Writing Tips

Usage of "Any" in Math

One must be careful when using the word “any” in mathematical writing because in certain contexts, “any” is ambiguous and can mean both “every” and “at least one”. Consider the following question:

Does $x^2 = 0$ hold for any $x \geq 0$?

There are two correct answers:

1. We can rewrite the question as, “does there exist at least one $x \geq 0$ such $x^2 = 0$?” The answer is “yes” because $x = 0$ satisfies $x^2 = 0$.
2. Alternatively, we can rewrite the question as, “does every $x \geq 0$ satisfy $x^2 = 0$?” The answer is “no.” If $x = 1$, then $1^2 \neq 0$.

Thus, do not use “any” when what you mean is “all.” Use “for all” or “for each” instead.

Avoid "Respectively"

In many cases, using the word “respectively” makes sentences hard to parse because it requires the reader to map each item in a list based on its position. For example, “Let $x$ and $y$ be the width and height of a rectangle, respectively,” can be more clearly stated, “Let $x$ be the width and $y$ be the height of a rectangle.” By placing the definition of symbol next to the symbol, you allow readers to glean the meaning of each withing needing to backtrack or remember the order that $x$ and $y$ were written.

Another example comes from an actual paper:

Consider a nonlinear system

$$\begin{aligned}
\dot{x} & =f(x) \\
y & =h(x)
\end{aligned}$$

where \(f\) and \(h\) are locally Lipschitz continuous and continuous, respectively.

This can be improved (and shortened) by removing “respectively”.

Consider a nonlinear system

$$\begin{aligned}
\dot{x} & =f(x) \\
y & =h(x)
\end{aligned}$$

where \(f\) is locally Lipschitz continuous and \(h\) is continuous.

Sometimes rewritten a sentence to remove “respectively” may lengthen it significantly or may introduce tiresome repetition. In such cases, use your judgement to decide whether the improved clarity is worth the increased verbosity.

The Best Notation is No Notation

As P.R. Halmos wrote, “the best notation is no notation.” If you can avoid introducing a new symbol, then the text is often more clear and the reader has one less thing to remember. A rule of thumb is to never use a letter only once. Consider the following definition:

A function $f$ is called a homeomorphism if it is continuous and has a continuous inverse.

The symbol $f$ serves no purpose since it is not used elsewhere in the definition. Thus, it can be omitted.

Make Your Document Color Accessible

When creating documents, make sure that graphs and uses of color are legible when viewed without color. Roughly 1 in 20 people are color-blind. And, regardless of a person’s ability to see color, it’s common to print papers in black and white. If your paper requires color to be understood, then you will lose readers!

For slides, color can play a more important role. You should make presentations accessible to color-blind people, but complete color-blindness is very rare. Most color-blind people are either unable to distinguish between red and green or between blue and yellow, but otherwise distinguish between other colors. Thus, if you pick your colors carefully, they can be seen by almost everybody.

See also:

• Online tool for checking color contrast
• Wikipedia’s color accessibility guide.

Don't Use Reuse Mathematical Symbols for Different Meanings

One of the surest sources of confusion in mathematical writing is to use one expression to mean multiple things. Authors often make this mistake with functions. To illustrate, consider a function $f$ from $\mathbb{R}$ to $\mathbb{R}$. The expressions “$f$” and “$f(x)$” have different meanings. The statement “$f(x)$ is monotonically decreasing” is meaningless because “$f(x)$” is a constant real number. Instead, write “$f$ is monotonically decreasing” or “$x \mapsto f(x)$ is monotonically decreasing”. The latter form is useful if you want to talk about the composition of functions without introducing a new symbol: “$x \mapsto f(1 + x^3)$ is monotonically decreasing”.

When is it OK to reuse a symbol?

A complete ban on reusing symbols in a mathematical text would cause you to quickly run out of symbols, especially in long texts. Similar to variables in a programming language, symbols have a particular scope.

The tightest scope is what I will call a bound symbol. A bound symbol is defined and used only in a single expression. The symbol $x$ is a bound variable in the following expressions: “$\forall x\ P(x)$”, “$\exists x\ P(x)$”, and “$x \mapsto f(x)$.” It is OK to immediately use the same bound symbol in multiple expressions, such as, “Consider the functions $x \mapsto f(x)$ and $x \mapsto g(x)$.” To aid the reader, however, each time a bound symbol is reused in a section of text, it should have the same type of meaning.

