Style - Some Writing Tips

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 $$2x^2-x-1$$ has the coefficient of the $$x^2$$ term positive.

GOOD:

The leading coefficient of the polynomial $$2x^2-x-1$$ is positive.

BAD:

The inverse of the matrix $$A$$ requires the determinant of $$A$$ to be non-zero in order to exist, but the matrix $$A$$ has zero determinant, and so its inverse does not exist.

GOOD:

The matrix $$A$$ 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 $$x$$ and $$y$$.

GOOD:

The function defined above depends on both $$x$$ and $$y$$.

·  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 $$x$$ 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].