LaTeX tips: Displayed Math

Another great reference page for math display tips:

http://www.math.uiuc.edu/~hildebr/tex/displays.html


Note: Most of the tips below require the amslatex macros. These are automatically loaded if you use the "amsart" documentclass, but if you use other documentclasses, such as "article", put the following in the preamble, after "\documentclass{...}": " \usepackage{amsmath, amsthm, amssymb} ". For more about the amslatex macros, and as a general reference for the tips below, see Gratzer's "Math into LaTeX".

Single line displays

• The double dollar symbol ($$). In Plaintex and Amstex displayed math was set by enclosing it between a pair of double dollar signs. In LaTeX the double dollar sign does not exist as a valid command, and most LaTeX books don't even mention it, since it is not supposed to be used. However, the dirty secret is that it works just fine in most situations, and many authors use it with impunity. Nonetheless, there are a few situations where it causes problems, and it may not work under future versions of tex. If you come from a Tex/Amstex background and are in the habit of typing $$'s, try to gradually switch over to using the backslash/bracket pair. If you are new to Tex/LaTeX, learn it right from the start (i.e., use the backslash/bracket method)

• The backslash/bracket pair, \ [ and \ ]. (without the space; I am using a space since MathJax picks up this as a LaTeX formula and until I can figure out how to block it when needed this is how I will work around it) Use this for all unnumbered displays. It's a good habit to put each of these pairs on a line by itself. Though Tex doesn't care, this makes displays stand out visually and makes editing and revising the tex file easier.

• \begin {equation}\label{...} ... \end {equation}. Use this if you want the equation to be automatically numbered. You can later reference the equation with "\eqref{...}", where the text inside the braces is the equation label. (Note that no parentheses are required with "\eqref".) For more on equation numbering, see below.

• \begin {equation*}... \tag{...}\end {equation*}. Same as the equation environment, except that automatic numbering is disabled, and the "\tag{...}" command generates an explicit equation number. For example, to get equation number (4.1a), use "\tag{4.1a}". (Again no parentheses are needed.)

Multiline displays

While single line displays are pretty straightforward to typeset, there are at least a dozen different constructs to generate multiline displays with the amslatex macros, and it's easy for a beginner to be overwhelmed by the multitude of options. Fortunately, you can get by knowning only two of these, namely the "align/align*" and "cases" environments. Unless you are a professional typesetter striving for 110% perfection, you don't need anything beyond that.

• \begin {align} ... \end {align} and \begin {align*} ... \end {align*}. Nearly all multiline displays can be typeset with the one of these two environments, which are derived from the amstex "\align ... \endalign" construct, and follow (essentially) the same syntax. The difference between "align" and "align*" is the same as between "equation" and "equation*": The starred versions don't automatically generate equation numbers. Use those versions, if you don't want equation numbers, or if you want to number equations manually.

• Equation numbering and labelling in align/align*. By default, every line gets numbered separately in the "align" environment. If you want just a single equation number for the entire display (which would be appropriate if the display consists of a chain of equalities or inequalities), pick a line in the middle of the display that is to get the number, and put "\notag" right before the linebreak symbol ("\\") on all other lines. On lines that are to be numbered, you can put the label command, "\label{...}", before the linebreak. With "align*", it works the other way: By default, none of the lines gets numbered, so to number a particular line you must put an explicit "\tag{...}" command at the end of that line, before the linebreak symbol.

• \begin {cases} ... \end {cases}. Aside from align, the only other multiline construct that you may need with any frequency is "cases". It works in much the same way as the amstex "\cases ... \endcases" construct. Note that the "cases" environment is a "subsidiary" math environement and has to be enclosed by another, top level, display math environent, such as equation or align/align*. Here is a typical example:

\begin{equation*}
|x|= 
\begin{cases} x & \text{if $x≥0$,}
\\
-x &\text{if $x\le 0$.}
\end{cases}
\end{equation*}

\begin{equation*} |x|= \begin{cases} x & \text{if $x≥0$,} \\ -x &\text{if $x\le 0$.} \end{cases} \end{equation*}

Note that the part after the alignment symbol usual contains some text and is best typeset using "\text{... }", and enclosing any math material within the braces in dollar signs.

