WRITING MATHEMATICS
by Tamara J. Lakins
Taken from Tamara's book "The Tools of Mathematical Reasoning":
http://bookstore.ams.org/amstext-26
--------
Chances are the mathematics course you are currently taking will require far
more "writing in English" than any other math course you have previously
taken, which may be surprising to you. Whereas in previous courses you may
have written solutions to problems by simply writing line after line of
formulas, with no English words at all, now you are required to write complete
English sentences and paragraphs. This does not mean that no formulas will
appear, but rather, formulas should be incorporated grammatically into
sentences.
In any subject, whether it be history, biology, economics, or mathematics, our
job is to communicate what we know, and how we know it, to others. Learning
to write mathematics well requires a lot of practice and can be difficult at
the beginning. The following guidelines for writing mathematics point out
some of the issues you'll want to keep in mind:
(1) All proofs (or solutions involving some sort of explanation) should be
written in grammatically correct, complete English sentences.
(2) Begin a proof by assuming the relevant hypotheses. End each proof with a
sentence that reiterates what has been proved. For example, if you are trying
to prove that the product of two odd numbers is odd, then you should begin by
saying,
"Let m and n be odd integers."
The last line of the proof might be something like,
"Hence, mn is odd, as desired."
(3) Proofs (or solutions involving some sort of explanation) should include
enough detail for the reader to understand your reasoning. Do not assume that
the reader knows what you are talking about. Assume that your reader has the
same mathematical background as you but does not know the proof you are
writing.
(4) Be sure that what you have written is mathematically precise.
Mean what you write, and write what you mean.
(5) Use proper mathematical notation and terminology.
* All variables must be explicitly defined.
For example, if you are trying to prove that the product of two odd numbers
is odd, then you should begin by saying, "Let m and n be odd integers." If
you are then tempted to write "m = 2k+1", then you should explicitly
identify what k is and explain why it exists.
* Mathematical symbols should not be confused with English words.
For example, the symbol "=" should be used only in mathematical formulas and
computations, not as the verb "is" in a sentence.
(6) Proofs should explicitly make reference to any definitions or theorems used.
(7) Proofs should not contain any scratchwork or work done in the margins,
nor any large sections of "crossed out" work.
(8) Proofread all solutions for correctness and clarity.
Recopying your solutions is one useful way to accomplish this.