Mathematical Writing - Vivaldi Franco 2014
Style
Some Writing Tips
A good sentence needs a lot more than grammatical correctness.
· Give priority to clarity over fancy language. Avoid long and involved sentences; break long sentences into shorter ones.
BAD:
We note the fact that the polynomial has the coefficient of the term positive.
GOOD:
The leading coefficient of the polynomial is positive.
BAD:
The inverse of the matrix requires the determinant of to be non-zero in order to exist, but the matrix has zero determinant, and so its inverse does not exist.
GOOD:
The matrix has zero determinant, hence it has no inverse.
· Place important words in a prominent position within the sentence. Suppose you are introducing the logarithm:
BAD:
An important example of a transcendental function is the logarithm.
In this classic bad opening ’an example of something is something else’, the focus of attention is the transcendental functions, not the logarithm.
GOOD:
The logarithm is an important example of a transcendental function.
GOOD:
Let us now define a key transcendental function: the logarithm.
Suppose you wish to emphasise the scalar product:
BAD:
A commonly used method to check the orthogonality of two vectors is to verify that their scalar product is zero.
GOOD:
If the scalar product of two vectors is zero, then the vectors are orthogonal.
· Prefer the active to the passive voice.
BAD:
The convergence of the above series will now be established.
GOOD:
We establish the convergence of the above series.
· Vary the choice of words to avoid repetition and monotony.3
BAD:
The function defined above is a function of both and .
GOOD:
The function defined above depends on both and .
· Do not use unfamiliar words unless you know their exact meaning.
BAD:
A simplistic argument shows that our polynomial is irreducible.
GOOD:
A simple argument shows that our polynomial is irreducible.
· Do not use vague, general statements to lend credibility to your writing. Avoid emphatic statements.
BAD:
Differential equations are extremely important in modern mathematics.
BAD:
The proof is very easy, as it makes an elementary use of the triangle inequality.
GOOD:
The proof uses the triangle inequality.
· Do not use jargon, or informal abbreviations: it looks immature rather than ’cool’.
BAD:
Spse U subs into T eq. Wot R T soltns?
· Enclose side remarks within commas, which is very effective, or parentheses (getting them out of the way). To isolate a phrase, use hyphenation—it really sticks out—or, if you have a word processor, change font (but don’t overdo it).
· Take punctuation seriously. To improve it, read [40] or [41].