing). The Fourier transform is, at least on a superficial level, useful for exactly the same reasons (for this particular application, at least).

Kosher Dill said:
Whereas on the other hand "{3, 5, 6} is a subset of 9" may or may not be true, depending on your model. So really, the interesting parts of math are very often the structures rather than the particular objects we use. It doesn't matter if my nine is different from your nine, or that neither of them are identical with the eternal quintessence of nineness. All we care about is whether an object is "nine-like" in the specific sense and context we define, and as it turns out that's plenty.
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All this is certainly true if you're working in the setting of set theory, but I'm told that type theory and that kind of thing is supposed to alleviate these kinds of issues with "nonsense questions" being possible to ask. With the caveat that it's autistic.
 
oksurealright said:
All this is certainly true if you're working in the setting of set theory, but I'm told that type theory and that kind of thing is supposed to alleviate these kinds of issues with "nonsense questions" being possible to ask.
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I think any mathematical foundation is going to come along with "nonsense" baggage though by the time you build up a model of the natural numbers. Just instead of a statement in ZFC set theory, perhaps you'd have something from the hundreds of impenetrable pages of "Principia Mathematica" that led up to defining the number 1. (and yes I know no one really uses PM)
 
ZFC is an easy to understand, intuitive and practical axiomatic system that does pretty much anything a mathematician might need, I never understood the sperging about other axiomatic systems.
 
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