• eqnarray/eqnarray* and array. Avoid these. Before the advent of amslatex, multiline displays and cases constructs had to be typeset with "eqnarray" and "array". However, the above (amslatex) environments "align/align*" and "cases" are easier too use, provide greater functionality, and generate better looking output. There is no reason whatsoever to use "eqnarray" instead of "align", and, except in very rare situations, "array" is unnecessary.

• Other amslatex display environements. Amslatex provides several other environments for multiline displays, such as "split, gather, gathered, multline, aligned, alignat," as well as starred version of most of these. These may do a marginally better job in some special situations, but the (mostly cosmetic) gain that these environments provide in those situations is not worth the effort of memorizing the various alternatives to the "align/align*" environments.

Writing tips

• How to number equations. By default, LaTeX will number equations consecutively, as (1), (2), etc., assuming you use the automatic equation numbering mechanism. If the paper is very short, or if there are only a few numbered equations, this is fine, but once the numbers get into the twenties and higher, a scheme that numbers equations by section, as in (1.1), (1.2), ..., (2.1), etc., is preferable. In fact, for the vast majority of journal articles, the latter is probably the best numbering scheme. To get equations numbered by section, just put the following into the preamble: " \numberwithin{equation}{section} ". For books, theses, or very long papers, an equation numbering scheme that is three levels deep (with numbers like (4.1.1), etc.) may be appropriate. To get this, just replace "section" above by "subsection", or the corresponding innermost level. The same mechanism works for other counters, e.g., theorem counters, instead of "equation".

• When to number equations. It is (generally) bad style to number every displayed equation indicriminately. This is what you get if you would use "\begin {equation} ... \end {equation}" throughout. The reason for numbering an equation is to enable subsequent references to it. Thus, in general only equations that are being referred to in the paper should be numbered. However, there are situations where one might number an equation even if it is not referred to in the paper. For instance, it might be appropriate to number all equations inside theorems, for consistency of style, and to make it easy to refer to these equations in subsequent work. Similarly, if a reference to an equation occurs right after that equation, a number may not be necessary; one could get by saying something like "Since f(x) is positive, the last linequality implies that ...".

• When to display a mathematical formula or equation. Formulas set inline are harder to spot, may cause problems with bad linebreaks (which one has to fix by inserting explicit linebreaks), and they may look poor, especially if they involve fractions, sums, integrals, or other "large" objects. On the other hand, one should not indiscriminately display every equation or formula. A judicious choice of what to display can make a a significant difference in the overall appearance and readability of a paper. Here are some guidelines. You should display formulas/equations in the following cases:

• A numbered equation. This is a no-brainer.

• A formula that is excessively long. If a formula takes up more than about half of a line of space, it would probably be good to display it. Formulas set inline should never be longer than one line (even though tex would accept multiline inline displays and probably do a reasonably good job in splitting the formula).

• A formula that takes up excessive vertical space. Formulas that involves complicated sums or integrals with multiple subscripts or superscripts, or stacked fractions, take up a lot of vertical space and cause subsequent lines to be moved down, if set as inline formulas. In these situations, it may be appriopriate to display the formula. Simple sums, integrals, or fractions, are usually not a problem; for example, an expression like "\sum_{k=1}^nk=n(n+1)/2" doesn't need to be displayed.

• A formula that you want to give special emphasis. Even if a formula would look fine inline from a typesetting point of view, it may be appropriate to display it for emphasis. This could be the case, for example, with an important definition, or a crucial condition/hypothesis in a theorem.

• How to break up long formulas. Breaking up overlong lines in displays can be very tricky, and requires a good knowledge of the underlying mathematics as well as a feel for good mathematical typesetting. For that reason, TeX doesn't break formulas, as it does with ordinary text. However, there are some general guidelines. Here are possible breaking points, in decreasing order of desirability:

• Right before an equal sign or equivalent (e.g., a "less than" sign). In this case, place the alignment symbol (ampersand) placed before the equal sign, i.e., " ... + x \\ &= ". The equal signs (or equivalent) should be aligned.

