What are /r/math's favorite Products & Services?

From 3.5 billion Reddit comments

https://mathpix.com/

> When I lost my .tex file to the Principia, I was devastated. Mathpix helped me effortlessly use equations from the Principia in my new work. I now have more time to stand beneath trees and get hit by apples.
- Isaac Newton

Worth noting that doing curves like this does not produce circle segments, these curves are 'steeper' (Imgur mirror).

The book i got this from was https://www.amazon.co.uk/Inside-Interesting-Integrals-Substitutions-Undergraduate/dp/1493912763/ref=sr_1_1?ie=UTF8&qid=1530548692&sr=8-1&keywords=inside+interesting+integrals

A full proof is in here which is just remarkable (if anyone can find it online please post it here).

I just thought it was great to see a integral where the result is just infinity and 0 plugged into the function; the possibilities for integrals with this are endless!

How is Common Core to blame? Here's the 1st grade Common Core guideline for how to teach equality, for instance: "Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2." (source, p. 15)

Nowhere in the standards does Common Core even insinuate that the equals sign means "all together." Some of the people who wrote the standards are in fact experts on this fallacy. (source, p. 20)

I used to have this problem...

I spent a lot of time at school / uni / work not being great at maths. I used to understand the concepts of algebra and pre-calc etc at a high level but really couldn't apply it. It just didn't make sense to me.

The turning point came when someone (my wife) pointed out that I seemed to be missing some of the fundamental experience with the basics of maths. Stuff that I felt I should just know. I was actually oddly unable to even admit to myself that i really didn't know some of the stuff that (in my mind at least) was trivial.

The way I solved this was to go right back to the beginning - learning very basic fractional manipulation, really simple algebra from simple linear equations, on to simultaneous equations, to quadratics and pre calc.

This was an eye-opener for me as once I started laying solid foundations to my understanding it was so much easier to build on.

Later in life I went on to study for an undergraduate degree in Maths. Something I'd never have done without someone pointing out to me that you can't build a house without good foundations.

Try this out http://www.khanacademy.org/ There's a Math(s) section and it's very good.

TL;DR: If you're struggling maybe you need to go right back to basics and build up your understanding. It worked for me.

He is clearly a very dedicated teacher, but he is not the only dedicated teacher in the Berkeley math department. I know this for a fact. One of these very dedicated teachers is Craig Evans, whose name Coward drags through the mud with out of context quotes.

I've talked about this a lot with people I know from UC Berkeley (mostly people who were grad students or postdocs there), and everyone I've spoken to believes that Coward is egregiously cherry picking and manipulating information, and that Coward's self-aggrandizing rhetoric is not particularly well founded. These are people I know care deeply about teaching, and have a history of working effectively for their students.

There are problems with undergraduate math education, and UC Berkeley might illustrate many of these problems. This is an important topic worthy of discussion. But in this particular situation, it just doesn't seem like that's what's going on. I'd keep in mind that professionalism will prevent the UC Berkeley math department from responding as loudly or as one-sidedly as Coward's attack, but that doesn't mean that he's giving you the full story, or even an honest representation of anything (see CowardGSI's comments and my comment below). I'd encourage people to keep fighting for better education, without jumping on the Coward bandwagon.

Jacques Hadamard wrote an entire book on this kind of phenomenon.

As far as anyone can tell, it's a natural part of learning math. A period of intense study is followed by a moment of clarity that occurs when one is *not* actively engaged in math, as though some kind of unconscious process were still working on math even when one's attention was focused elsewhere. Though this moment doesn't happen during the period of intense study, that is still a key component of the process.

This is really cool! The reason why this works is as follows. Your expression consists of two parts: a matrix and a vector, and they are multiplied. First, the matrix is a so-called rotation matrix (see https://en.wikipedia.org/wiki/Rotation_matrix). When pre-multiplying a vector, it produces a rotation of that vector by 2pi/n floor(t) radians. In other words, this rotation matrix takes the integer part of t and rotates the input t times by 2pi/n radians. Second, the vector (cos(pi/n), sin(pi/n)(2t - 1) for 0 <= t <= 1 defines a line segment between two points on the unit circle, the first point being pi/n radians clockwise from the point (0, 1), and the second point being pi/n counterclockwise from the point (0, 1). See https://www.desmos.com/calculator/dy8eywwsdw for an example. (Note also that 2t - 1 for 0 <= t <= 1 varies from -1 to 1.)

You can therefore think of the "drawing process" in Desmos as follows: starting at t=0 and going towards t=1, the vector produces a line segment; the matrix for 0 < t < 1 is an identity matrix, meaning that the line segment remains unrotated. As t goes from t < 1 to t=1, the vector "starts over" from the starting point, and it produced the same line segment again, while the rotation matrix rotates these points by 2pi/n radians. At jumps from t < 2 to t=2, the vector again starts over, but now the line segment is rotated by 2 * 2pi/n radians. And so forth. The result is a sequence of line segments, each of which is the same line segment but rotated 0, 2pi/n, 4pi/n, 6pi/n, 8pi/n, ... radians. In particular, if n is irrational, this produces an infinite number of distinct rotations, i.e., you never see the same rotation twice.

I created a vector function that can plot regular polygons and star shapes a while back, I have since been reminded of it, and decided to clean it up. Here is a Desmos link if you want to play around with it: https://www.desmos.com/calculator/s1tisuromo?fbclid=IwAR2bPPc64pERn8aZQ-WEJMz97_H1fWjuxBsDPCKL5q8Eosw6p9nalDaPOPA I hope you guys enjoy :)

For real analysis I really enjoyed Understanding Analysis for how clear the material was presented for a first course. For abstract algebra I found A book of abstract algebra to be very concise and easy to read for a first course. Those two textbooks were a lifesaver for me since I had a hard time with those two courses using the notes and textbook for the class. We were taught out of rudin and dummit and foote as mainly a reference book and had to rely on notes primarily but those two texts were incredibly helpful to understand the material.

