# Math: Basics

## General Tips

### Get Gratzer's book "Math into LaTeX".

This book is a must for anyone doing a significant amount of mathematical typesetting. It is the only LaTeX book that fully covers the enhancements provided by the amslatex package. If you buy any book at all, get this one. If you don't want to spend the money on the book, download a pdf file of the first chapter. (This is based on an older edition, but still useful.)

### Use the amslatex packages.

Amslatex is a collection of packages that greatly enhance the mathematical capabilities of Latex and have become indispensible for typesetting documents with significant mathematical content. The principal packages are amsmath and amsthm. To make these packages available, add "\usepackage{amsmath, amsthm}" near the beginning of the document, right after "\documentclass{...}". (With the "amsart" documentclass, these are automatically loaded, but there is no harm in leaving the \usepackage{...} instruction in.) Most of the tips below take advantage of the features provided by these packages and assume that you have loaded the packages, as shown above.

### Use the math mode of TeX correctly: enclose all mathematical material in dollar signs (or equivalent environments), while leaving any nonmathematical material outside dollar sign pairs.

This might seem like a an obvious rule, but it is commonly violated. Here are some examples:
• Enclose variables and numbers embedded in regular text within dollar signs. For example, in a phrase like "Let F be a field" "F" represents a mathematical quantity and thus should be enclosed in dollar signs: "Let $F$ be a field." The same goes for numbers, if used in a mathematical context: Instead of "In dimension 3 we have ..." use "In dimension $3$ we have ...". (However, in textual contexts such as "Section 3" or "Theorem 1", the numbers should be left in text mode. The same goes for item labels, e.g. "by part (i) of Theorem 1", etc.)
• Leave punctuation signs outside (inline) mathematical expressions. A surprisingly common mistake is to include punctuation signs within the dollar signs delimiting a math formula. Punctuation signs are not part of the formula (they belong to the surrounding text), and therefore should not be enclosed within the dollar signs. For example, in the phrase "let $f(x)= cx$, where c is a constant", the comma is not part of the math expression and therefore should be outside the dollar sign pair. It would be wrong to set this as "let $f(x)= cx,$ where c is a constant".
• Use mode-specific font commands. Most font changing commands come in two versions, one for ordinary text (e.g., \textbf{...}), and the other for mathematical material (e.g. \mathbf{...}). Use the version appropriate for the mode, e.g.: " a \textbf{field} is ...", "... a vector $\mathbf{v}$ ...".
• Enclose text material inside displays in \text{...}. \text{...} causes the expression enclosed in braces to be typeset in text mode. This is useful in displayed formulas that involve some textual material. For example, in the expression "f(x)= \sin x and g(x)=\cos x ", the word "and" is ordinary text and thus should be typeset in text mode: "$f(x)=\sin x \quad \text{and}\quad g(x)=\cos x$". (Note here that the text is separated from the surrounding formulas by a "\quad" spacing command.)
• Don't italicize words by placing them inside $...$. The letters do come out italicized, but the spacing looks awful, since it is optimized for mathmode and the letters will be typeset as if they were mathematical variables, multiplied together.
• Use \operatorname for words acting as mathematical operators. For example, the correct way to typeset the phrase "Let rank A denote the rank of a matrix A" is "Let $\operatorname{rank} A$ denote the rank of a matrix $A$". The second occurrence of "rank" is part of the ordinary English text and thus should be left in text mode. By contrast, in the first occurrence ("... rank A ...") "rank" is mathematical operator that should be enclosed within the dollar sign pair. The "\operatorname{...}" command (see below) ensures that it gets typeset with the spacing and fonts appropriate for a "log like" operator.

### Use spacing commands in math mode sparingly; if you do need explicit spacing, use standard spacing commands such as "\quad", "\qquad", rather than multiple instances of single blanks ("\ \ \ "), or ties ("~~~~").

While there are spacing commands like "\," "\!" available to finetune the spacing in math mode, TeX usually gets the spacing right on its own, so those commands should be used very sparingly. In my experience, the vast majority of authors who use such manual spacing commands do so inappropriately, sometimes to compensate for other coding errors (e.g., not using the "\operatorname{...}" construct for a math operator). This usually results in a typeset output that looks worse than the output that TeX would have generated without such forced spacing. If in doubt, trust TeX on getting the spacing right.

One situation where explicit spacing commands are needed is in displayed formulas, e.g., to offset an expression such as "(i=1,2,...,n)" or "for i=1,2,...,n" from a formula, or in multiline displays to cause continuation lines to be shifted a bit to the right. In those instances I would use "\quad" or "\qquad" (the latter only in multiline displays) to obtain the appropriate amount of space. I recommend against using multiple explicit blanks (e.g., "\ \ \ " or "~~~~") since that makes for ugly looking and hard to read and maintain code, and haphazard spacing. By sticking to standard units (quads) in the spacing, one achieves a more uniform look.

### Keep in mind that in math mode TeX ignores all spaces (except for a blank line). Thus, you can break up complex formulas by inserting blank spaces and linebreaks at appropriate places in the code, in order to improve the readability of the code.

This is especially useful in complex, multiline displays. I often see such displays set as a single, very long line. However, this makes the editing of the displays a tricky and potentially risky undertaking (since editors typically operate in line mode, a single incorrect editing command may mess up the entire line), and it makes hard to track down errors and spot such things as forgotten parantheses.

