Commonly Accepted Principles of Writing Mathematics Well
1. General Remarks on Mathematical Content and Style
a. Always be precise and mathematically
correct, yet concise. For more details, suggestions, and
examples, see the guidelines for Mathematics students at MIT: Writing
a Math Phase Two Paper .
b. If using the first person, use "we"
instead of "I".
c. State a theorem (or
proposition, lemma, corollary) before proving it. State the
theorem concisely, completely, and correctly. (Hypotheses and
d. Do not begin a sentence with a
e. Introduce a theorem appropriately
before proving. If appropriate, give an example of the use a
theorem before giving the proof.
f. If a complicated proof is
involved, it may be helpful to give an outline of the method of the
proof prior to the body of the proof.
g. Use the active voice instead of passive
voice where possible.
h. Elaborate results with examples,
applications, and diagrams.
k. Make sure the level of the writing is
appropriate to the target audience.
i. For formal mathematics (such as
a paper for a course or a publication), do not give just lists
of definitions and theorems. Elaborate (as
suggested above above) with examples and exposition between the
definitions and theorems.
k. Do not just list a sequence of
equations or inequalities as the body of a proof. Guide the reader
with transitional phrases.
2. Guidelines for the Establishing the Content of
Expository Mathematical Text:
1) What is
the purpose of this theorem, result, algorithmic method, or definition?
2) What does
this theorem or definition really mean?
3) Why is this
result or definition important?
4) What is the
central idea in the proof of the theorem?
5) How are other
theorems or lemmas used to prove this theorem?
6) Is this
theorem a step towards a major result or is it the major result?
7) How does one
apply this theorem? What is an example of a mathematical object
satisfying (or not satisfying) the definition or result?
8) Why are
the hypotheses of this theorem so important? (How will the theorem fail
if the hypotheses are not satisfied.)
9) Is this
theorem the strongest possible result? Does it have any important
10) Does the
converse of this theorem hold? If not, why (provide an example
2. Guidelines for Appropriate Formatting of Mathematical Text.
a. Use standard and accurate notation. For
example, the symbol R is commonly understood to represent the set
of real numbers; C the complex numbers
(although in handwritten format one normally uses what's called
"blackboard bold" to write the symbols for the more common sets of numbers - I'll
demonstrate in class);
real variables; u, v vectors; a, b, c
real constants; k, n, m
integers; f, g, h functions, p, q
for polynomials. Note how symbols are used in your
mathematical texts and refer to them if you have doubts.
b. Theorems, definitions, examples are offset
from the rest of the text. Start a new paragraph and normally indent the
theorem, precede by a blank line. If numbering your results or naming
them, they should be bold faced. Be consistent in the style
c. Put a long or complicated expression (equations,
formulas, etc.) on a line by itself -- normally centered and offset by
blank lines. If you will need to refer to it later, you should use
a consistent numbering scheme. If your paper is divided into
numbered sections, it is helpful to incorporate these into your
d. Leave a space between symbols and adjacent
text. (Some suggest a double space.) Also use
appropriate, conventional formatting of mathematics ( such as
italicizing variables). Example:
Let z = 2 + 4i.
e. If you mark the ends of proofs use
accepted notation such as QED. Alternatively, appropriately
format the proof so the beginning and end of the proof are easily
identified. Whatever style you use, be consistent.
f. Do not use a symbol instead
of a verbal description unless it is necessary (and helpful) for later
g. Define every symbol before it is used.
h. Use different symbols for different
i. Do not use abbreviation symbols such as =,
>, in text (as opposed to within formulas).
j. Use mathematical terms correctly. Commonly
confused are functions, values of functions, equations, and expressions.
Whenever the least upper bound is equal to the maximum of the set ...
Whenever the least upper bound = the maximum of the set ..
k. Do not write equality ( = ) unless you mean
Important Note: One of the best ways to improve your style
of writing mathematics is to READ well written mathematics. Pay
attention to the wording and style of writing in your mathematics
texts. Note the customary formatting. Refer to these when
making decisions about the content and style of your mathematical
Web sites with help for writing mathematics:
thoughts on writing mathematics
How to Use the Equation Editor for Microsoft Word
a Math Phase Two Paper
Assignment Using the handout and the suggestions above,
look at the sample Word document which contains
a number of violations of accepted mathematical writing style. Find as
many as you can and mark appropriate corrections. Then edit the
document in Word incorporating your suggestions. Save under a new file
name and submit to me via email. You may work with a partner if
you wish, and submit your work jointly.