## Writing Mathematics – Part II

Learn to Communicate Mathematically.

Read, write, and orally communicate mathematical concepts.

A Few Basic Rules

No significance should be attached to the order in which these basic rules are presented.

“The fact that some mathematics conventions have been universally adopted around the world suggests that they accomplish something important.” [Maurer]

For that reason it is unwise for a novice to vary from standard conventions, even though they are not absolute rules.

• Write with a pencil so that errors may be erased and replaced with correct work.
• Use complete sentences, correct grammar, and correct spelling.
• Symbols (like the + symbol) that have a specific mathematical meaning are reserved for mathematical use.
• Do not submit your first draft.
• Many mathematical adjectives and nouns have precise mathematical meanings,
• Emulate the writing style found in the textbook and/or lectures.
•  Start each sentence with a word, not a mathematical symbol.
• An English synonym will not serve as a replacement.  For example, “element” and “part” are not interchangeable when referring to an element of a set.
• Honor the equal sign.
• Avoid the use of imprecise terms.
• Two mathematical expressions or formulas in a sentence should be separated by more than just a space or by punctuation; use at least one word.
• Words have meanings: be aware of them. For example, an equation has an equal sign in it.
• Do not use abbreviations.
• Do not end a line with an equal sign or an inequality sign.
• Insure that every statement is mathematically correct.
• Strive for a good balance between words and symbols.
• Use different letters for different things.
• Remember mathematics is case sensitive.
• Define any terms or variables which you use.
• Once a variable has been assigned a meaning, do not re-use it with a different meaning in the same context.
• There is a distinction between the definite article (“the”) and the indefinite articles (“a” and “an”).
• Do not start a sentence with a formula.

Examples of Incorrect Mathematics Writing

• It is incorrect to speak or write of the quotient of a and b. When speaking or writing about division, statements must make it clear which is the divisor and which is the dividend.
• It is incorrect to speak or write of the difference of a and b. When speaking or writing about subtraction, statements must make it clear which is the subtrahend  and which is the minuend.
• It is incorrect to connect several different equations with equal signs, where the intermediate equal signs are intended to convey “equivalent to”.  For example, x = y – 3 = x + 3 = y is very confusing and altogether wrong.
• It is incorrect to speak of “moving a number or variable”; there is no mathematical operation called “move”.
• Equations can be equivalent but cannot be equal.
• Writing “solve” when the action is “compute” or “evaluate”.
• Writing f = x + 1 instead of  f(x) = x + 1.
• Writing  n = even = 2n instead of “If n is even, then n = 2k for some k”.
• Writing n2 = 16 = n = ±4 instead of n2 = 16 implies n = ±4
• Writing k = k+ 1 instead of “replace k by k + 1”.
• Writing (2, 3, 8) instead of {2, 3, 8}.
• Writing  $x\subset A$ instead of $x\in A$.
• Writing “length + area” instead of “length and area”.
• Writing a decimal approximation instead of an irrational number.
• Confusing the words equation, expression, and function.
• An expression is an algebraic combination of terms containing no verb.
• An equation is a mathematical statement which contains an equal sign.
• A function consists of a domain, range, and rule.
• Confusing the terms polynomial, polynomial equation, and polynomial function.

Common Syntax Errors to Avoid

• It is incorrect to write 3 + – 4.                   Correct syntax is 3 + (- 4)
• It is incorrect to write 3 ¸- 4.                    Correct syntax is 3 ¸ (- 4)
• It is incorrect to write 3 – – 4.                    Correct syntax is 3 – (- 4)

It is incorrect to write two operation symbols next to each other.

Never Violate Long-Standing Word Usage

“The fact that some mathematics conventions have been universally adopted around the world suggests that they accomplish something important.” [Maurer]

• It is correct to write: divide both sides of the equation by 10
• It is incorrect to write: divide 10 to both sides of the equation
• It is incorrect to write: divide 10 by both sides of the equation
• It is correct to write: subtract 5x from both sides of the equation
• It is incorrect to write: subtract 5x to both sides of the equation
• It is incorrect to write: subtract both sides by 5x
• It is incorrect to write: minus 5x from both sides of the equation
• It is incorrect to write: – 5x from both sides of the equation

References:

[Lee] “A Guide to Writing Mathematics”

http://www.cs.ucdavis.edu/~amenta/w10/writingman.pdf

[Lee] “A Mathematical Writing Checklist”

http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/writingcheck.pdf

http://www.swarthmore.edu/documents/WRITE_PRIMUS.pdf

[Berry] “Writing Mathematics”

http://agora.cs.wcu.edu/~jlawson/teaching/writing/writing.pdf

[Grayson] “Course Description for Math 248 at The University of Illinois”

http://www.math.uiuc.edu/~dan/Courses/2003/Fall/248/

[McCain Library – Agnes Scott College] “Writing in Math is Integral”

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### 3 Responses to “Writing Mathematics – Part II”

1. Bob Chaffer Says:

I think that items one and four in the rules list are a bit out of date. Word processing and document processing software accomplish the purpose of the using a pencil advice and the notion of “first draft” is somewhat more obscure. The point is to re-read and improve the product.

Writing is an easier task in this era. I recall that, as a student, I found the amount of time and labor required to recopy a long document with a few corrections was a disincentive to creating a second draft (although that was directed more often in non-mathematics classes.)

2. drdelmath Says:

Bob I would agree completely with you if we were not talking about mathematics. I have yet to encounter a student who used a computer to write mathematics homework.

As you observed the idea of writing multiple copies of a long document is not pleasant, but I know no way of avoiding it if you do your work in ink. It is not acceptable to submit a disorganized paper with various items scratched out. That kind of work is not acceptable outside the classroom and should not be acceptable in the mathematics classroom.

For you and I writing in this era is easier, but for the majority of students writing mathematics is no different today than it was 50 years ago.