Symbols introduced in a proof should be restricted to that proof (many readers skip proofs on their first read of a text), so we can treat a proof as a symbol scope. If we introduce “$\varepsilon > 0$ and $\delta = 2\varepsilon$” in the proof of Theorem 1, then it is OK to define “$\delta \in (0, 1)$ and $\varepsilon = \sqrt{\delta}$” in the proof of Theorem 2.

Symbols introduced in definitions and theorems are not scoped in the same way.

Don't Start a Sentence With a Mathematical Expression

Don’t start a sentence with a mathematical expression (numerals, variables, or equations). Instead, start every sentence with a word. Consider, for example, that you have a set $A$. It’s confusing to say: “$A$ is …” because the symbol “$A$” looks like the word “A”. Similar confusion can ensue if a sentence ends with a mathematical expression and the next starts with one. For example, “The value of $X$ is $2$. $5$, $6$ and $7$ are numbers.” It looks, at first blush, that $X = 2.5$. When a clause ends with a mathematical expression followed by a comma, avoid placing another mathematical expression immediately after the comma.

There are several ways to avoid starting with a mathematical expression.

• Say the type of the object before the symbol: “The set $A$ is…,” “The expression $\sigma(2 + x)$ represents…”, “The inequality $a^2 \leq b^2$ holds for…”.
• Say the name of the object or short description: “The empty set $\emptyset$ is…”, “The union $A \cup B$…”.
• Reorder the sentence.

Correctly Format Text in Equations

When writing a mathematical equation that contains a word or other string of characters that is a unified block, place those characters inside a \text{...} macro.

To understand why this is important, suppose you have defined \(a,\) \(n,\) and \(t\) as real numbers. What does the expression \(tan(a)\) mean? Anybody experienced with trigonometry, would read this as the tangent of \(a,\) but because juxtaposition indicates multiplication, the expression is ambiguous—it could also be read as \(t\) times \(a\) times \(n\) times \(a.\) To avoid this ambiguity, write \(\tan(a)\) ($\tan(a)$ or $\operatorname{tan}(a)$).

The particular way that you format a string depends on the context.

• \operatorname{} to define functions like $\sin$ and $\cos$ that consist of several characters. Based on the context, \operatorname{} inserts the proper spacing before the text.
• \text{} to insert plain text that matches the formatting of the surrounding text environment, such as \text{if } ... \text{otherwise} ... in a cases environment.
• \textup{} to insert upright text. Useful for equations that appear in italicized environments, like theorems.
• \texttt{} to format programming variables and other code-like text.
• \textsc{} to format text using “small caps”.

If the string of characters appears repeatedly, define a macro to make it easier to typeset.

Initial Value Notation

When writing the initial value of a dynamic system, the initial value should be typeset using a subscript zero (“0”) not the letter oh (“O”) or (“o”). To illustrate, the initial value of a state \(x(t)\) should be written as \(x_0\) (“x_0”) not as \(x_O\) (“x_O”) or \(x_o\) (“x_o”).

Dashes for Parenthetical Statements

When emphasizing a parenthetical statement, an m-dash should be used. In LaTeX, an m-dash is typeset using three hyphens: ---. There should not be spaces around the m-dash.

Incorrect: This is a sentence --- but I stop to make a parenthetical statement.
\(\text{This is a sentence — but I stop to make a parenthetical statement.}\)

Correct: This is a sentence---but I stop to make a parenthetical statement.
\(\text{This is a sentence—but I stop to make a parenthetical statement.}\)

Be careful when writing an m-dash at the end of a line of LaTeX code because LaTeX will inject a space at each line break. You can suppress the added space by ending the line with “%” immediately after “---”:

% Incorrect
This is a sentence---
but I stop to make a parenthetical statement.

% Correct
This is a sentence---%
but I stop to make a parenthetical statement.

Presenting a Sequence of Manipulations to an Expression

When presenting a series of operations performed on an expression, it can be clearer to state the step-by-step recipe of how to perform the manipulations, than to show only the results of each step. The clearest, of course, would be to provide a step-by-step guide that shows the result from each step. Unfortunately, space constraints often prohibit such spacious exposition.