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If any undergrads are struggling with those two courses I would highly recommend you check out those two textbooks. They are by far the easiest introduction to those two fields I have found. I also like that you can find solutions to all the exercises so it makes them very valuable for self study also. Both books also have a reasonable amount of excises so that you can in theory do nearly every problem in the book which is also nice compared to standard texts with way too many exercises to realistically go through.

The best differential equations book that I have used was my first ODE course in Undergrad, by Blanchard, Hall, and Delaney. It was a textbook that didn't take a lot of time going into the solutions of ODEs, but the modeling and interpretation aspect much more. The book treats you like a human too and it reads so well. Highly suggest it.

I'd personally recommend learning Python. It's a lot more noobie friendly than C will be and gets you in the habit of writing 'pretty' code. Using the NumPy and SciPy libraries will give you the tools you need to solve harder math problems.

C is a favorite of mine due to its speed and flexibility but it's not very friendly to beginners (pointers in particular). Fortran is also very good to use for math, but I personally don't care much for it.

Also, if you haven't heard of it yet, you should check out Project Euler. It's a collection of mathy problems that either require programming to solve or are greatly facilitated by it. I'd check out some tutorials in python and try to solve some of the problems on that site.

I don't know if I'm a good example, I'm not a mathematician, just a humble 55 yo researcher in computer science, but I'm really greateful there is not a dress code in my line of work (for example, I do not own suits or ties. I despise ties).

Currently I own 3 pairs of shoes like this one which differ in color and in thickness (one is also rainproof).

I do not really like shoes. I wear shoes mainly if it rains heavily and I have to walk around the city, or if I go walking in the woods or on the beach. Most of the times (all year around) I wear short socks and Birkestock sandals (model "Milano"). I don't like to wear long socks, nor to be barefoot in sandals.

Since I stopped wearing shoes all the day about 4 or 5 years ago my feet are much happier. Here in Tuscany (Italy) the weather is rather mild and my office is always quite hot. In winter, if it is very cold (say, 24F-40F) sometimes I wear two pairs of socks in such a way I can take one pair off when I'm inside.

For a feature rich 3D gravity simulator that uses real units and includes 50+ simulations (from our solar system to galaxy collisions) check out Universe Sandbox.

http://universesandbox.com/

What started as a personal project has become my obsession and sole source of income. I've been working on it for over 2 years and now work on it full time.

I just released the 14th update to Universe Sandbox 2 yesterday and have more updates planned throughout 2011. Let me know how I can make it better.

I'm in literally the same boat with you. Full time software engineer and I graduated almost a year ago.

I think two things motivate me to self-study:

1. It's something I enjoy

2. It's someting I'm excited to apply to some project (personal or professional)

To a degree, 2 is kind of a subset of 1. Anyway, when I come home I just find an hour to set aside and work through a book, ebook, or video to develop the skill in question. Recently it's been this book. And someday's I just wanna play video games and be lazy. So I think a large part of it is understanding that it's a slow process for the most part.

If you, or both of you, are into programming, you might look into Project Euler.

I personally also enjoy optimizing/computing stuff from games. This can be done for mostly any game that involves numbers or randomness in some way. (Experience vs cost efficiency/economics/optimal strategy.) (MMO)RPGs and /r/incremental_game s are pretty good targets for these kind of things. Games actively requiring hard mathematics are pretty rare as they have a pretty narrow userbase.

Here's free and frequently updated book: Foundations of Data Science by Blum, Hopcroft, Kannan

Also, check out:

Probability and Computing: Randomization and Probabilistic Techniques in Algorithms and Data Analysis by Mitzenmacher and Upfal.

You would think that. And you would be wrong. We keep making the same mistakes over and over.

More RNG fails:

In 1998, the Arizona Pick 3 lottery (Pick 3 digits, 0 - 9) ran daily for over a month before someone noticed that it never picked 9. http://catless.ncl.ac.uk/Risks/19.83.html#subj5

Scientific Games' quick pick system never picks the last horse: http://www.zdnet.com/blog/projectfailures/software-bug-prevents-big-brown-quick-pick-racing-payoff/789

The answer is approximately 4.29733.

Use this image:

http://i.imgur.com/DqqUq.jpg

We've got two similar right triangles.

ABC is similar to EFC

BCD is similar to BFE

So using this and the Pythagorean formula we can come up with enough equations to eliminate our variables:

10^2 = x^2 + y^2

12^2 = x^2 + z^2

u^2 = w^2 + 5^2

v^2 = (x-w)^2 + 5^2

u/10 = w/x = 5/y

v/12 = (x-w)/x = 5/z

And it's a straightforward, but long, solution from there. So I just plugged it into Wolfram Alpha:

http://www.wolframalpha.com/input/?i=10^2%3Dx^2%2By^2%2C+12^2%3Dx^2%2Bz^2%2C+u^2%3Dw^2%2B5^2%2C+v^2%3D%28x-w%29^2%2B5^2%2C+u%2F10%3Dw%2Fx%2C+u%2F10%3D5%2Fy%2C+v%2F12%3D%28x-w%29%2Fx%2C+v%2F12%3D5%2Fz

https://www.geogebra.org/ geogebra has some great tools for that. Play around with it on desktop if you can!

