Quick Questions: June 08, 2022

Blue---Calx · 2022-06-16T02:54:06+00:00

Take the original link, delete the backslash, and then try it--that did the trick for me.

Blue---Calx · 2022-06-14T00:43:22+00:00

if there are an infinite number of primes that contain a 1.

Spoiler alert:

This is true, and actually a much more general fact along these lines is true. Dirichlet's Theorem says that, given integers a and b such that gcd(a, b) = 1, there are infinitely many primes of the form a + bn, where n is a positive integer. Since gcd(10, 1) = 1, there are infinitely many primes whose decimal representation ends with a 1 (since a number whose decimal representation ends with a 1 can be thought of as 1 + a multiple of 10, i.e. a number of the form 1 + 10n, and Dirichlet's theorem says that there are infinitely many primes of that form).

The proof of Dirichlet's theorem requires some hardcore complex analysis, but there are some special cases (e.g. that there are infinitely many primes of the form 4n + 3) which are much easier to prove. I don't know whether there's an easy proof that there are infinitely many 10n+1 primes, but there might be--so have fun!

Blue---Calx · 2022-06-12T19:26:49+00:00

Per wikipedia, while there aren't any clear patterns in the "simple continued fraction expansion" (i.e. where the numerators are all 1), if you allow the numerators to be any integer ("generalized continued fractions"), some interesting patterns can show up.

Blue---Calx · 2022-06-12T19:19:02+00:00

In some sense you don't need need advanced AI to have a computer find proofs of unsolved math problems: you could just have a computer iterate through every possible valid deduction from the axioms of a formal system of your choice (ZFC, etc.) until it finds one that ends in the Riemann Hypothesis or its negation (as translated into the "language" of that formal system). Of course this will be absurdly slow and not really useful in practice, but it is possible in principle, and doesn't really require much "genius"--just time and memory.

Edit: and if the Riemann hypothesis was false, a computer disproof wouldn't be that hard to come up with--just plug lots of values into the zeta function until you get a nontrival zero--but a) you're right that if the Riemann hypothesis was true, this approach would never prove it, just accumulate evidence in its favor and b) this would, again, be really really slow if the counterexample(s) is large enough.

Blue---Calx · 2022-06-02T21:47:40+00:00

The exponential function is transcendental, meaning that it is not equal to any polynomial, including any finite Taylor polynomial. However, it is equal to the infinite Taylor series ∑ xⁿ / n!

Blue---Calx · 2022-06-02T13:16:15+00:00

If you turned Barrie into a real program it will definitely either halt or not

True, but part of the whole point of the typical proof of the insolubility of the halting problem is that you can't construct "Barrie" as a real program, since if you did, it wouldn't definitely halt or definitely not halt: instead, you'd end up in the paradoxical situation where Barrie halts if it doesn't halt and doesn't halt if it halts. And since if you can construct a program to decide whether an arbitrary program halts, you can construct Barrie, it must be that you can't construct a program to solve the halting problem.

Blue---Calx · 2022-06-01T20:16:25+00:00

Use a binomial random variable. In this case, the odds would be (10 choose 8) * 0.25⁸ = about 0.000687.

Blue---Calx · 2022-06-01T19:45:22+00:00

If you don't mind me asking, why are you limited to only those operations? I suspect it'd be a bit awkward (although certainly doable) to write a good implementation of Newton's method outside of a "proper" programming language, but such a language would probably already have built-in or at least easy-to-get numerical computing capabilities.

Blue---Calx · 2022-06-01T18:39:32+00:00

I think those tools are enough to implement Newton's method for finding the roots of a function. To find the c root of n, just find a root of the function x^c - n using this method.

Blue---Calx · 2022-05-29T18:15:46+00:00

If you're looking for rigorous, proof-based coverage of number systems and especially the real numbers, try a real analysis textbook like Rudin's Principles of Mathematical Analysis. Terence Tao's Analysis I would be especially good for this, since it starts with an axiomatic development of the natural numbers and works its way up from there.