Suppose we want to solve \((a + x)/2 = 3\) for \(x.\) In the following table, the left shows a purely symbolic approach, which leaves the reader to fill in the gaps. The right column describes what is required but does not show the result of every step.

"Minimize" vs. "Min" in Optimization Problems

When writing an optimization problem, write “minimize” and “subject to” instead of “min” and “such that” or “s.t.”.

The problem with writing “$\operatorname{min} f(x)$” instead of “$\operatorname{minimize} f(x)$” is that the expression “$\operatorname{min} f(x)$” represents the minimal value of $f$ whereas “$\operatorname{minimize} f(x)$” is a statement of the goal of the optimization problem.

We use the following code when writing optimization problems. In the preamble, we define \minimize, \maximize, and \subjecto macros, as follows:

% Usage: \minimize{\x \in \reals}
\newcommand*{\minimize}[1]{\operatorname*{minimize}_{#1}\quad} 
% Usage: \maximize{\x \in \reals}
\newcommand*{\maximize}[1]{\operatorname*{maximize}_{#1}\quad} 
\newcommand{\subjectto}{\textup{subject to}\quad}

The starred version of the \operatorname macro causes subscripts to be placed below the text in display equations.

In the document body, we write an optimization problem as

\begin{align*}
    \minimize{x \in \reals^n} & f(x) \\
    \subjectto & Ax \leq 0 \\
               & Bx = 0.
\end{align*}

If space is a concern, you may abbreviate “subject to” as “subj. to” (\textup{subj.\ to}).

Use Nonbreaking Spaces

In LaTeX, the character ~ inserts a non-breaking space. As a rule of thumb, use ~ in the following situations:

• between a number and its unit of measure: 1000~miles, 183~billion
• before each citation (e.g., ...as was shown by Smith~\cite{smith_2020})
• before the last word of a paragraph to prevent it from being placed on a line by itself, especially if the word is short.

To prevent line breaks in the middle of an equation, place curly brackets “{...}” around the equation. For example ${a \in A}$ will always be displayed on one line, whereas ${a^2 + b^2} = {c^2 + d^2}$ may have a line break at the equal sign but nowhere else.

Puncuation in Equations

When writing mathematics, punctuation should be placed in equations at the points where punctuation would occur if the equation was converted to text, as though spoken.

Repeat Yourself

Despite our best efforts, it is inevitable that we, as authors, inadvertently write things that are ambiguous or otherwise difficult to parse. When saying something important and difficult to understand, find ways to repeat yourself without saying exactly the same thing twice. While we often try to be as concise as possible and thus avoid redundancy, a degree of redundancy can be a lifesaver for readers when they encounter a statement that is confusing. Similar to how redundancy is vital in safety-critical control systems to ensure robustness to failures, including redundancy in writing can offer guardrails that protect readers from misunderstanding.

Here are key principles to doing this well:

1. When repeating yourself, the repetition should convey the information in a completely different way. Strategies for this are presented below.
2. When using information that was stated long before, reiterate the fact (perhaps in an abbreviated version) along with a note of where it was first introduced. This can save readers from needing to flip through previous material.
3. Not everything needs to be repeated. Trust the intelligence of your readers. If a fact is absolutely clear with a single statement, then it does not need to be repeated. If it is obvious or common knowledge, then you may omit it completely.

Methods for Strategic Repetition

1. After introducing a concept, provide an example that illustrates it. Ideally, pick one or more examples of the concept along with a non-example that the readers might mistakenly expect be an example.
2. Before an equation, say in words what the equation says in math. For example, instead of writing, “We have that $\dot T = -\lambda T$,” we might write, “The rate of change of the temperature is $\dot T = -\lambda T$.” (This also allows you to avoid starting sentences with an equation without resorting to filler phrases like, “We have that…”.)
3. Choose notation that reinforces the meaning of the symbols. For example, using $r$ for radius or consistently using bold letters for vectors (e.g., $\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}$) are subtle ways to indicate to readers what their meanings are.
4. In computer code, include comments that say what the code does.
5. When space permits, include diagrams. “A picture is worth a thousand words.”
6. Before or after a technical theorem, write an informal description of its important aspects.
7. Simply including a proof or derivation after a mathematical statement serves to reiterate the statement by building it up piece-by-piece.
8. For statements that are composed of a mix of logical operators and quantifiers, it can sometimes be helpful to readers if you reiterate claims into logically equivalent statements. For example, “all $x$, $P(x)$ holds” can be rewritten as “there does not exist any $x$ such that $P(x)$.” Similarly, “If $P$, then $Q$” can be written as its contrapositive.