• Before a plus or minus sign. In that case, the continuation line should be shifted to the right by a \qquad. If the continuation line is very short, two \qquad's might be better.

• Between two large "chunks" that are multiplied. The chunks could be large parenthesized expression, sums or integrals. This should be used only as a last resort, and in that case the preceding line should end with an explicit multiplication symbol ("\times"), the continuation line should begin with "\times" symbol and be shifted as far to the right (by preceding it with a few "\qquad"'s after the alignment symbol).

• Punctuation in displays. Displayed mathematics should be properly punctuated. For example, if a displayed formula occurs at the end of a sentence, the display should be terminated by a period. No additional spacing (such as "\," or "\ ") is needed to separate the formula from the punctuation sign.

A good way to get the punctuation right is to replace the display by a place holder, such as "[DISPLAY]" and punctuate accordingly. Thus, for example, if the sentence surrounding the display is "Since f is monotonic, we have [DISPLAY], where ...", then no punctuation symbol is needed before the display, but the display should be terminated by a comma, since the phrase introduced by "where" following the display calls for a comma. A comma before the display would be wrong in this case (as would any other punctuation sign, such as a colon), since it would interrupt the continuity of the sentence.

Here are additional hints for some special situations:

• Punctuation in cases constructs. If a formula is set via a "cases" construct, each case ( = line) should be terminated with a comma; if the display occurs at the end of the sentence, the final comma should be changed to a period.

• Puntuation for "parenthetical" matter. Expressions indicating the range of validaty of a formula, or a limiting process can be set either in parenthesis with no punctuation separating the expression from the formula, or separated from the formula by a comma. The following examples illustrate the two options:

f(x) = \sqrt{1+x} \quad (x \ge -1)
f(x) = \sqrt{1+x}, \quad x \ge -1
f(x) \sim x^2 \quad (x\to\infty)
f(x) \sim x^2, \quad x\to\infty

It would be wrong to use both a comma and parentheses. (Note that, depending on the context, a punctuation sign at the end of the display may also be called for.)

Additional hints

• Spacing in displays. Usually it's best to leave the spacing up to TeX. However, if explicit horizontal spacing is needed (for example, to set an expression like "(n \to \infty)" apart from the rest of the display, or to separate two equations on the same line), "\quad" in most cases generates the right amount of space. Don't try to create spacing with a bunch of explicit spaces ("\ "); the spacing generated in this way is usually not optimal, and the explicit spaces will likely have to be removed (and possibly replaced by "\quad") when the paper is typeset at the publisher's end.

• Avoid blank lines before or after a display, unless you really want to start a new paragraph: It is tempting to surround displayed math material by blank lines in the source file, to make them stand out and easier to locate. However, this is usually wrong, since blank lines are interpreted as paragraph breaks, may generate some additional vertical spacing and cause the next line of text to be indented - something you usually don't want. If you want to set off displays in your source file, do so by inserting a line with comment symbols, such as

%%%%% equation 3.1 %%%%%%%%%%%%%%

before and/or after the display.

• Correcting underful vboxes and bad page breaks. By default, TeX does not break pages inside display environments such as align. If a paper has many multiline formulas, this may cause "underful vboxes", which can look very poor if the badness (reported by tex in the log file) is a few thousand. One way to remedy this is to put the command "\allowdisplaybreaks" in the preamble. However, this may cause some poor pagebreaks which may have to be fixed. A compromise solution is to use "\allowdisplaybreaks" locally, on the individual displays that cause problems: "{\allowdisplaybreaks \begin {align} ... \end {align} }." The best solution is to prevent this problem by avoiding excessively long displays (say, with five or more lines). It is often not hard to break up a very long display into two, for example by inserting a phrase like "By the Cauchy-Schwarz inequality, the last expression is at most" in the middle of the overlong display, followed by the remaining lines of the display.

• Multiline subscripts on sums or integrals. Use the "\substack{...}" command, which works much like the "\sb ... \endsb" pair in amstex. It is much easier to use, and produces better looking output than an array environment or a construct using "\atop" (derived from plain tex).