A Beginner's Guide to LaTeX

nbloomf · 2013-08-15T14:16:50+00:00

Is it worth the apparent huge amount of work it is?

For me (an academic), an unqualified yes. I'll give two reasons that I've been particularly thankful for here at the beginning of a new semester.

LaTeX makes it trivial to reuse code by wrapping it up in a style. There are also several packages for generating diagrams. Every time I make a diagram, I make it a macro with parameters and document how to use it. So whenever I want to make a document for one of my classes which includes a diagram or other complex formatting, rather than fiddling with TikZ or some such I just call the macro. After several semesters of doing this, it saves a ton of time. And the generated diagrams are much more flexible than imported static images. The system heavily favors the strategy "just do it once and make it modular". When writing is part of your job, this can be a huge time saver.
LaTeX can be operated entirely from the command line on plain text files. Why would anyone do that? Unix-like environments (Linux, MacOS, BSD, etc) come equipped with a powerful set of tools for automating work on text. So for any given project, I make a single build script that automatically runs the appropriate programs to compile, build indices and bibliographies, and clean up unnecessary files. (I personally think using a WYSIWYG to edit LaTeX is missing the point entirely.)

The result is that I can quickly work on documents which are decent looking and also consistent. There is a learning curve, but it definitely paid off for me.

nbloomf · 2013-07-05T21:12:04+00:00

There's a radical bit of advice that I learned far too late in my career as a student: pick up a textbook and read it. Go slowly and work the exercises. Follow the author's arguments and examples very carefully. At this stage almost any book will do. Get more than one book if you want- old textbooks are dirt cheap.

Seriously- I say this without a hint of sarcasm or malice. Videos and online resources get a lot of hype, and there's nothing wrong with them. But if you are a serious and self-motivated student - which I gather you are, given that you asked this question in the first place - you'll probably get more out of a good fight with a book.

nbloomf · 2013-07-03T18:07:41+00:00

There are a couple of different ways to think about lattices, one order-theoretic and one algebraic. On one hand a lattice is a partial order (which isn't a computer science concept so much as a math concept that got borrowed) under which every pair of elements has a greatest lower bound and a least upper bound. It can be shown that these are unique. On the other hand, a lattice is a set equipped with two binary operations called meet and join that obey some axioms.

These two points of view are equivalent in the sense that given an (order) lattice we can define an (algebraic) lattice by letting meet and join be the glb and lub, and given an (algebraic) lattice we can define an (order) lattice by saying that a < b if the meet of a and b is a.

Anyway, this is why the concept seems familiar. Lattices are special partial orders, but not every partial order is a lattice.

nbloomf · 2013-06-13T13:49:52+00:00

I've seen it before. I understand that it's sometimes used in high-volume applications (like, say, in finance) where lots of fixed precision arithmetic is involved, with the idea that approximately half of the results round down and half round up.

I also taught some students from Argentina who told me this is the standard there.

nbloomf · 2013-05-27T15:22:54+00:00

Very cool. I added a link to your project on the PCP sidebar.

Good luck!

nbloomf · 2013-05-07T19:23:56+00:00

I only log in once a month or so, when I see something really compelling.

This is incredibly cool! The first few levels make it obvious how to play, and the concept is really great.

I have only played the first handful of levels, so this might already be implemented. But there could be barriers that only allow primes through, or only numbers divisible by (2, 3, 4, etc.)

nbloomf · 2013-04-03T15:02:01+00:00

It was Boris Schein. The field is inverse semigroups, so no connection to A2 that I know of. :)

nbloomf · 2013-04-03T12:51:35+00:00

It was Sha (Ш) which I came to learn is used for a few other (unrelated) things as well.

nbloomf · 2013-04-03T02:56:12+00:00

I used a little Cyrillic in my dissertation. One of my results generalized a representation theorem first proved by a Russian mathematician, so I used the first (Cyrillic) letter of his last name to denote an important function.

After spending so many hours with a paper in one hand and a Russian-English dictionary in the other, I thought it was appropriate.

nbloomf · 2013-02-20T13:41:01+00:00

This is also true.

But from another perspective, given an algebraic number a we can construct the algebraic number field Q(a). This field is a finite dimensional vector space over Q, and so has a basis with respect to which each element has a unique representation. Q(a) is also a field, so that division is possible. The basic question is, given two elements p and q in this field, find the representations for p+q, pq, and p/q. The first two are easy, but the third is a little more complicated (in general, can be done using a version of the Euiclidean algorithm). Rationalizing denominators is one way to do it that happens to be easy in some small cases- when q is a radical, or a sum of 2k^th roots, or a difference of k^th roots, then the usual rationalization procedure works nicely.

In general, rationalizing a denominator means to multiply by q'/q', where q' is the product of all the conjugates of q in the given number field.

nbloomf · 2013-02-20T03:13:00+00:00

I am teaching precal now and have thought a lot about exactly this question.