Thus, I recommend breaking up long formulas into shorter "chunks", separated by (single) linebreaks. This has no effect on the output since TeX ignores spaces (including linebreaks) in math mode, but it greatly improves the readability and maintainability of the code. For example, a fraction set with "\frac{....}{....}", with complex expressions in the numerator and denominators, becomes more readable if the second pair of braces is placed on a separate line by itself.

The only exception to the spacing rule for math mode is a blank line (more precisely, two consecutive linebreaks, or two linebreaks separated only by other spacing commands). This is not allowed in math mode and will cause an error message.

### Avoid forcing displaystyle with \displaystyle or \limits for inline math material.

TeX typesets fractions, sums, integrals, and similar "large" expressions, differently depending on whether they occur "inline" (i.e., embedded in a paragraph of regular text), or in a displayed formula. For example, in sums occurring inline the summation limits are set as subscripts or superscripts to the summation symbol, while for sums in displayed equations these limits appear above and below the summation symbol. In fractions set inline the numerator and denominator are reduced, while in displayed fractions numerator and denominator appear in normal size. These and similar typesetting conventions ensure that mathematical expressions set inline do not protrude too much into the surrounding text or consume an excessive amount of vertical space.

One could override this default behavior of TeX and force display style on inline math material with the \displaystyle command, or, in the case of sums, use the \limits command to force the summation limits to appear above and below the summation symbol. However, this is almost always a bad idea, as the resulting output would look very poor. If the expression is complex enough (e.g., one involving multiple levels of subscripts or fractions) that it would become too small when set in inline style, it should probably be set as a displayed equation, or rewritten, e.g., using slashed fractions or other notational devices (e.g., rewriting a fraction using negative exponents).

### Set lengthy formulas or "tall" mathematical expressions as displays, rather than inline.

Complex mathematical material usually does not look good inline. Moreover, even though TeX can break lines in inline math material, there are usually far too few good spots for linebreaks; as a result, one often has to deal with overfull or underfull boxes, and poor linebreaks. Furthermore, trying to fix such linebreaks may introduce additional bad linebreaks or overfull/underfull boxes further down in the paragraph.

These problems can be avoided by setting larger, more complex expressions as displays. What should be displayed depends, of course, on the context, but as a general rule, if an equation takes up more than half a line when set inline, I would consider using a display for it.

### Use "slashed fraction" notation for fractions set inline, occurring in a subscript or superscript context.

Fractions set in the standard "stacked fraction" notation do not look good when they occur in inline math mode, or in a context (e.g., a superscript, subscript, summation or integration limit) where the size of a fraction is reduced. In almost all of those situations, it is usually best to rewrite the fraction as "slashed fraction": For example, in inline mode, replace "\frac{1}{2}" by "1/2", and "\frac{q}{1-p}" by "q/(1-p)". In a display, replace "\sum_{\frac{x}{y}\le n \le x}f(n)^{\frac{1}{3}}" by "\sum_{x/y\le n\le x}f(n)^{1/3}".

### Use "\operatorname" to define new "log like" math operators.

For example, "\newcommand{\Ext}{\operatorname{Ext}}" causes "\Ext" to behave much like "\sin" or "\log". This ensures that the spacing and fonts come out right. This is preferable to manual constructs with "\mbox", "\text", etc.

## Miscellaneous tips.

### Use "\substack", not "\atop", for multiline summation conditions.

The \atop command is a relic from plain TeX and has been rendered obsolete by the \substack construct provided by the amsmath package. Using \atop in a document with the amsmath package loaded generates a warning message.

### Use "\mid", not "|", for the vertical bar in the set notation.

The proper way to code the vertical bar in a set "{ ... | ....}" is with "\mid", rather than the vertical bar character ("|"). This generates a vertical bar, just the bar character, but it also adds an appropriate amount of space to the left and the right of the bar symbol.

### Use "\Vert" (or "\|"), not "||", for a double bar indicating a norm.

With two separate bar symbols, the bars would be spaced out too much. The "\Vert" macro (or equivalently, "\|") generates a "double bar" with the correct amount of spacing for a norm.

### Use "\dots", not "...", for ellipsis in math mode.

Three single periods ("...") are appropriate for an ellipsis in text mode, but not in a mathematical context, such as a "k=1,2,...,n", or "1 + 2 + ... + n". In the latter cases, the "\dots" command is appropriate. It generates not only the right amount of spacing between the dots, but it also places the dots at the appropriate level, namely on the baseline in the first context, and centered in the second context. Instead of "\dots" one could have used the more explicit commands "\ldots" (lower dots) in the first case, and "\cdots" (centered dots) in the second. However, using "\dots" has the advantage that one doesn't have to worry about the placement of the dots since the TeX gets it right in almost all cases. The only exception is the case of terminating dots that should be centered (e.g., in "f(1)+f(2)+..."). In this case, an explicit "\cdots" is needed.

### Use "\binom" for binomial coefficients.

A surprisingly common mistake, even by otherwise TeX-literate authors, is to generate binomial coefficients in a cumbersome manner, e.g., using a matrix or array environment (which generate inferior output), or using the "\choose" construct (a relic from plain TeX, which generates a rather uninformative warning message when used in a document with the amsmath package loaded.) The correct way to typeset binomial coefficients is with the "\binom{...}{...}" macro, which has the same syntax as the fraction macro "\frac".