Mathematica is good too, though I've only ever tried the free version they package with the raspberry pi os for desktop use... I think you might otherwise have to pay or get a license through a university.

I always liked The Art of Computer Programming which, in an effort to establish the scale of problem difficulty and to discourage people from trying to do all the exercises, has the third exercise be (paraphrased) "Show that there are no positive integer solutions to a^n + b^n = c^n for n ≥3."

I just finished flatland. There's a good librivox recording of it if you commute. Some interesting mind expanding ideas in it and suitable for younger readers if you want to get them interested in math

Audio Book

Gutenberg

If anyone with a wear OS device is interested in an (almost) unreadable watchface, download the app facer, and my watchface is here

I think you'd be doing yourself a favor if you learn how to use LaTeX before you start your classes. You might find some of your classes require your assignments be formatted in TeX. Others won't, but your professors/TAs will be thankful for how easy it is on their eyes.

I have no specific useful tips (nor do I claim to be particularly good at math, only interested), but something that really helped me was Khan Academy. All the lectures are in video with pertaining exercises. He is really good at explaining and giving examples so it is easy to understand. If you're having trouble with multiplication and division, maybe the arithmetic-section is something for you to start with?

They should have linearly interpolated between the terms. It gives a really cool effect that the function is winding around the cosine.

EDIT: here it is.

Yep, the editor DID ask for text explaining the pictures (source), but Conway refused. The editor eventually caved, because Conway.

The paper is purposely unhelpful just so that Conway could claim a "record".

There's this great math textbook that I used called "Introduction to Analysis and Abstract Algebra" by a guy named Hafstrom that I found in the basement of my university library.

It was really cool because it treated the two subjects simultaneously, each supporting the other, whereas normally they're kept separate. It's been a few years since I've read it but as I recall the theoretical development starts with explaining what a field is (groups and rings come later) and explaining how the real numbers form a field with just a few extra axioms. It's a very different approach than what I encountered in my analysis and abstract algebra courses.

https://www.amazon.com/Introduction-Analysis-Abstract-Algebra-Hafstrom/dp/0721644554

You get a vertical asymptote wherever the denominator equals zero. You can do that arbitrarily.

The real trick is getting two horizonal asymptotes. The quickest way to do that is to use the absolute value function on either the top or the bottom of a rational function, where both are of the same degree.

Sooo...without actually checking (I'll do that shortly), I would guess that y = |x^3 | / (x + 1)(x + 2)(x + 3) would do the trick.

EDIT: And it works!

I haven't watched any of the Khan videos before, so I decided to pick one at random and record my thoughts. I picked the video on inverse sine

http://www.khanacademy.org/math/trigonometry/v/inverse-trig-functions--arcsin

It starts off pretty well, though I would have chosen an angle other than pi/4 since it has the same sine and cosine. It would have been harder to derive the sin of (say) pi/6, but that should be something all students would be able to do by the time they got to learning about inverse trig functions.

When he starts talking about the domain of arcsin, he doesn't explain why the domain is what it is. Why not 0 to pi? Why not 0 to 2pi? These are natural questions students ask when learning this concept.

In general the video seems unpolished and stream-of-consciousness, which is fine, but you can tell that he isn't an experienced math teacher. His refusal to use parentheses is aggravating to see, especially when he writes stuff like sin^-1 -sqrt(3)/2. There are several ways students will commonly misunderstand that collection of symbols, and he breezes right over it.

My conclusion is that this video would be a useful resource for a student learning this material, but not really a replacement for a well-trained math teacher/professor. Granted, many students have a teacher/professor who is not well-trained, in which case these videos may be the best resource they have available.

I wonder if Khan has considered hiring a professional mathematician/physicist/etc. to improve his videos? Given the popularity of the videos, I would think he'd be able to afford it.

I like this page from the Wolfram Tones FAQ, which slips this bit in:

> systems known as one-dimensional cellular automata (now often called Wolfram automata)

..."often" called Wolfram automata on websites owned by Wolfram, maybe.

Python+NumPy+SciPy+matplotlib is an amazing combo for scientific computing. Whenever I have the freedom to choose, I use that over MATLAB. Python is a much nicer language to work in. Plus, NumPy's idioms are fairly easy to pick up if you already know MATLAB: http://www.scipy.org/NumPy_for_Matlab_Users

As for Octave, it works though as a programming environment it is pretty average. It is a good choice if you need to reuse an existing library from MATLAB. Unfortunately, Octave's implementation tend to be on the slow side—i.e., it is generally much slower than Python's or MATLAB's.

I have never used it, but I've heard good things about Julia: http://julialang.org/. The syntax of Julia is similar to MATLAB, but it has extra features to make general programming easier. Plus, they have a fast JIT implementation.

Anyhow, it is hard to make more specific recommendations without knowing your needs. If you are just looking to do straightforward numerical linear algebra, then above are all pretty good choices. For more complex tasks, like the ones we can find in computer vision, machine learning and material sciences, there are better alternatives.

> The primary source is the book by Goldstein & Goldstein.

Goodstein & Goodstein.

"Goldstein" is this classical (in every sense) mechanics text, which also touches upon planetary motion.

It's still available. https://www.amazon.com/Dubbs-Arithmetical-Problems-Pupils-Part/dp/B000Q7XX0G

..and there's a PDF too. https://vinairemath.files.wordpress.com/2015/10/1893-dubbs-arithmetical-problems.pdf

Yes. You can find more details in:

>Schmelzer, T., & Baillie, R. (2008). Summing a curious, slowly convergent series. The American Mathematical Monthly, 115(6), 525-540. <strong>Link</strong> (no paywall).

