Numbers and Symbols - Some Writing Tips

Mathematical Writing - Vivaldi Franco 2014

Numbers and Symbols
Some Writing Tips

Effectively combining numbers, symbols, and words is a main theme in this course. We begin to look at some basic conventions.

·  A sentence containing numbers and symbols must still be a correct English sentence, including punctuation.

BAD:

$$a<b$$ $$a\not =0$$

GOOD:

We have $$a<b$$ and $$a\not =0$$.

GOOD:

We find that $$a<b$$ and $$a\not =0$$.

GOOD:

Let $$a<b$$, with $$a\not =0$$.

BAD:

$$x^2-7^2=0$$. $$x=\pm 7$$.

GOOD:

Let $$x^2-7^2=0$$; then $$x=\pm 7$$.

GOOD:

The equation $$x^2-7^2=0$$ has two solutions: $$x=\pm 7$$.

·  Omit unnecessary symbols.

BAD:

Every differentiable real function $$f$$ is continuous.

GOOD:

Every differentiable real function is continuous.

·  If you use small numbers for counting, write them out in full; if you refer to specific numbers, use numerals.

BAD:

The equation has 4 solutions.

GOOD:

The equation has four solutions.

GOOD:

The equation has 127 solutions.

BAD:

Both three and five are prime numbers.

GOOD:

Both 3 and 5 are prime numbers.

·  If at all possible, do not begin a sentence with a numeral or a symbol.

BAD:

$$\rho $$ is a rational number with odd denominator.

GOOD:

The rational number $$\rho $$ has odd denominator.

·  Do not combine operators ($$+$$, $$\not =$$, $$<$$, etc.) with words.

BAD:

The difference $$b-a$$ is $$<0$$

GOOD:

The difference $$b-a$$ is negative.

·  Do not misuse the implication operator $$\Rightarrow $$ or the symbol $$\therefore $$ . The former is employed only in symbolic sentences (Sect. 4.2); the latter is not used in higher mathematics.

BAD:

$$a$$ is an integer $$\Rightarrow $$ $$a$$ is a rational number.

GOOD:

If $$a$$ is an integer, then $$a$$ is a rational number.

BAD:

$$\Rightarrow x=3$$.

BAD:

$$\therefore x=3$$.

GOOD:

hence $$x=3$$.

GOOD:

and therefore $$x=3$$.

·  Within a sentence, adjacent formulae or symbols must be separated by words.

BAD:

Consider $$A_n, n<5$$.

GOOD:

Consider $$A_n$$, where $$n<5$$.

BAD:

Add $$p$$ $$k$$ times to $$c$$.

BAD:

Add $$p$$ to $$c$$ $$k$$ times.

GOOD:

Add $$p$$ to $$c$$, repeating this process $$k$$ times.

For displayed equations the rules are a bit different, because the spacing between symbols becomes a syntactic element. Thus an expression of the type

$$ A_n=B_n,\quad n<5 $$

is quite acceptable (see Sect. 6.3).