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How to write mathematics

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It’s not just mathematicians who need to “write math.” Mathematics is fundamental to all forms of science, engineering, and technical endeavors. As Albert Einstein said, “Pure mathematics is, in its way, the poetry of logical ideas.”

And like any form of written communication, math-oriented writing benefits tremendously from care and attention to quality. Well-written math concepts are more likely to be correct, more likely to help the author accurately articulate the concepts they are exploring, and – most importantly – more likely to educate and inform the reader effectively.

Yet incorporating math into a written work doesn’t simplify the task; it complicates it. Here are some best practices to improve math-related writing.

1: Follow the rules of grammar in math formulas and expressions.

Think of mathematical formulas as sentences, with different components corresponding to speech elements. For example, in the phrase 1 + 1 = 2, the equal symbol acts like a verb binding the subjects (one and one) to the object (two). When you incorporate mathematical formulas into a written work, by ensuring they act as complete sentences with a conventional grammatical structure, you will make the concept clearer and more understandable for readers. True, some math will defy typical construction, just as some writing does, but most formulas should adhere to a clear, complete, and expected construction. For example:

8x2z – 3y8x2z > 10y
This expression is subject only, without verb or object.A complete expression with subject – verb – object.

2: Use symbols correctly and appropriately

For instance, you might use the “=” symbol to mean many different things. In addition to “equal to,” the symbol could mean “the next step is.” However, the more meanings you try to attribute to a single symbol, the more confusion you risk creating in the reader. If there’s no specific symbol for what you want to say, split out the equations onto different lines; you might even write out the words rather than symbols, if necessary. Dr. Kevin Lee, a professor at mathematics at UC Davis and author of “A Guide to Writing Mathematics,” offers the following example of this principle:

32x – 2(3x) = -1 = (3x)2 – 2(3x) + 1 = 0 = (3x -1)2 = 0 = 3x = 1 = x = 0We want to solve for x in the equation
32x – 2(3x) = -1.
We can rewrite this equation in terms of 3x:
(3x)2 – 2(3x) + 1 = 0.
After factoring, this becomes
(3x -1)2 = 1
and it follows that 3x = 1, or x = 0.

3: Identify, define, and describe variables and assumptions.

Mathematics can be a highly conceptual and sometimes abstract field. Further, math proofs may require the author to make some assumptions and incorporate variables that may not be familiar to the reader. Always make sure you notate any assumptions you’ve made (e.g., did a rate of change remain steady? Was a function linear or continuous?). Similarly, define each variable as you introduce it, and be as specific as possible to avoid unnecessary questions. For example, Lee suggests:

Let D(t) be the distance at time t.Let D(t) be Agnes’s distance from the arena in miles at t hours after the riot began.

4: Follow the normal principles of good writing.

Math writers face two major writing challenges. The first is if the actual math in the piece is correct, clear, and understandable. The second is if the piece effectively and accurately communicates the underlying ideas. The latter point depends on the fundamentals of good writing. This is easily its own topic, and writers who must deal with mathematical subject matter might find their work improves when they brush up on their writing-specific skills. Additional training, practice, or research into writing principles can empower authors to make their points in a clear, meaningful, and impactful way to their intended readers.

How to write mathematics

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