Convergence is proved in Theorem 1 and the rest of the paper answers the immediate follow-up:

>Once a series is known to converge, the natural question is, “What is its sum?”

The circumference is fixed at 1, so each side of the polygon has length a = 1/N. A regular polygon can be broken up into N isosceles triangles with one angle of alpha = (2pi/N) and two angles of (pi - 2pi/N)/2. The side opposite the angle of alpha = 2*pi/N will have length a. Break each of these isosceles triangles again into two right triangles, such that tan(alpha/2) = (a/2) / b, where b is the height of the right triangle. Then b = (a/2) / tan(alpha/2). The area of each isosceles triangle is then A = (1/2) * a * b = a^2 / 4 / tan( alpha/2 ). Putting a = 1/N into this gives:

A = 1 / (4N^2 ) / tan( pi / N ).

Edit: Left off the total area: The total area is then N * A = 1 / (4N) / tan( pi/N ).

Also, note that as N goes to inf, the above approaches 1/(4pi), the area of a circle with circumference 1.

Plot.

I needed to learn LaTeX a couple of months ago and started with this. It is great for learning with since it has live compiling, you get to see what changes you make immediately.

However for larger documents (the one I was working on came up >100 pages) I found ShareLaTeX to be better, particularly when working with multiple people on the same document

Not me, but I've seen this possibly-apocryphal story somewhere. (Likely in this book by Paul Halmos, but I'm not certain whether that's where I first saw it.)

It's a standard exercise in elementary linear algebra to prove that C is a vector space over R. Apparently some careless professor interchanged the roles of C and R, leaving the students horrified.

Perhaps even more interesting, depending on what structures of R as a C-vector space we'd want to preserve, this actually can be done! Basically, the idea is that R and C are isomorphic as additive groups, since both have vector space bases over Q of the same cardinality. The question is then isomorphic to asking whether C can be made a vector space over C, which is trivially true.

you should definitely give harder problems multiple tries, you might not be able to solve them right away. Go back and re-learn the concepts the problem needed. Sometimes you might need to use a concept which you are not familiar with at the moment. I recommend reading "How to Solve it " by G. Polya https://www.amazon.com/How-Solve-Mathematical-Princeton-Science/dp/069116407X

It covers different problem solving approaches in a agnostic fashion.

You don't need to write it as an explicit function equation to use it as a function, just differentiate x^(2)+f(x)^(2)=5^(2) through by x, which is valid since everything is a function of x. In this way, the Implicit Function Theorem is justification to things like "Differentiate xy+sin(y^(2)x)=4 through with respect to x."

Though you do have to treat differentation as an operator and that is something you may not be ready for in your videos yet, as it would have to happen after all the limit/approximation stuff has been fleshed out.

Anyways, I was just curious, since explicitly discussing the total differential of a 2D function is generally not the route taken to introduce implicit differentiation.

EDIT: Sidenote, the graph to that random equation is actually pretty cool.

you may find help here in this massive amount of free instructional video published at Khan Academy

http://www.khanacademy.org/

>Topics covered in the first two or three semesters of college calculus. Everything from limits to derivatives to integrals to vector calculus. Should understand the topics in the pre-calculus playlist first (the limit videos are in both playlists)

Knuth dedicated The Art of Computer Programming:

> This series of books is affectionately dedicated to the Type 650 computer once installed at Case Institute of Technology, in remembrance of many pleasant evenings.

There is a bijective map between the natural numbers and the rational numbers, but this map is not a homomorphism. That is, we may have f : Q -> N, but we won't have f(p*q) = f(p)*f(q) and f(p + q) = f(p) + f(q), where the first operations are in Q and the second are in N.

It occurs to me that maybe you mean, "forget about the operations in N and just use whatever they are from Q to force it into a field." Well, you could do that sure, but that's a little silly because then all you have is the rationals where every element has a different name. There's not really any point to it.

When people say the naturals don't form a field, they mean the naturals with the operations they've been endowed with, not some arbitrary operations you give them.

One of the most nerdy gifts I've received is a version of this clock which I will admit I quite enjoy. I'm cringing at 3(pi-0.14) though.

I remember reading A Brief History of Time while in middle school. I picked it up out of the public library on a whim. I was surprised at how easy of a read it was for a topic that is so complex. It was at that point I understood that the most complex topics in human history were easy to understand at a high level if explained simply, that the knowledge was easily accessible to someone like me.

It's one of the few books that I can point to that I can say legitimately changed my life.

I know Geogebra is free and really user-friendly. It's great for high school level math, and there's also a free tablet version available as well. I'd highly recommend it for your student.

My understanding is that some of the 3rd party sellers on Amazon use algorithms to automatically set and adjust prices. They tend to work pretty well and be stable if Amazon is also selling the book, since these prices tend to depend on what other people are selling for and Amazon's prices set a more reasonable and stable baseline.

There was a story about a textbook being sold for something like $32 million because two third party sellers were in an unintentional arms war to be the second cheapest seller. So the book started off at, say $100, but then they both kept increasing the price by, say, $1 each time the other one adjusted theirs. If that's not bad enough, imagine the price being incremented by a percentage with no cap, then you have exponential growth and we're all doomed.

This isn't a perfect example, but take a look at these colored pencils. They were sold by Amazon itself (not FBA) and were something like $12 or $13. Since then, they sold out. Although I can't figure out when exactly that was (other than between Oct 30th and earlier this week), this price tracker shows some minor instability (probably caused by inventory fluctuations), followed by a huge jump to a price no one would pay for those colored pencils even accounting for scarcity.

