B Mathematical Writing

This appendix is designed to give you helpful hints for the writing required of the homework and the projects. You will find that mathematical writing is different than writing for literature, for general consumption, or perhaps other scientific disciplines. Pay careful attention to the conventions mentioned in this chapter when you write math.

A few words of advice:

Do all of the math first without worrying too much about the writing.
When you have your mathematical results you can start writing.
Write the introduction last since at that point you know what you’ve written.
You will spend more time creating well-crafted figures than any other part of a mathematical writing project. Expect the figures to take at least as long as the math, the writing, and the editing.

B.1 The Paper

Write your work in a formal paper that is typed and written at a college level using appropriate mathematical typesetting. This paper must be organized into sections, starting with a Summary or Abstract, followed by an Introduction, and ending with Conclusions and References. Each of these sections should begin with these headings in a large bold font (using the LaTeX \section and \subsection commands where appropriate). Within sections I would suggest using subheadings to further organize things and aid in clarity.

B.2 Figures and Tables

Figures and tables are a very important part of these projects. Never break tables or figures across pages. Each figure or table must fit completely onto one sheet of paper. If your table has too much information to fit onto one sheet, divide it into two separate tables. In addition to the figure, this sheet must contain the figure number, the figure title, and a brief caption; for example “Figure 2: A plot of heating oil price versus time from Model F1. We see that the effects of seasonal variation in price are dominated by random fluctuations.” In the text, refer to the figure/table by its number. For example in the text you might say “As we see in Figure 2, in model F1 the effects of seasonal variation in price are dominated by random fluctuations.” Every figure and table must be mentioned (by number) somewhere in the text of your paper. If you do not refer to it anywhere in the text, then you do not need it, and subsequently it will be ignored.

Think of figures and tables as containing the evidence that you are using to support the point you are trying to make with your paper. Always remember that the purpose of a figure or a table is to show a pattern, and when someone looks at the figure this pattern should be obvious. Figures should not be cluttered and confusing: They should make things very clear. Always label the horizontal and vertical axes of plots.

B.3 Writing Style

The real goal of mathematical writing is to take a complex and intricate subject and to explain it so simply and so plainly that the results are obvious for everyone. Your paper should demonstrate that you not only did the right calculations, but that you understand what you did and why your methods worked.

Write math using the pronoun ‘we’ instead of ‘I.’ For example: “First we calculate the sample mean.” This ‘we’ refers to you and the reader as you guide the reader through the work that you’ve done. The “we” gives your writing a guiding tone and should invite the reader to work along with you.

B.4 Tips For Writing Clear Math

B.4.1 Audience

The following suggestions will help you to submit properly written homework solutions, papers, projects, labs, and proofs. The goal of any writing is to clearly communicate ideas to another person. Remember that the other person may even be your future self. When you write for another person, you will need to include ideas that may be in your mind but omitted when you are writing a rough draft on scratch paper. If you keep your intended audience in mind, you will produce higher quality work. For a course in mathematics, the intended audience is usually your instructor, your classmates, or a student grader. This implies that your task is to show that you thoroughly understand your solution. Consequently, you should routinely include more details.

One rule of thumb must prevail throughout all mathematical writing:

When you read a mathematical solution out loud it needs to make sense as grammatically correct English writing. This includes reading all of the symbols with the proper language.

Don’t forget that mathematics is a language that is meant to be spoken and read just like works of literature!