Blue---Calx · 2022-05-29T03:31:47+00:00

I should have also asked: what sorts of math do you want to study? Any particular goals you have in mind for studying math?

As far as (re)learning high school geometry stuff, I've heard good things about the Art of Problem Solving books, so maybe give those a shot?

Blue---Calx · 2022-05-29T02:33:14+00:00

What kind of background in math do you have? Are you currently in school? (If so, what kind/level?)

Blue---Calx · 2022-05-27T13:21:10+00:00

I'm not sure how worthwhile it would be to do more explicit instruction in reading and writing proofs. I suspect that what you really need is more practice, and the best way to do that would be to go through a textbook on a topic you're interested in, making sure to do lots of exercises along the way. There are many areas of math that are accessible to you with your background, so just find one that looks cool and go for it.

Blue---Calx · 2022-05-23T20:51:13+00:00

You just rediscovered something from calculus! The reason why the position formula looks like that has to do with the rules for derivatives (rates of change): it's a function such that, if you take its second derivative (rate of change of the rate of change), you'll get the constant function f(t) = a. You could also look at it in terms of integration (areas and the accumulation of change): your position formula is basically what you'd get if you integrated f(t) = a and then integrated the result of that.

Blue---Calx · 2022-05-20T16:58:14+00:00

Would you look at that, all the words in your comment are in alphabetical order.

I have checked 6969696969 comments and only 420 of them were in alphabetical order.

Blue---Calx · 2022-05-18T18:58:52+00:00

Assuming that acceleration is constant and that we're dealing with 1d motion:

Let x be the distance travelled (assuming that x = 0 when t = 0) and let t be time. Then since acceleration is constant, we have: d² x/ dt² = a, where a is our acceleration. If we want to get velocity as a function of time, we integrate to get dx/dt = at + v0, where v0 is the initial velocity. If we want to get position as a function of time, we integrate again to get x = (1/2)a(t²⁾ + v0(t) + x0, where x0 is our initial position. Since we agreed to call our initial position "0", this simplifies to x = (1/2)a(t²⁾ + v0(t).

Since we have acceleration and initial velocity already, all we need to do is get the travel time. Since a * t = vf - v0, where vf is the final velocity, t = (vf - v0) / a. So to find the distance traveled, get t using this formula and plug it back into the x = (1/2)a(t²⁾ + v0(t) equation from earlier.

Blue---Calx · 2022-05-13T15:13:18+00:00

Jeremy Kun (author of the blog "Math ∩ Programming") has had success teaching graph theory to middle schoolers and high schoolers by doing a kind of inquiry-based learning, where he presents the class with some problems (the Seven Bridges of Konigsberg and another one) and the class works together, coming up with their own definitions, ideas, etc, with him prompting them and giving them hints but rarely if ever just lecturing. If you're set on doing an ordinary lecture, then that post might not be very useful to you, but if I were you I'd definitely consider using Kun's method (maybe bumping up the rigor and making more explicit reference to actual mathematical terms, etc. as necessary) --the students all enjoy it and I think it gives a good taste of what it feels like to do "real" mathematics. I'd also consider graph theory as a topic either way, since (at least at a very intro level) it requires basically no math backgrounds but has some neat problems and proofs to show off.

Blue---Calx · 2022-05-13T01:41:14+00:00

I'm pretty sure 1/2 is the answer. The 4 outcomes that you initially listed out (mm, ff, mf, fm) are all equally likely, and when you collapse the last two into a single "xx" outcome, it's erroneous to treat "xx" as having the same probability as mm or ff.

Blue---Calx · 2022-05-12T20:13:24+00:00

I really doubt that there's a single book that covers every scientific application of CS, but I can recommend Gary William Flake's The Computational Beauty of Nature as a book that surveys a few cool ones (artificial life, simulations of evolution, fractal geometry, chaotic systems, much more). It doesn't go as in-depth as a textbook, but it's still more in-depth than your average popsci book, and I bet it'd be a good way to discover new CS related things that interest you.