This is also what's going on when you see something going for $50 and with "9 used from $78.00."

I've heard it can help to message sellers and tell them that the price is ridiculous, because they could have very well not noticed what happened and will fix it.

(a*b)^n =a^n * b^n is only a property when you have positives (technically nonnegative reals), as the "primary" solution to these problems is always positive (ex. square roots of positive numbers are always positive).

When a and b can be nonpositives, then your solutions can start branching out in different directions, which causes some trouble when you multiply them back together.

Consider, for example, that (-1 * -1)^(1/2) != (-1)^(1/2) * (-1)^(1/2), as the former is just the square root of 1 (which is 1) and the latter is i^2 = -1.

Since WolframAlpha uses a nonreal principle branch for radicals of negative numbers (there are reasons why this makes sense), you end up getting a similar phenomenon with your inputs.

For those interested in a popular overview of the topic, I recommend Chaos: Making a New Science by James Gleick. It does a pretty good job of popular explanations of the theory, and talks about a lot of key people in the field.

Grinches equation pre Christmas
&
Grinches equation post Christmas
What did that prove? That the Grinches parametric equation that represented his heart grew by a factor of 10^5 that day.
x^2+(y-(x^2)^(1/3))^2 = 1

I have an engineering (research) background so my answer is biased in that regard but I’d say: - Fractional calculus: fractional operators, fractional optimization, fractional order control systems, operational method. Basically, there have been some new developments in the application of special functions in fractional calculus which apply to many sciences. - Differential inclusions. - Control Theory. - Boundary Value Problems. - Plemelj-Privalov theorem (Clifford analysis). - Subdiagonal algebras.

There’s a dope book on the subject you’re asking about that I recommend. It’s brand new (I got an advanced copy through my institution earlier this year, yay). Check it:

https://www.amazon.com/Current-Mathematical-Analysis-Interdisciplinary-Applications/dp/3030152413

Do you mean the Russian texts? The one I remember is of Polyanin.

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But I would suggest that GF Simmon's classic book on ODEs is much better.

I don't think they look very similar.

On the other hand, tan(x) does look a lot like x+x^(3)/3. This is because that's the third-degree Taylor polynomial for tan(x), centered at x=0. To learn more about this, take calculus.

Try moving the b slider around here: https://www.desmos.com/calculator/aydeqmaz0a

I set the value of x to 1. The value a is the vertical length of the rectangle. b represents the angle of the arc, and R represents the radius of the circle.

The length of the red curve is always 1, which you can verify if you want.

Pyramid Power by Toth and Nielsen. I only managed to get through one chapter in the bookstore (hard to read when you're constantly struggling to contain your laughter), but the high points were where the authors showed that the Pyramid is the perfect shape (It focuses the energies of the square, a shape that is perfect in every way aside from not being a pyramid, to a point, QED), and then use this to prove that black holes don't exist.

I wish I could find a pdf so that I could ~~laugh at it more~~ brush up on the proofs, but I haven't been able to find one and god knows I refuse to buy it

> Unfortunately a lot of it is out of reach of even an undergraduate Senior math major.

This is true, but other developments happen that make these same results more approachable. For example, you can learn computational geometry as an undergraduate that was graduate level or above as little as 20 years ago.

New results sometimes make it easier to explain old results, or provide intuitions that did not exist previously.

I don't know if this is such a good example. The Pythagorean theorem is extremely simple and elegant to prove geometrically. The algebraic proofs seem way more ad-hoc and complicated in this case.

https://brilliant.org/wiki/proofs-of-the-pythagorean-theorem/

See the proof by similarity and rearrangement on that page. Maybe you know a simpler proof than the algebraic one on that page but anyway, the geometric proofs are still great!

Another 20th century female mathematician who isn't mentioned much is Julia Robinson. She played a central role in resolving Hilbert's 10th problem about Diophantine Equations (polynomials restricted to the integers), which you should actually be able to explain to your students! The statement of the problem is entirely elementary, goes back to antiquity and the very beginnings of number theory and symbolic algebra, and yet the solution is intimately connected to modern subjects like compatibility and incompleteness.

Let me also suggest the book 'The Honors Class - Hilbert's Problems and their Solvers' by Yandell. It has brief biographies of loads of 20th century mathematicians, and also a description of what the problem was about and what was involved in solving it.

https://www.amazon.com/Honors-Class-Hilberts-Problems-Solvers/dp/1568811411

Well, if you saw some of my proofs on my algebra final, BS could mean either one!

I've just been teaching myself via trial and error and googling things as I get stumped. I have a chromebook, so I use ShareLaTeX, which is both free and browser based.

I specifically recommend to anyone who is going to purchase Alice In Wonderland, to buy the 150th anniversary edition with illustrations done by Salvador Dali, who himself loved to incorporate mathematical ideas into his paintings. This specific copy is prefaced with an introduction by Thomas Banchoff, a geometer at Brown, who was friends with Salvador Dali. Banchoff describes various works of Dali and their mathematical inspirations, as well as the friendship they had.