B.4.2 How To Make Mathematics Readable – 10 Things To Do

When read aloud, the text and formulas must form complete English sentences. If you get lost, say aloud what you mean and write it down.
Every mathematical statement must be complete and meaningful. Avoid fragments.
If a statement is something you want to prove or something you assume temporarily, e.g., to discuss possible cases or to get a contradiction, say so clearly. Otherwise, anything you put down must be a true statement that follows from your up front assumptions.
Write what your plan is. It will also help you focus on what to do.
There must be sufficient detail to verify your argument. If you do not have the details, you have no way of knowing if what you wrote is correct or not. Keep the level of detail uniform.
If you are not sure, even slightly, about something, work out the details on the side with utmost honesty, going as deep as necessary. Decide later how much detail to include.
Do not write irrelevant things just to fill paper and show you know something. This includes avoiding writing about dead ends, failed attempts, or broken code.
Your argument should flow well. Make the reading easy. Logical and intuitive notation matters.
Keep in mind what the problem is and make sure you are not doing something else. Many problems are solved and proofs done simply by understanding what is what.
The state of mind when you are inventing a solution is completely different from the mode of work when you are writing the solution down and verifying it. Learn how to go back and forth between the two. The act of typing your solutions forces you to iterate over this process but remember that the process isn’t done until you’ve proofread what you typed.

B.4.3 Some Writing Tips

Use sentences:

The feature that best distinguishes between a properly written mathematical exposition and a piece of scratch paper is the use (or lack) of sentences. Properly written mathematics can be read in the same manner as properly written sentences in any other discipline. Sentences force a linear presentation of ideas. They provide the connections between the various mathematical expressions you use. This linearity will also keep you from handing in a page with randomly scattered computations with no connections. The sentences may contain both words and mathematical expressions. Keep in mind that the way your present your solution may be different than the way that you arrived at the solution. It is imperative that you work problems on scratch paper first before formally writing the solution.

The following extract illustrates these ideas.

Let $n$ be odd. Then Definition 3.10 indicates that there does not exist an integer, $k$, such that $n = 2k$. That is, $n$ is not divisible by $2$. The Quotient– Remainder theorem asserts that $n$ can be uniquely expressed in the form $n = 2q + r$ , where $r$ is an integer with $0 \le r < 2$. Thus, $r \in \{0, 1\}$. Since $n$ is not divisible by 2, the only admissible choice is $r = 1$. Thus, $n = 2q + 1$, with $q$ an integer.

Read out loud:

The sentences you write should read well out loud. This will help you to avoid some common mistakes. Avoid sentences like:

Suppose the graph has $n$ number of vertices.
The piggy bank contains $n$ amount of coins.

If you substitute an actual number for $n$ (such as 4 or 6) and read these out loud they will sound wrong (because they are wrong). The variable $n$ is already a numeric variable so it should be read just like an actual number. The correct versions are:

Suppose the graph has $n$ vertices.
(Read this as: “Suppose the graph has en vertices.”)
The piggy bank contains $n$ coins.

You should also avoid sentences like:

From the previous computation $x=5$ is true.

A better way to say this is:

From the previous computation we see that $x=5$.

When you read the equal sign as part of the sentence you realize that there is no reason to write “is true.”

$=$ is NOT a conjunction:

The mathematical symbol $=$ is an assertion that the expression on its left and the expression on its right are equal. Do not use it as a connection between steps in a series of calculations. Use words for this purpose. Here is an example that misuses the $=$ symbol when solving the equation $3x=6$: \[\text{Incorrect! } \qquad 3x = \underbrace{6 = \frac{3x}{3}}_{\text{false!}} = \frac{6}{3} = x = 2\] One proper way to write his is:
$3x=6$. Dividing both sides by $3$ leads to $\frac{3x}{3} = \frac{6}{3}$, which simplifies to $x=2$.

“$\implies$” means “implies”:

The double arrow “$\implies$” means that the statement on the left logically implies the statement on the right. This symbol is often misused in place of the “$=$” sign.

