Fractions Plain and Fancy (12/27/99)
Prepared by:
Joseph Malkevitch
Mathematics and Computing Department
York College (CUNY)
Jamaica, NY 11451
Email: malkevitch@york.cuny.edu (for additions, suggestions, and corrections)
Fractions are first introduced in primary school, yet, simple issues involving fractions are still being pursued at the research level. Here we will just give a taste of many delightful areas that a study of fractions leads one to.
We are all aware that before the consolidation in relatively modern times to represent numbers using the Hindu-Arabic system that many experiments with ways to represent numbers (e.g. XXXIV for thirty-four in Roman times) were developed in the ancient world. Especially interesting in retrospect was the practice in Egypt of representing numbers between 0 and 1 which were rational (i.e. a ratio of two integers) by writing the fraction as the sum of fractions with numerator 1, and distinct denominators. Below I will use the phrase "unit fraction" or Egyptian fraction to mean a fraction of the form 1/n for some integer n>1, and the Egyptian expansion of a fraction will refer to the writing of the fractions as the sum of Egyptian fractions with distinct denominators. (The only exception to this general rule was the Egyptians did have a special role for the fraction 2/3).
In the mathematical papyri that have come down to us there are tables which indicate that the Egyptians had a sophisticated understanding of the way that fractions could be use in computation. However, they did not have the advantage of the a/b notation for fractions, and we are not exactly sure how they worked with fractions and conceptualized about them! Looking at these questions with a modern eye leads to many instructive and interesting issues.
Consider for example the question:
Given a/b (a, b positive integers, a < b) can one always write a/b as the (finite) sum of unit fractions with distinct denominators?
In modern terminology this provides an example an existence question. It is conceivable that one might be able to give a proof that it was always possible to do this without in fact displaying a procedure (algorithm) for actually producing the unit fractions that do the job. If in fact it is always possible to find an expansion of any a/b as a sum of Egyptian fractions, what are procedures for finding the "Egyptian expansion" of a/b? Furthermore, for any particular algorithm for finding the Egyptian expansion, what is the computational complexity of finding the Egyptian expansion? (Sometimes one may wish not to allow the Egyptian expansion of a/b to use the fraction 1/b.)
Some other issues will emerge by first looking a few examples.
2/7 = 1/5 + 1/13 + 1/115 + 1/10465.
3/7 = 1/7 + 1/8 + 1/9 + 1/56 + 1/57 + 1/72 + 1/3192.
3/10 = 1/4 + 1/20 = 1/5 + 1/10.
Try to find an Egyptian fraction expansion of 21/23. Can you find a solution where there are only 5 terms? Can you prove that there can be no expansion with only 4 terms? (There is a five term expansion but no 4 term expansion.)
From these examples one can see that there can be tradeoffs in finding Egyptian expansions between how many factions are used, how large the largest denominator gets, and other criteria of "goodness" for an expansion.
How can we be systematic in constructing an expansion of a/b?
One approach is to use the very nice, but simple observation that:
Tool 1: 1/n = 1/(n + 1) + 1/[n(n + 1)]
Proof: Merely add the two fractions on the right and verify that one obtains the sum on the left.
We can use this observation to try to expand a general fraction:
Example:
3/5 = 1/5 + 1/5 + 1/5
= 1/5 + 1/6 + 1/30 + 1/6 + 1/30
= 1/5 + 1/6 + 1/30 + 1/7 + 1/42 + 1/31 + 1/930
It is not so clear in fact that this method will always terminate, since at latter stages two fractions with the same denominator may emerge from different fractions farther back.
Tool 1 is a special case of a more general and lovely way to construct Egyptian fraction expansions, that seems to go back to ancient Egypt (as called to my attention by Milo Gardner):
Tool 2: 1/nm = 1/n(n + m) + 1/m(n + m)
Note that Tool 1 arises from Tool 2 by setting m=1, and applies in particular when n is a prime.
As an example consider how to expand 1/8.
We can write 1/8 = 1/4(2) and apply Tool 2: 1/8 = 1/4(6) + 1/2(6) = 1/24+ 1/12 = 1/ 12 + 1/24.
This tool can also be adapted to when the numerator of a fraction is not 1. Thus, to expand 2/7 we would get 2(1/7(1)) = 2(1/7(8) + 1/1(8)) = 2(1/56 + 1/8) =2/56 + 2/8 = 1/28 + 1/4 = 1/4 + 1/28. You may want to compare this expansion with the four fraction expansion for 2/7 listed earlier!
Here is another approach to trying to find an Egyptian expansion for a fraction:
Tool 3: 1/xy = 1/z times z/xy.