"So That" vs. "Such That"

Despite sounding similar, the phrases “so that” and “such that” have different meanings. Using the wrong phrase can cause mathematical statements to be incorrect. The phrase “so that” indicates motivation whereas “such that” imposes a constraint.

We may, for example, say, “Pick \(x \in (-5, 5)\) so that \(x^2 < 25\).” In this case, we are alerting the reader to the fact that it is important that \(x^2 < 25\) but not adding any new information. In other words, replacing the sentence with “Pick \(x < 5\)” does not change the meaning. If, instead, we wrote, “Pick \(x \in (-5, 5)\) such that \(x^2 < 25\)”, the mathematical meaning changes to “Pick \(x\) such that \(x \in (-5, 5)\) and \(x^2 < 25\).” This, of course, is equivalent, but may confuse readers.

The consequence of using “such that” when we mean “so that” is—at worst—moderate confusion, but the result of using “so that” instead of “such that” is logically incorrect statements. For example, suppose we write, “Pick \(x, y \in \mathbb{R}\) such that \(x + y > 0\).” This imposes a constraint on \(x\) and \(y\); namely, their sum must be positive. The sentence “Pick \(x, y \in \mathbb{R}\) so that \(x + y > 0\)” is mathematically incorrect because picking \(x, y \in \mathbb{R}\) does not always result in a positive sum.

Often, readers can figure out your intended meaning if you mix up the two phrases, but doing so forces them to do extra work to infer what you, as the author, should have made explicit.

Size of Spaces of a Period

When LaTeX sees a period followed by a space, “. “, it assumes that the period is the end of a sentence, so it makes the space wider. When using a period that is not at the end of a sentence, the following space should be typeset using “\ “ instead of a normal space “ “. This ensures size of the space after the dot is correct.

My name is P.\ Wintz

My name is P. Wintz

The difference is subtle and the actual spacing will depend on the settings for a particular document, but the difference helps readers discern whether a period ends a sentence.

Best Practices for Formatting Tables

• Eliminate unnecessary rules (lines) between columns and rows.
• Don’t repeat information. If some expression (such as units) occurs in every cell in a column or row, move it to the header.
• Don’t use a table if a plot would be better. If you have a large amount of information and readers are not interested in exact values, it is often better to display it graphically.
• Round aggressively. Readers rarely need more than one decimal point.
• Consider rearranging or transposing the table. If readers are likely to compare two quantities, place them in adjacent columns (it is slightly easier to compare elements between columns than between rows).
• Include a reference to the table in the text of the document and highlight important features.

Use Precise References to Locations in Documents

LaTeX is a powerful tool for cross-referencing.

Use it.

Instead of saying, “In a later section…”, “In the next section…”, or “Below…”, say, “In Section 4…”. This allows readers to quickly pinpoint the thing you are referencing. If you have TeX configured correctly (using hyperref), then references will automatically include a internal link to the given place in the document, allowing readers to immediately jump there.

This advice also applies when citing other works. If a citation is, say, an entire book, then the reader must spend time searching through it to find what you were talking about. When citing in TeX, the \cite macro has an optional argument for giving a locator. For example, to cite Theorem 3.4 in khalil_nonlinear_2014, use \cite[Theorem 3.4]{khalil_nonlinear_2014}.

Some examples

• “Results presented in the next section…” \(\implies\) “Lemma 3 in Section 4…”
• “In the above inequality…” \(\implies\) “In inequality (4)…”
• “This theorem is given by Kahlil in [4]” \(\implies\) “This theorem is given by Kahlil in [Theorem 3.4, 4]”.

Including Links

% Add support for mixing colors, such as "black!30!blue"
\usepackage{xcolor}

% Enable hyperlinks
\usepackage[]{hyperref}
\hypersetup{
    final,                   % Include links even if document is in 'draft' mode.
    hidelinks,               % Prevent boxes from being drawn around links in some editors.
    colorlinks=true, 
    linkcolor=black,         % TOC, links to labeled equations and environments
    urlcolor=black!30!blue,  % URLS including links in references  
    citecolor=black!30!blue, % In-line citations  
}