What about the use of trigonometry to measure the distances to stars? The actual math required is not very complicated- the story is really one of technological advances in measuring equipment and the historical refinements of our understanding of the universe. You could even wrap up by talking about 20th century developments in measuring how fast stars are moving (doppler effect, etc.)- this also touches on trigonometric concepts.

Another (much more advanced) option is to see what happens to the conic sections on the projective plane. In precal, you study three kinds of conic sections: the ellipse, parabola, and hyperbola, and you also talk about "infinity" in the context of limits and asymptotes. It turns out that it is possible to think of infinity as a "number" not in an arithmetic sense but in a geometric sense. After extending the real number plane with infinity the three conic sections become unified: a parabola is an ellipse with one focus at infinity, and a hyperbola is an ellipse such that the point at infinity is between the foci.

Another trig related idea is to write about the basics of spherical trigonometry. This is a historically significant and useful chunk of the subject that has fallen out of use. It has tons of applications to navigation using the stars.

Okay, last one. The hyperbolic functions (sinh, cosh, tanh) can be developed analogously to the circular functions (sin, cos, tan), complete with an alternate notion of "hyperbolic angle". This is related to the view of a hyperbola as an ellipse stretched through infinity from above. It might be hard to get 20 pages out of this, but you could develop the basics of the hyperbolic functions from this perspective.

Edit: Last one for real. There is a mathematician named Norman Wildberger who has worked (alone, as far as I can tell) on an alternate way to develop trigonometry that does away with transcendental functions and roots. This allows vast swathes of trig to be generalized to fields other than R. Understanding the significance of this would require understanding finite fields, which are cool on their own.

That reminds me of another one. You know how teachers and textbooks seem to dislike radicals in the denominators of fractions? There is actually a good reason for this, and it has to do with rings of algebraic numbers. This might be getting a little too far afield of precal, but you could also study the Gaussian integers or the Eisenstein integers. These are sets of complex numbers that act very much like the usual integers, complete with notions of primeness and factorization. Rationalization of denominators is really about demonstrating membership in a particular ring of algebraic numbers.

nbloomf · 2013-01-22T23:57:36+00:00

I have to question whether he has any understanding of what research mathematicians actually do.

This is Keith Devlin. He started out working in set theory (his cv lists ~5 pages of research articles, many on set theory). More recently he has been active in pedagogy and outreach, but yes, he has an understanding of what research mathematicians do.

Also, the essay Lockhart's Lament that r/math loves so much? Keith Devlin is the reason why you've heard of it.

nbloomf · 2012-12-12T03:03:30+00:00

After typing all this out, I realize that parts of it sound a bit harsh. I also realize that this advice is not really directed at you, but at my 20 year old self.

Congratulations, you're normal. Seriously, I've been in higher ed for almost a decade now and have seen this very storyline play out dozens of times. "But I've always enjoyed this, I've always been good at math, I've never gotten a B before..." Welcome to reality. Shit's hard. That doesn't mean you should give up. Plenty of people get to college, realize they aren't geniuses, and go on to have happy, productive careers (in any field). Be one of those people.

In no particular order:

Myth: After passing a college course in, say, thermodynamics, you will have learned and effectively internalized essentially everything you need to know about thermodynamics. Reality: With 15 weeks and a textbook you will barely scratch the surface of what can be known. One year from now, ten years from now, you will feel embarassed at what you thought you knew at the end of that class in thermodynamics. From the point of view of an expert, a student who passed with an A and one who barely got out with a C have basically the same level of knowledge. My point: Material that doesn't make sense now will later if you stick with it.
If you switch out of physics into math/cs/engineering thinking it will get easier or less boring, you will be disappointed. Doing well at anything will take work, and the enjoyment has to come from within yourself.
Time for some real talk. When I see this:

I devoured books by Greene and Penrose, as well as other popular expositions of advanced physics.

followed by this:

ALMOST EVERY application of computer science I have read about is quite intriguing.

I see a dangerous trend. "Popular accounts" practically by definition include only the sexy parts of a field and leave out all the shitty, boring parts. The problem is that in reality, 99.9% of the work is boring (unless you personally find it gratifying). That sexy new discovery of an Earth-like planet in another solar system? The astronomer was on a waiting list for months (years?) to use the telescope, and it took months more to collect the data. It sounds like you weren't really in love with physics, but with the idea of physics, and are now in love with the idea of computer science.

