E2. A Puzzling Fraction Representation

Show that any positive rational number can be written in the form

where a, b, n, and k are non-negative integers and (nines) represents a string of one or more 9s.

A1. Relative and Absolute Extrema

In finding minima and maxima, first-year calculus students often use the fact that, for a function differentiable on the entire real line, if the function has exactly one relative extremum, that extremum is an absolute extremum. The proof is simple: WLOG, say the function has a relative maximum at a. If a is not an absolute maximum, then there is a point b where f(b) > f(a). Then on the closed interval with endpoints a and b, f has a minimum value. Since the minimum is less than f(a), it is not attained at either a or b, so it is in the open interval, and thus is a relative minimum.

Does the above result generalize to R²? In particular, say f is differentiable on the entire plane, and has a relative maximum at (0,0), and no other relative extrema. Must the function have an absolute maximum at (0,0)? Why or why not?

E1. A Chemist's Question

Years ago, a friend who was a Ph.D. chemist asked me the following question. I thought the question was very easy. If you find it too easy yourself, you might pose the question to a child of the appropriate age.

The chemist knew that certain fractions, such as 1/4, can be expressed as a terminating decimal, while other fractions, such as 1/3, can only be expressed as a non-terminating, repeating decimal. The chemist wanted to know how to tell whether a given fraction can be expressed as a terminating decimal.