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E4. Very Proper Fractions

A proper fraction is one where the numerator is no larger than the denominator. We'll call a fraction with denominator 1 a VPF (very proper fraction). Imagine a civilization where the only way of representing fractions is as a VPF or a sum of (two or more) VPFs. To keep things interesting, they don't allow using the same denominator twice. So, for example, they can't write 2/5 as 1/5 + 1/5. However, 2/5 CAN be represented as a sum of distinct VPFs: 1/3 + 1/15. Since the denominator of these fractions are all 1, we can simplify the notation by simply writing the numerators. For example, we've shown that we can convert from our system to their system by 2/5 = (3,15).

Problem: Write 1143/1170 as it would be expressed in this civilization.

Extra challenge: Describe an algorithm for expressing any proper fraction as a sum of distinct VPFs, and prove that your algorithm works. (In particular, show that it terminates in a finite number of steps.) This challenge is an advanced problem.

E3. A Fraction Sequence

Let S be the sequence of all proper fractions with denominator ≤ 20, arranged in order from smallest to largest, so the first fraction is 0/1 = 0 and the last one is 1/1 = 1. Represent each fraction in lowest terms. Define the gap between two consecutive fractions in this sequence to be the difference of the smaller from the larger.

  1. Determine the smallest gap in the sequence, and find two fractions that have this gap between them.

  2. The gap between the first two fractions is 1/20 – 0/1 = 1/20, as is the gap between the last two fractions. Not counting these fractions, find two fractions that have a gap that is as large as possible.

Personal Thoughts on Math Research

I publish math research once in a blue moon, and always with co-authors. I've just finished some new research. There are a few things I like about it. (1) The result was surprising, and the work was challenging, but not too difficult. (2) It is a very concrete result in elementary geometry, that I can explain to most anyone. I think the result is nifty. (3) It was a family affair, as my co-authors are my brother Marshall and his son, Michael. It was a nice division of labor, with me doing the geometry part and Marshall and Michael doing the calculus part. Michael's TeX expertise came in handy, as well. (4) It is the first paper that I have contributed to that depended on mathematical software for its solution.

See for the paper, titled "Constructing a quadrilateral inside another one". Be sure to look at version 3, which is much improved over the earlier versions. It's only 9 pages, and a pretty easy read, as math papers go. We are at work on a new version, which should improve the exposition and simplify the proof of the main theorem a bit, but I think the current version is actually publishable quality. We'll be submitting the new version to The Mathematical Gazette.

This paper is an example of old-fashioned mathematics done with modern tools. There is nothing here that couldn't have been done in the 18th century, but without Geometer's Sketchpad I would never have come across the problem or have been able to obtain experimental evidence for our conjecture, now a theorem. And without Maple (or the reincarnation of Euler) we couldn't have done the calculations involved.

Technical production notes: Marshall wrote up the result using Scientific Workplace. Since I don't have a copy, I sent him my corrections as Word documents (using the MathType math editor), and produced wmf format graphics with Sketchpad. It would have really simplified our collaboration if we had the same software. What are other people using for collaborations?

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.