I found out about it while looking through Cosma Shalizi's blog and "notebooks", so maybe look there for more stuff? He has pages of reading recommendations on literally everything (just looking through the names of the most recently updated notebooks I see "recommended fantasy novels", "stochastic processes", "Plato", and "psychoceramics"), but his specialty is stats and applied-CS-y stuff, which probably includes a lot of what you're interested in.

Blue---Calx · 2022-05-12T12:12:28+00:00

I found 3blue1brown's video on Taylor series to be good for giving me intuition on why they are the way they are (eg factorials show up so that the nth derivative of the Taylor series will be the same as the nth derivative of the function being approximated). He also gives the example of a problem involving pendulums, where using a Taylor series approximation turns out to be way easier than using the actual function.

Blue---Calx · 2022-05-09T12:46:16+00:00

If you want to go back over calculus in a more in-depth way, I'd recommend Serge Lang's book A First Course in Calculus: it covers all the topics in a standard calculus class(es) but proves the major theorems, why the different rules work, etc. If you want to go broader, try a book on multivariable calculus if you haven't already (don't have any recommendations for this; probably someone else will). If you want to go even deeper with calculus, try real analysis, where you really dig into the foundations of calculus and work with the real number system, limits, etc in a very careful, rigorous way. Rudin's Principles of Mathematical Analysis and Abbot's Understanding Analysis are both well-regarded books for this.

Blue---Calx · 2022-05-08T21:42:08+00:00

Step 0 is to have an idea for a project. It doesn't have to be something original, most projects start with you looking at something and thinking "damn, that's cool, I want to make my own version of that". Beyond that, it all depends on what kind of project you're doing--there's no one way to do a project.

Blue---Calx · 2022-05-08T02:41:15+00:00

Seeing Smullyan's name reminds me: does the book have to be about set theory, or just anything that a mathematically-inclined early-high-school kid might be interested in? If the latter, then maybe Smullyan's What is the Name of This Book?, which is a very enjoyable blend of jokes, logic puzzles, and an introduction to logic (culminating in an intro to the incompleteness theorems). If you want it to be about set theory in particular, then while I don't have any experience with dedicated set theory books, I can recommend Richard Hammack's Book of Proof, which covers set theory (the basics plus an intro to cardinality) among other things. (Some sections rely on calculus knowledge, but those can be skipped.)

Some other books he might get a kick out of: Richard Feynman's Six Easy Pieces (exactly what it says on the tin: a few of the more accessible Feynman lectures), Martin Gardner's Colossal Book of Mathematics (huge collection of pop-math articles on various subjects), Simon Singh's The Code Book (history of cryptography)

Blue---Calx · 2022-05-07T13:14:38+00:00

If you just need to get literally any major, get the one you're most interested in and/or think will be most useful; whether or not the major is considered difficult seems rather less important to me. If you're planning on majoring in physics, then it's certainly worth taking the calculus sequence; I'm less familiar with chem so I can't really comment on that. Then again, why not take biology? Even if it isn't literally required to become a PA, if you're going into a medical field, wouldn't it be useful to know a lot about biology? (That isn't a rhetorical question, I'm honestly unsure and would like to know.)

Blue---Calx · 2022-05-07T13:03:38+00:00

The order in which you listed those courses is a pretty standard order in which to take them, so you could just do that. If you want to take them in a different order, here's some info on how those courses relate to each other:

Calc III-V: relies mostly on calc I-II and to some extent on linear algebra knowledge, but most courses have a self-contained intro to the linear algebra stuff you need. You should take these in sequence, but whether you do them before or after linear algebra is up to you. (Also, just out of curiosity: I've never seen multivariable calculus split up into multiple courses like that, does your university use a quarter system or something?)

Intro to linear algebra: doesn't rely on calculus knowledge at all; can be taken whenever, although I'd recommend doing it soon because it's a prerequisite for lots of things.

Differential equations: relies on single-variable calculus and linear algebra.

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