FAQ

Some common questions that come up all the time in #math.

What does this symbol mean?

Mathematical symbols don't make sense by themselves. They need context. Take for example a simple "+" symbol. What does

$a+b$

mean? Here are some possible interpretations:

• The sum of two integers
• The sum of two equivalence classes modulo $n\in\mathbb{Z}$
• The indirect sum of vector spaces $a$ and $b$
• The direct sum of vector spaces $a$ and $b$ (why not, maybe whoever you're reading chose different symbols).
• The composition of $a$ and $b$ in an abelian group
• The composition of $a$ and $b$ in a non-abelian group (sure, why not, it's not set in stone that + should denote an abelian operation).
• The disjunction of propositions $a$ and $b$ (i.e. "OR").

and so on...

The point is, it doesn't make sense to walk into #math, spit out some random formula without context, and ask people to figure out what the symbols mean. You are the person who is most likely to know the context of what you're reading, so you are the person most likely to be able to figure out the symbols. Here are some hints on how to do it:

• Mathematics keeps building on previous results, so almost every mathematical exposition works in the same way. If you cannot figure out what the symbol means, look through the exposition you're reading through and search backwards through it until you find the author's definition of the symbol. You may have to search far, perhaps you skipped ahead to Chapter 5 of some book and the symbol was defined in the book's Preface, or perhaps it was just a few paragraphs before. Binary search can help you here, skim back to a section where the symbol doesn't appear frequently, skim forwards to a place where it does; the symbol is very likely to be defined near the point where it is first used.

• Some books actually have a table of notation as an appendix or at the beginning of the book. Go look there.

• The symbol actually has a common meaning in the subject area you're looking at, and the author didn't think it was necessary to define it again, because you would know what its usual meaning in this context was. If you don't, consult a more elementary exposition in the general subject area of whatever you are reading. If you cannot figure out the meaning of a common symbol in this subject area, you probably have other more elementary gaps anyways, so reading a more elementary exposition cannot hurt.

• If it's an online work (e.g. Wikipedia) there is very good chance that there are hyperlinks to more context in case the current article doesn't explain the meaning of the smbol.

• Sometimes Detexify can help you at least give the symbol its LaTeX name, and this can sometimes help. Beware, because sometimes, it won't because the LaTeX name is whimsical, e.g. $\cap$ and $\cup$ are "cap" and "cup" respectively, and this won't help you figure out they usually mean set intersection and union.

As a final remark, remember that symbols are the medium, not the message. If you are finding that reading the mathematical notation is the hardest part of reading some exposition you're looking at, you probably are missing more fundamental familiarity with the subject's concepts.

Here are some numbers, what's the formula?

Here are some numbers,

1 2 4 8 16

what's the formula? If you are thinking $2^x$, no, I have a surprise for you. The next number in the sequence is 31, which is the maximum number of regions into which a disc is cut by joining 5 points on its boundary.

Mathematicians usually think of this question this way. In other words, not too kindly. Why should there be a formula? Formula for what? For generating those numbers? It's always trivial to do polynomial interpolation, or to come up with any sort of crazy "rules" that you think the numbers should obey, but without context, it's all just meaningless faff.

Furthermore, people often phrase this question without realising that they are asking about interpolation and regression. Roughly speaking, interpolation is finding a function (hopefully simply expressed) that goes through all of the points. Regression is for when you only want to go approximately through the points (e.g. the line of best fit) and you have some reason to believe that there is a simple rule going through the points but it's muddled with some sort of random noise. Which one of bthose do you want? What makes you believe either one of those should work?

That being said, you can try the following ideas:

• If your sequence is made up of integers, it may be of general interest. Look it up in Online Encyclopedia of Integer Sequences to see if there's anything there.

• Look for more context to your problem. Where did you see these numbers, and why should we care? It's terribly boring to come up with a bunch of random-looking integers and then asking the channel to see if there should be some concise description that fits those numbers. Perhaps the context itself has the answer, not the numbers by themselves.

• If you decide you should do regression, then pick appropriate software for it. Octave and R are general purpose numerical computing environments that can do many kinds of regression. Most spreadsheet software can also do regression.

Don't be surprised if your question is met with some ridicule. It is quite possible that sequence simply has no simple description and mathematicians generally hate make grand extrapolations based on a few scant generally meaningless numbers.