From:chavey@beloit.edu (Darrah Chavey)Date:Fri, 15 Nov 1996 10:36:09 -0600Newsgroups:sci.mathSubject:Re: Egyptian Fractions

In article <56aark$cdr@gza-client1.cam.ov.com>, don@cam.ov.com (Donald T. Davis) wrote: >Le Compte de Beaudrap <jd@cpsc.ucalgary.ca> writes: >> >> what is an Egyptian Fraction? > >egyptian scribes did arithmetic calculations in a >seemingly bizarre way. when they had to handle a >fractional quantity, they represented it as a sum >of an integer and several "unit fractions," each of >the form 1/n. so, for example, they handled 4 5/6 >as 4 + 1/2 + 1/3. fractions with big denominators >were very cumbersome in this system, and both addition >and multiplication of fractional quantities required >a lot of table-lookup, so as to reduce 2/n terms to >standardized sums of distinct 1/m terms. They did have a special number for 2/3-rds. >no-one knows why the egyptians found this style >necessary; it may be that they just couldn't conceive >of a better way, or that they found it more practical >for the problems that they had to solve. ... There are a couple of reasons. First, they wrote a number 1/n as the number n with an oval above it. This makes it difficult to come up with a notation for m/n without really inventing a completely new notation, which is always difficult. Second, the unary fractions come up naturally in their method for division. For example, to divide 53 by 8, they would proceed as follows: Begin with two columns, one headed by "1" and the other by "8". Double each column until just less than 53: 1 8 2 16 4 32 Now start over from the 1 and 4 and halve each number until the second column is "1". I use () to stand for the oval that should really be above the numbers in the first column: 1/2 = () 2 4 1/4 = () 4 2 1/8 = () 8 1 Now mark the numbers in the second column which add up to 53 (by the notion of binary expansion of a number, there will be a unique way to do this) 1 8 2 16 * 4 32 * 1/2 = () 2 4 * 1/4 = () 4 2 1/8 = () 8 1 * The corresponding marked numbers in column 1 give you your answer: 53 / 8 = 2 + 4 + ()2 + ()8, or 2 + 4 + 1/2 + 1/8. Thus the notion of unary fractions, 1/n, comes up naturally from this algorithm for division. (Division by numbers other than powers of 2 make life more interesting for the Egyptians.) Dominic Alivastro, "Ancient Puzzles", suggests a third reason why this use of unary fractions is good. Consider the problem Ahmes poses of dividing 3 loaves of bread between 5 people. We would answer "each person gets 3/5-ths of a loaf". If we implemented our solution, we might then cut 2 loaves into 3/5 | 2/5 pieces, with bread for 3 people; then cut one of the smaller pieces in half, giving the other two people 2/5 + 1/5 pieces. Mathematically acceptable, but try this with kids and they will insist that it is not an even division. Some have larger pieces, some have smaller. Ahmes would calculate 3/5 as : 3/5 = ()3 + ()5 + ()15 [ = 1/3 + 1/5 + 1/15 ] Now cut one loaf into fifths, cut two more into thirds, then take one of the 1/3-rd pieces and cut it into 5-ths (for the 1/15-th pieces), and you can now distribute everyone's 3/5-ths share in a way that _looks_ equal, since they will have exactly the same size pieces. (And no, I don't want to argue about the crust.) Darrah Chavey -- I'm dead, Horatio. (Hamlet) Math/CS Dept. -- He's dead Jim! (Not quite Hamlet) Beloit College 700 College St. Beloit, WI 53511