The piece by Banchoff can actually be found online by searching for "Salvador Dali and the Fourth Dimension", but if you're going to buy Alice In Wonderlnad might as well get this copy https://www.amazon.com/dp/0691170029/ref=cm_sw_r_cp_apa_i_S0EPCbQGYDQMN

The Mis-Education of Mathematics Teachers made a huge impression on me, in particular its emphasis on content knowledge and the fundamental principles of mathematics. More recently, the following comment by Ian Stewart has persuaded me to put more emphasis on the visual aspects of the subjects I teach:

> One of the saddest developments in school mathematics has been the downgrading of the visual for the formal. I'm not lamenting the loss of traditional Euclidean geometry, despite its virtues, because it too emphasised stilted formalities. But to replace our rich visual tradition by silly games with 2x2 matrices has always seemed to me to be the height of folly. It is therefore a special pleasure to see Tristan Needham's Visual Complex Analysis with its elegantly illustrated visual approach. Yes, he has 2x2 matrices―but his are interesting. (Ian Stewart, New Scientist, 11 October 1997) (source)

Besides the Napkin Project I mentioned, which is a genuinely good resource? I got a coordinate-free treatment of linear algebra in my school's prelim. abstract algebra course. We used Dummit and Foote, which must be prescribed by law somewhere because I haven't yet seen a single department not use it. However, in reviewing abstract algebra I instead used Hungerford, which I definitely prefer for its brevity. But really, you can pick any graduate intro algebra text and it should teach this stuff.

> It would be like comparing wiles proof for FLT to an entire book about Modular Forms, Elliptic Curves and Galois Representations.

I have (mostly) read that book. It's nothing like Mochizuki's work.

The issue isn't that the theory has a bunch of definitions, most theories do. It's that the entire theory seems to be a bunch of definitions with no nontrivial work or minor applications, except one incredibly grand claim. It this is actually true, there would be nothing even remotely like that in the history of mathematics.

Edit:

> He has made an extremely terrible job communicating with the westerners yeah, but it does not seem to be the case for other japaneses as one of his colleagues wrote a 300 page summary on it

He's convinced a few people, maybe like 10-20, but that's not the same thing as convincing the entire japanese community. And the people he's convinced have also been incapable of explaining it to their colleagues (and not for lack of trying, it seems), which kind of makes it questionable whether they actually understand it, or just think they do. My understanding is that the 300 page summary didn't really help much.

Lander and Parkin's 1966 disproof of Euler's conjecture on sums of like powers is only 2 sentences

http://www.openculture.com/2015/04/shortest-known-paper-in-a-serious-math-journal.html

You can try plotting a bunch of logarithm functions y = log*2* x, y = log*3* x, y = log*2.5* x, etc.

Now find the one with a slope of 1 at x=1.

https://www.desmos.com/calculator/kdlavyjawk

Well, here are some video lectures by Harvard on Abstract Algebra. They use the 1st edition of Artin's "Algebra" as the class text, if you happen to have it and want to follow along. I really would recommend getting your hands on a copy and doing all of the homework assigned for the class. Only then can you begin to learn it.

Harvey Mudd College has an excellent video lecture series on Real Analysis (Lecture 21 is missing from the playlist as it had to be recorded in a later semester, so you'll have to go on their channel to watch that one). The textbook used for this course is "Principles of Mathematical Analysis" by Walter Rudin (affectionately called Baby Rudin by many Analysts since Rudin also has "Real and Complex Analysis", Big Rudin), covering chapters 1-5.

I'm currently using Harvard's video lectures to keep myself busy over the holidays and to help with my goal of completing Artin in this semester. I wish they had video lectures for Algebra II (which finishes Artin) online, but I can't complain about such high quality free education. :) I'll be using the Harvey Mudd lectures for next semester when I plan on tackling Baby Rudin.

This entire article acts as if people are claiming that factoring is definitely hard (which would be wrong, but no one in the field to my knowledge is doing that), so he counters by doing the exact opposite: claiming it is easy (which is equally misleading and useless).

> Then the number of people who have seriously tried must be on the order of magnitude of 100.

So he gives absolutely no reasoning and then concludes that it "must" be on the order of magnitude of 100? I would be absolutely shocked if it's that low (hell, the single paper "PRIMES is in P", which is closely related to this problem and was published in 2006, has been cites 736 times itself -- yet we're to believe only 100 people have worked on one of the most famous number theory problems of the past 40 years?). But I'm not going to say it "must" be on the order of 1000 or 10000 or any other number, because it would be pulled entirely out of my ass.

> but factoring is almost undoubtedly not NP-hard.

"Undoubtedly"? Based on what? Your gut feeling?

> Of course, I have no real evidence for my views;

You don't say. Trying to dispel uninformed yet strong opinions by providing the exact same? Wow.

There are at least two books containing stories similar to those in this thread. Mathematical Apocrypha and Mathematical Apocrypha Redux.

Move on to Galois theory. If he has a good background in algebra, he should be able to manage it fine. The advantage of Galois theory over other things like module theory or whatever is that Galois theory proves statements that you would be interested in even without knowing any Galois theory (constructibility of polygons, abel ruffini etc) and moreover, it is an absolutely beautiful subject to learn even without these applications.

I don't know if he knows any group theory but you don't need much to get started - basic definition of a group and some easy theorems like lagrange's theorem and the relation between normal subgroups and homomorphisms.

Get a good Galois theory text book and as him to work through the proofs. Since you have so much time, you can ask him to read the definitions, the statements of the proofs and try and prove them himself. If he gets stuck for a long time, tell him to look at a little bit of the proof till he gets a new idea and then try to complete the proof.

I think this is absolutely the best way to learn math but it takes a fair bit of time - shouldn't be a problem in your case.

I learnt Galois theory from this text: https://www.amazon.com/Galois-Theory-Fourth-Nicholas-Stewart/dp/1482245825 and it is still one of my all time favorite books in math.

The Man Who Knew Infinity had accurate whiteboards. Ken Ono and Manjul Bhargava were consultants on the film, and made sure the math was relevant and taken directly from Ramanujan's notebooks.