Do not merge steps:

Suppose you need to calculate the final price for a $20 item with 7% sales tax. One strategy is to first calculate the tax, then add the $20. Here is an incorrect way to write this.
\[\text{Incorrect! } \qquad 20 \cdot 0.07 \underbrace{=}_{false!} 1.4 + 20 \underbrace{=}_{false!} \$21.4.\] The main problem (besides the magically-appearing dollar sign at the end) is that $20 \cdot 0.07 \neq 1.4 + 20$. The writer has taken the result of the multiplication ($1.4$) and merged directly into the addition step, creating a lie (since $1.4\neq 21.4$). The calculations could be written as: \[\$20 \cdot 0.07 = \$1.40 \text{ so the total price is } \$1.40 + \$20 = \$21.40\]

Avoid ambiguity:

When in doubt, repeat a noun rather using unspecific words like “it” or “the.” For example, in the sentences

Let $G$ be a simple graph with $n \ge 2$ vertices that is not complete and let $G$ be its complement. Then it must contain at least one edge.

there is some ambiguity about whether “it” refers to $G$ or to the complement of $G$. The second sentence is better written as “Then G must contain at least one edge.”

Use Proper Notation:

There are many notational conventions in mathematics. You need to follow the accepted conventions when using notation. For example, A summation or integral symbol always needs something to act on. The expressions \[\sum_{i=1}^n \qquad \qquad \int_a^b\] by themselves are meaningless. The expressions \[\sum_{i=1}^n a_n \qquad \qquad \int_a^b f(x) dx\] have well-understood meanings.

As another example, \[\underbrace{\lim_{h \to 0} = \frac{2x+h}{2}}_{\text{incorrect!}} = \frac{2x}{2} = x\] is incorrect. It should be written \[\lim_{h \to 0} \frac{2x+h}{2} = \frac{2x}{2} = x\]

Parenthesis are important:

Parenthesis show the grouping of terms, and the omission of parenthesis can lead to much unneeded confusion. For example, \[x^2 + 5 \cdot x-3 \quad \text{ is very different than } \quad \left( x^2 + 5 \right) \cdot \left( x-3 \right).\] This is very important in differentiation and summation notation: \[\frac{d}{dx} \sin(x) + x^2 \quad \text{is not the same as} \quad \frac{d}{dx} \left( \sin(x) + x^2 \right)\] \[\sum_{k=1}^n 2k+3 \quad \text{is not the same as} \quad \sum_{k=1}^n \left( 2k+3 \right)\]

Label and reference equations:

When you need to refer to an equation later it is common practice to label the equation with a number and then to refer to this equation by that number. This avoids ambiguity and gives the reader a better chance at understanding what you’re writing. Furthermore, avoid using words like “below” and “above” since the reader doesn’t really know where to look. One implication to this style of referencing is that you should never reference an equation before you define it.
Incorrect:

In the equation below we consider the domain $x \in (-1,1)$ \[f(x) = \sum_{j=1}^\infty \frac{x^n}{n!}.\]

Correct:

Consider the summation \[\begin{aligned} f(x) = \sum_{j=1}^\infty \frac{x^n}{n!}.\end{aligned}\] In this equation we are assuming the domain $x \in (-1,1).$

“Timesing”:

The act of multiplication should not be called “timesing” as in “I can times 3 and 5 to get 15.” The correct version of this sentence is “I can multiply 3 and 5 to get 15.” The phrase “3 times 5 is 15,” on the other hand, is correct and is likely the root of the confusion. The mathematical operation being performed is not called “timesing.” It seems as if this is an unfortunate carry over from childhood when a child hears “3 times 5,” sees “$3 \times 5$,” and then incorrectly associates the symbol “$\times$” with the word multiply in the statement “I can multiply 3 and 5 to get 15.”