1/8 = 1/6 times 6/8 = 1/6 (1/2 + 1/4) = 1/12 + 1/24 (*)
Trying the same approach using z = 5 (instead of z equal to 6) we get:
1/8 = 1/5 times 5/8 = 1/5 (1/2 + 1/8) = 1/10 + 1/40.
It may seem that this approach is rather ad hoc, but setting z = x + y we obtain: 1/xy = 1/(x + y) times (x + y)/(xy) = 1/(x + y) x (1/y + 1/x) as a systematic way of seeing what is happening in (*) above. Note that this is merely another way of rewriting Tool 2.
When I first saw (*) I thought the pattern was based on the use of choosing z so that z/xy exceeded 1/2 and taking advantage of this observation, to write z/xy as a 1/2 plus another fraction, ideally one with unit denominator. This explained why I tried the expansion based on 5 in the same way. It is not clear, however, whether this observation has any value! Put differently, unless perhaps there is a nice way to choose z as a function of x and y it is not clear what can be accomplished.
Trying it again with z = 7 yields 1/8 = 1/7 (7/8) = 1/7 (1/2 + 3/8) = 1/7(1/2 + 1/4 + 1/8) = 1/14 + 1/28 + 1/64. In this case, the expansion fails to yield a two fraction expansion as in the other two tries. The average denominator is (14 + 28 + 64)/3 = 106/3. Note that in comparing these two Egyptian expansions both only use two terms but in one the average denominator is 18 and in the other 25.
Using the approach above, we get 1/8 = 1/9 + 1/72 which has an average denominator of 40.5. Could there be a 3 fraction expansion with a better average denominator than 2 fraction expansion? Note that the average denominator for some 3 fractions expansions is better than the average denominator for at least one 2 fraction expansion.
A variety of patterns can be looked for here: patterns for fractions of the form 1/p where p is a prime versus patterns for 1/x where x is a composite but has certain properties. One can also look for patterns in expanding k/x for fixed k bigger than 1 where k and x are relatively prime (i.e. have no factor in common between the numerator and denominator) and where x might be a prime, or composite with particular properties.
There are surprisingly simple problems for which answers have not yet been forthcoming:
Conjecture (Paul Erdös and E. Straus): 4/n can always be written as the sum of three unit fractions.
Conjecture (W. Sierpinski): 5/n can always be written as the sum of three unit fractions.
Other interesting questions concern fractions representable by unit fractions with only odd denominators. It is known that such an expansion is always possible. Even the special case where the sum is one is interesting. For example: 1/3 + 1/5 +1/7+1/9 + 1/15 + 1/21 + 1/27 +1/35 + 1/63 + 1/105 +1/135 = 1.
Another interesting vantage point for fractions is to look at the fractions (in lowest terms, that is there is no common factor between the numerator and the denominator) listed in increasing order whose denominator is no larger than a fixed integer n. The fractions arranged in this way are usually referred as being the Farey Sequence of order n. It is traditional to include 0 and 1 in each list, in the form 0/1 and 1/1. (Farey was a French mineralogist who described some of the properties of what are today referred to as Farey Sequences in 1816, though he does not seem to have been the first to consider them.)
0/1 1/1
0/1 1/2 1/1
0/1 1/3 1/2 2/3 1/1
0/1 1/4 1/3 1/2 2/3 3/4 1/1
0/1 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1/1
The Farey sequence of order n obeys a large number of remarkable properties:
a. if r/s, t/u, and v/w are three consecutive fractions in a Farey sequence of order n then: t/u = (r + v)/(s + w)
(It is a worthwhile exercise to verify that if a/b is a fraction which is less than c/d that the fraction (a +c)/(b + d) is a fraction which is bigger than a/b but smaller than c/d, and certainly not the fraction that represents the sum of a/b and c/d!)
b. If r/s and t/u are consecutive Farey fractions in the sequence of order n, then s + u is larger than n.
c. If r/s and t/u are consecutive Farey fractions in the sequence of order n, then ru - st = -1. (One way to remember this result is that if one forms a determinant whose columns are the two fractions this result says that this determinant has value -1).
Problem: Can you find a formula for the number of terms in the Farey sequence of order n?
Problem: Look for a pattern in the absolute value of the differences between the denominators of the sequence of fractions in the Farey Sequence of order n?
Problem: Will there always be a fraction in the Farey Sequence of order n+1 which arises from two consecutive fractions in the Farey Sequence of order n?
Problem
References:
Beck, A., and M. Bleicher, D. Crowe, Excursions into Mathematics, Worth Publishers, New York, 1969.
Bleicher, M., A new algorithm for the expansion of Egyptian fractions, J. Number Theory 4 (1972) 342-382.
Campbell, P., Bibliography of algorithms for Egyptian fractions, (preprint), Beloit College, Wisconsin.
Graham, R., On finite sums of unit fractions, Proc. London Math. Soc. 14 (1964) 193-207.