" On March 22, 1977, as I was drafting Section 7.1 of The Art of Computer Programming, I read four papers by Peter van Emde Boas that turned out to be more appropriate for Chapter 8 than Chapter 7. I wrote a five-page memo entitled Notes on the van Emde Boas construction of priority deques: An instructive use of recursion,'' and sent it to Peter on March 29 (with copies also to Bob Tarjan and John Hopcroft). The final sentence was this:Beware of bugs in the above code; I have only proved it correct, not tried it.''

From http://www-cs-faculty.stanford.edu/~uno/faq.html

A mathematical book I read when I was undergraduate is "The Man Who Knew Infinity". It's quite a brilliant book about the Ramanujan, the great Indian mathematician. I recently saw a couple of formulas of his while doing some probability theory... incredibly beautiful and incredibly deep.

Khan Academy is without a doubt the website you want to use.

It goes thru alot of maths and even economic ideas, that will definitely help you start. you can even login in and track which videos you have reviewed and track your progress

http://www.inference.phy.cam.ac.uk/dasher/MoreScreen.html

This thing might give you a seizure, or you might really like it. If you're savvy, you can train or modify it to use an alphabet of LaTeX or math symbols.

This is a cool visualization; and being able to physically manipulate it is cool. I like this representation better though: https://www.desmos.com/calculator/88vvqohfik

b moves the parabola's vertex along the path of a reflected parabola with the equation y=-ax^(2)+c. Mess with the sliders! It's pretty neat.

> the plan is to provide support for WebAssembly

Great idea, both security-wise and performance-wise. Also, maybe GPU.js could be a nice addition too.

Something like Arithmetica is exactly what I was looking for: BOINC is great, but setting up a project always felt a little tedious for developers and sometimes even end-users. This seems to simplify the workflow enormously. I wish you best of luck in the project.

I don't think that is a very compelling argument, unless we believe mathematicians can do no notational wrong :-) The imprecise, ambiguous, sometimes obfuscatory notation that arises in multivariable calculus and the calculus of variations is a well known and frequently discussed issue. I think we underestimate the difficulty it causes to students, especially to students coming from other disciplines who aren't steeped in the mathematical vernacular.

It's been problematic enough that there are some high profile and semi-accepted attempts to refine the notation, such as the functional notation used in Spivak's Calculus on Manifolds, which is based in an earlier attempt from the 50s if I remember correctly. Another presentation of physics motivated in large part by fixing the notation is Sussman & Wisdom's Structure and Interpretation of Classical Mechanics which adopts Spivak's notation, and also uses computer programs to describe algorithms more precisely.

A million random digits.

There is a nice book in Springer's GMT series by Steinberg called something like Representation Theory of Finite Groups. It was a very accessible into to how Fourier analysis is done on finite abelian groups.

Edited with a link.

It's useful for almost everyone. Ramanujan used it - he did extensive calculations to 'get a feel' for equations (see The Man Who Knew Infinity). These days you can just use a computer to experiment e.g. with Python.

Indeed, he's Computer Science and he's specializing in Graph Theory; Graph Theory is a long ways from modern number theory in my experience.

He has a reasonable looking publish history google scholar but only a single paper on anything relating to number theory. That plus the lack of a proof and just a 'publication pending' makes me very, very skeptical.

A combination of the Knapsack problem (as csticky said) and 2D packing.

Which links to an example of a rectangle packing problem: here. Though it's not trying to achieve the same thing.

In general, there's not going to be a fast algorithm for this kind of problem. This is NP-complete I believe. So while there is no general all-purpose algorithm for "weighted 2D-rectangle packing" or whatever it would be called, you could design your own based on the problem.

For example, in the case of Diablo, you could probably run through any combinations which maximise the knapsack problem, and then test for a packing. Usually in Diablo you have enough 1x1 items to make packing doable.

Alternatively, you could try a greedy algorithm: Sort by value per weight, then pack until nothing more fits. This would fail when, for example, you put a 1x4 instead of two 2x2 items which would be worth more together.

In fact, for Diablo, the maximum width of an object is 2, so this gives you another hint for optimisation.

Basically, it's not possible to construct an polynomial-time* algorithm which will guarantee you maximal/optimal usage of the inventory due to the nature of the problem. But specific algorithms can obviously be designed which will give you solutions to the problem. The performance on these depends on the problem faced.

* Thanks gman2093

I just bought The Code Book over a week ago along with a few others. People in /r/math were talking about the documentary based on the book The Man Who Knew Infinity and how the book is better and less sensational. Through that I came across Fermat's Enigma, also by Simon Singh and which I'm currently reading, and The Code Book, as well as Journey Through Genius, which is about many mathematicians throughout the years and seems to be a mini-biography of each. Also just finished re-reading The Drunkard's Walk and convinced my mom to start reading it since I'm reading a book she bought for me. So there's some recommendations for anyone looking for some reading material.

Thanks for getting me excited to read The Code Book. I'll make sure it's next on my queue!

Hey, you can solve this with a spreadsheet. Check it out. This is an exact (numerical) solution, not based on simulation.

Because the table for 100 pills is kind of large, there's a simpler page where the problem is worked out for 5 pills, and then the full table for 100 pills in on the third page.

The first page, labeled 5 days no constraint, tells you the probability of reaching any situation at some point in the course of emptying the bottle. For instance, you can see that when total pills = 3 and whole pills = 2, the probability is 0.47. This means that there's a 47% chance that at some point you'll have 2 whole pills and 1 half pill in the bottle. This is calculated simply by looking at the cells up and to the left. This is because each day you either move one cell down or one cell right, so you can calculate it from them.

