Mathematics for Every Day

Ideally, authors should write with a target audience in mind. If a writer is skillful, this will increase the enjoyment of the reader. It is, perhaps, nowhere more important to keep the audience in mind than when writing mathematics.

The more mathematically sophisticated the intended reader, the more information which can be communicated in a short space. Consider the difference beween
\[ {{10}\choose{2}}\qquad \frac{10!}{2!8!}\qquad 45 \]

The expressions decrease in clarity of intent from left to right. The binomial (the left-hand expression) expresses clearly that the result is the number of pairs possible from a grouping of 10 choices. The central expression is slightly more ambiguous as to intent: it might be choosing 2 of 10, 8 of 10, or not the result of a choice at all. The right expression is devoid of intent.

However, the expressions increase in clarity of value from left to right. Less and less mathematical knowledge or calculation is required to discern the basic value of the expression.

It is a tension between these two goals, clarity of intent and clarity of value, that makes writing mathematics, aside from doing mathematics, such a difficult endeavor.