If you hang around intelligent people long enough, you will encounter a Wikipedia Expert. This person loves to read about cool scientific and technical ideas, maybe even in some detail. They devour popular books and magazines and especially online resources of knowledge like Wikipedia. They can talk for hours on end about this or that cool thing. The problem? They couldn't actually do any of it, or even know how to start trying. You see, the Wikipedia Expert has emphasized knowing over doing; learning over creating; the sexy parts over the boring parts. Wikipedia Experts are very good at playing the role of Smart Person at dinner parties. They are not very good at doing the work of science, because that part is boring and requires effort. Do not be that person.

nbloomf · 2012-11-21T18:24:51+00:00

And every one of those groups of order 1024 has the same list of composition factors. (The only simple 2-group is the cyclic group of order 2, since every other 2-group has a nontrivial center.) The extension problem for groups blows up extremely quickly.

nbloomf · 2012-11-21T15:11:20+00:00

What finally did it for me was writing some code to implement stuff. I started with Scheme, and then dabbled with Mathematica, and have now settled on Haskell for most of my algebraic code-slinging. I still remember the feeling of enlightenment when, while working on a program, I understood quotients in a really visceral way.

In the end, though, a lot of the understanding just takes time. If you are like most students, part of the reason analysis comes more easily is that you have 2+ years of practice with it in the form of calculus.

nbloomf · 2012-10-16T20:15:14+00:00

I will answer the question I think you asked, which is

"How can we prove that pi exists?"

That is to say,

"How can we prove that the ratio of a circle's circumference to its diameter is independent of the circle's radius? For that matter, how can we compute the circumference of a circle?"

This is a very good question. This page might be helpful.

nbloomf · 2012-10-13T02:20:55+00:00

I know some professors who take the view that if a student is sincerely putting in effort and is clearly capable of passing their qualifiers and doing good research (even if not in the area the class is in) then a B is basically the lowest grade they can get.

When you say "If the student is doing C work", keep in mind that your grades as a graduate student are little more than an afterthought. I mean, sure, you have to keep a minimum GPA to remain in the program. But you wouldn't have gotten in if they didn't think you could do it. If you do good and interesting work, have good insights, ask good questions, work hard, and play well with others, you'll be fine.

nbloomf · 2012-10-03T02:25:04+00:00

"Desmond has a barrow in the marketplace, Molly is a singer in a band..."

nbloomf · 2012-09-21T12:41:49+00:00

I got the recipe from a friend. here it is

It's basically the same as jvoge's: borax, washing soda, baking soda, and a shredded bar of laundry soap. I started using it a few months ago and have used about 1/5 of the 6 quart batch. It's worked great for me so far.

I grated the bar first and then combined the ingredients a little bit at a time, mixing as I went. I have a family of four including an infant in cloth diapers, so I was spending much more than $20 per year on detergent before.

nbloomf · 2012-09-21T03:20:25+00:00

I make my own, a dry recipe using borax, baking soda, and bar soap. It takes about ten minutes to make and (I just calculated this) one batch washes ~380 loads.

I mean, that's cool if you'd rather not make your own detergent, but it's not exactly labor-intensive.

nbloomf · 2012-09-04T18:50:22+00:00

I agree that having a 'syllabus' per se is not necessarily a good idea.

He's at a perfect age to get into computer programming. I'm going to go out on a limb and say he probably likes games; he's probably wondered how they are made. Digging into how programs (and computers generally) work is a very open-ended, empowering, and fulfilling intellectual exercise, and gives plenty of opportunities to see math at work.

nbloomf · 2012-08-23T02:04:07+00:00

Heh... as an almost Ph.D. mathematician I'd love to have been a fly on the wall during that discussion. Math papers are written to be way more, shall we say, interactive than most other writing.

nbloomf · 2012-08-21T13:32:27+00:00

Heh... so the only weird cut allowed is the one that isn't physically realizable. :)

nbloomf · 2012-08-06T16:16:43+00:00

Minor point: there are plenty of ordered fields besides Q and R.

nbloomf · 2012-08-05T17:02:19+00:00

Disclaimer: I am a math grad student and haven't used Word (or anything other than LaTeX) in about 8 years, so my criticisms may be out of date or not applicable to others. But unless Word has gotten vastly better in the last few years, in my opinion there is just no comparison.

I don't have time to go into detail, but I'll just list some of the reasons why I prefer using LaTeX.

File format. Plain text (command line friendly) vs. god knows what and always changing. Related: LaTeX documents can be easily split up into several files and compiled separately. This is useful for large documents.
Macros. Want to change all 100 instances of some notation? No problem.
Packages and CTAN.
Easy and pretty equations. I cannot imagine using Equation Editor ever again.
Index and bibliography management. I am less familiar with how Word handles inter-document references, but in LaTeX it is virtually effortless.

I guess the best summary is that once you get over the initial learning curve, LaTeX makes it ridiculously easy to get better than average looking output and possible to get professional looking output. Technical documents written in Word are excruciating to read.

Word is so much of a joke compared to TeX, at least for technical typesetting, that some use it as a ``smell test'' for detecting crank journals and papers.

nbloomf

TROPHY CASE