This, however, is not exactly what you want, because we want to stop counting once you hit the first time you get >50% chance of getting a half pill. The second page, labeled 5 days solved, shows this constraint taken into account. For instance, the probability of getting to 1 whole pill and 4 half pills is 0, because you would have stopped the previous day when you were at 2 and 3.

On this page, each of the 5 possible finishing conditions are listed along the bottom, along with the probability of each one. For instance, the center column says you have a 22% chance of reaching the finishing condition after 6 days. The average number of days to reach the finishing condition is 4.64, computed by calculating the weighted average over these.

The third page shows the same calculation for 100 pills. The solution given at the bottom is 89.5 days.

Calculus is the obvious one, as integration and differentiation are used in a lot of other classes.

Linear Algebra and basic matrix operations. Pretty much every field in mathematics uses this as a base.

Most universities have a course on basic set theory and proof structures. It sets you up for the kind of thinking that is considered "real math".

Analysis would be the next step, the theorems are simple and intuitive but learning to prove them will develop your skills greatly.

Also, obligatory link to Khan Academy

You do this by web scraping. There are many tutorials online for web scraping. Personally, I have used python to do this but this can be done using javascript and in many other languages. Here is a tutorial in python: Web Scraping Tutorial

I hope this helps!

If it makes you feel any better: it comes from the Latin "semen", meaning seed (source ). The English word semen comes from the same Latin root, but seminal does not derive from the English word.

Other derivatives of the Latin "semen" are "seminary" and "disseminate". I don't think any of these words should be discriminated against just because of their more uncouth cousin.

Not to be too nitpicky, but I think you're actually wrong about most of this.

I can't find a modern source, but here is an article that surveys implementations of division in floating point hardware as of a couple decades ago. While there is a minority that does division by successive approximation (e.g., Newton-Raphson or Goldschmidt), the most common algorithm is SRT, which is a subtractive technique that is actually best viewed as just long division, with a few tricks to save some computation.

More importantly, regardless of the implementation, the IEEE floating point standard requires that results of division be exact to the precision of the representation. So it's not the algorithm for division that is responsible for any error here, but rather the limited space to store the answer. It's not at all important to be aware of which division algorithm your CPU uses.

Note that your example of poor floating point precision involves subtraction, which is not surprising. Subtraction often gives differences whose scale is vastly different from the inputs, and this is THE dominant reason for numerical instability. On the other hand, multiplication and division are very safe, because there's no added inaccuracy at all coming from working with numbers even of very difference scales.

Norwood, Rick; Poole, George (2003), &quot;An improved upper bound for Leo Moser's worm problem&quot;, Discrete and Computational Geometry 29 (3): 409–417, doi:10.1007/s00454-002-0774-3, MR 1961007.

Those numbers were either a) calculated to give you some seudo-random in your calculations, or b) derived from some process like geiger counters to make them truly random.

If you need something TRULY random, nowadays you can go to random.org, and get you some. Pick some lottery tickets while you are at it.

Couldn't find an exact match, but it looks to be about $40 I mean it's print from the 1800s, not exactly an original manuscript.

This type of question is exactly what projecteuler.net is all about. I would suggest you check it out as it challenges you to learn programming and math. Python is a great and easy language to learn for a site like this.

Sadly that book is already fading in the collective memory. As I recall, he wrote a later book called Silicon Snake Oil. If he thought it was bad then, I could imagine his opinion now.

You are incorrect about Riemann's numerical work. He did find at least the first nontrivial zero along the critical line, in unpublished notes found after his death. See https://books.google.com/books?id=kBsHBgAAQBAJ&amp;pg=PA323&amp;lpg=PA323&amp;dq=riemann%27s+work+nontrivial+zero+14.13&amp;source=bl&amp;ots=d81xzyOUfo&amp;sig=Xe-N4gMQU5XiC-OiPQVf1fxpQSs&amp;hl=en&amp;sa=X&amp;ei=6QxIVbPEOY2syASUtYHQBg&amp;ved=0....

Check out Overleaf. It's a free online LaTex editor with great community-made note-taking templates. Since it runs in the browser it works on any OS.

Hopefully someone else has recommendations on stuff that runs offline.

Aperiodic bubble formation from a submerged orifice

Why are there so many papers, theoretical and practical, on this subject?

Not a mathematician, but... :-)

It all started quite innocently. One of the first things I built during my PhD was a clock synchronization system. Our measurements showed some behaviour that we could not explain. So, in order to study it better, I built a simulation system. For the simulation, I needed to generate noise that fit real systems as closely as possible, including flicker noise. It turned out, that this "solved problem" was harder than I expected it to be, and wasn't solved at all. In order to get a better simulation, i tried to understand the mathematical description and soon found out that neither description we had was complete, nor that, which we EE considered to be true, was actually true. So started my journey to learn measure theory, fractional calculus and stochastic calculus (not necessarily in that order).

This is quite similar to a project my friends and I did over the past few months. Using the digits 1-5 exactly, once, we tried to make as many integers between 0 and 1000 as we could. We got all but 50 numbers before becoming bored. If anyone cares, our work can be found here.

The website abebooks sells the "eastern" or "asian" editions of textbooks at a much lower price. Basically all textbooks go for like $20 plus $20 shipping.

The textbooks contain the same material but are printed softcover on sometimes lower quality paper. In my experience they have been of generally good quality.

I've always ever used GraphViz for my graph-drawing needs. It's got several different programs that do layout in different ways, and I usually find one that works for my dataset. Input is in a pretty simple text file.

I do this kind of things more as a system administrator to display network diagrams and the such, and it's very convenient that a lot of tools out there will output data in the GraphViz text format.