B.4.4 Mathematical Vocabulary

Function:: The word function can be used to refer just to the name of a function, such as “The function $s(t)$ gives the position of the particle as a function of time.” Or function can refer to both the function name and the rule that describes the function. For example, we could elaborate and say, “The function $s(t) = t2 - 3t$ gives the position of the particle as a function of time.” Notice that both times the word function is used twice, where the second usage is describing the mathematical nature of the relationship between time and position. (Remember that if position can be described as a function of time, then the position can be uniquely determined from the time.)
Equation:: To begin with, an equation must have an equal sign ($=$), but just having an equal sign isn’t enough to deserve the name equation. Generally, an equation is something that will be used to solve for a particular variable, and/or it expresses a relationship between variables. So you might say, “We solved the equation $x + y = 5$ for $x$ to find that $x = 5 - y$,” or you might say “The relationship between the variables can be expressed with the following equation: $xy = 2z$.”
Formula:: A formula might in fact be an equation or even a function, but generally the word formula is used when you are going to substitute numbers for some or all of the variables. For example, we might say, “The formula for the area of a circle is $A = \pi r^2$. Since $r = 2$ in this case, we find $A = \pi 2^2 = 4\pi$.” The bottom line: If you’re going to use algebra to solve for a variable, call it an equation. If you’re going to use it exactly as it is and just put in numbers for the variables, then call it a formula.
Definition:: A definition might be any of the above, but it is specifically being used to define a new term. For example, the definition of the derivative of a function $f$ at a point a is \[f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}.\] Now this does give us a formula to use to compute the derivative, but we prefer to call this particular formula a definition to highlight the fact that this is what we have chosen the word derivative to mean.
Expression:: The word expression is used when there isn’t an equal sign. You probably won’t need this word very often, but it is used like this: “The factorization of the expression $x^2 - x - 6$ is $(x - 3)(x + 2)$.”
Solve/Evaluate:: Equations are solved, whereas functions are evaluated. So you would say, “We solved the equation for $x$,” but you would say “We evaluated the function at $x = 5$ and found the function value to be $26$.” Avoid the words “plugged in,” such as “we plug 5 in for $x$,” when you actually mean that you are doing substitution.
Add Subtract vs Plus Minus:: The word subtract is used when discussing what needs to be done: “Subtract two from five to get three.” Add is used similarly: “Add two and five to get seven.” Minus is used when reading a mathematical equation or expression. For example, the equation $x - y = 5$ would be read as “$x$ minus $y$ is equal to five.” Plus is used similarly. So the equation $x + y = 5$ would be read as “$x$ plus $y$ is equal to five.” Some things we don’t say are “We plus 2 and 5 to get 7” or “We minus $x$ from both sides of the equation.”
Number/Amount:: The word number is used when referring to discrete items, such as “there were a large number of cougars,” or “there are a large number of books on my shelf.” The word amount is used when referring to something that might come in a pile, such as “that is a huge amount of sand!” or, “I only use a small amount of salt when I cook.”
Many/Much:: These words are used in much the same way as number and amount, with many in place of number and much in place of amount. For example, we might say, “There aren’t as many cougars here as before,” or “I don’t use as much salt as you do.”
Fewer/Less:: These are the diminutive analogues of many and much. So, “There are fewer cougars here than before,” or “You use less salt than I do.”

B.5 Code and Mathematical writing

Generally speaking you will need to write code to do some of the math required of whatever assignment you’re working on. It is very rare, however, that you would need to include the actual code in the written version of the paper. No one is going to read printed code, and you cannot reasonably expect the reader to have any way to execute it or to conveniently debug it. If you feel that you must include code then provide a permanent link to a shared document with the proper viewing privileges (e.g. a Google Colab document that is set to view only permissions).

The big takeaway: you will almost never include your code in your paper since the reader won’t read it and the code likely only makes your central mathematical points less clear to the reader. This is sometimes rather emotionally difficult since the bulk of your time will be spent writing your code, but the bulk of your writing will not be about the code – it should be about solving the problem at hand. Remember that well-crafted plots along with an explanation can capture all of the details of your code while simultaneously being very clear to the reader.

In the case that that you are writing about implementing an algorithm in a particular language, be sure that the code is written in a different font. In LaTeX consider using the verbatim environment to set your code apart from the paragraphs and to give a typewriter-style font that reminds the reader that they are reading code.