Egyptian FractionsDate: 03/21/99 From: Anonymous Subject: Egyptian Fractions Consider the fraction 6/7. The book I was reading gave this explanation: 1 7 (I assume they use 7 here because it is the denominator) 1/2 3+1/2 (halving) 1/4 1+1/2+1/4 (halving) But why do they do the next step? Why not halve again? 1/7 1 1/14 1/2 1/28 1/4 Now they note: 1/2+1/4+1/14+1/28 = 6 therefore 6/7 = 1/2+1/4+1/14+1/28 I understand that Egyptian Fractions are unit fractions, that there are never more than 4, and that no two are the same size, but I cannot follow line 4. Thanks for your help. Date: 03/21/99 From: Doctor Rick Subject: Re: Egyptian Fractions I am wondering about the context of this problem. Is it purporting to explain exactly how the Egyptians worked with fractions, or is it a modern set of rules for working with unit fractions in a more-or-less Egyptian manner? A book I have (Boyer and Merzbach, _A History of Mathematics_, 2nd edition, 1988) gives some examples from a table of Egyptian fractions in the Rhind Papyrus, written by the scribe Ahmes in 1650 BC. The authors are not able to reach a conclusion about why one decomposition into unit fractions (for instance, 2/15 = 1/10 + 1/30) was preferred out of the many possible decompositions (such as 1/15 + 1/15, or 1/8 + 1/120, or 1/12 + 1/20). Apparently there are no known examples in the Egyptian papyri of the method they used to decompose a fraction into unit fractions. The papyri do contain examples of how to multiply using a similar technique (successive doubling, rather than halving); See our archives: http://mathforum.org/dr.math/problems/wihart.6.4.96.html But wherever the rules and the method in your book came from, I think I can see how the method works. Let us go through it. First, what we want to do is to find a number n (expressed as an Egyptian fraction) such that 6 = 7 * n. This number n will be equal to 6/7. We build a table in which each number on the right is 7 times the number on the left, by starting with two numbers that obviously have this relation, and successively dividing both sides by 2: n 7n -------------- 1 7 1/2 3+1/2 1/4 1+1/2+1/4 For the Egyptians to do this, they would need to understand that half of 1/n is 1/(2n). We cannot take this for granted, but my book says they did understand this. Now look at the terms on the right. Our goal is to find numbers in the righthand column that add to 6. Then we can add the terms on the left to get our fraction. If each term on the right is 7 times the number on the left, then any sum of terms on the right is 7 times the sum of the corresponding terms on the left (by the distributive property). We have two terms that add to between 5 and 6, namely, (3 + 1/2) + (1 + 1/2 + 1/4) = 5 + 1/4. All we need now is 3/4, which is less than 1. So let us start over, with 1/7 in the lefthand column and 1 in the righthand column. The righthand column is still 7 times the lefthand column. n 7n -------------- 1/7 1 1/14 1/2 1/28 1/4 We can stop here because the smallest fraction we need is 1/4. It will add to the 5 + 1/4 we have already, bringing the total to 5 + 1/2; then we add 1/2, bringing the total to 6 (that is what we want). So, we mark off the rows that add up to 6 in the right column, and add only these rows in the left column as well: n 7n --------------------------------------- 1 7 1/2 3+1/2 X 1/4 1+1/2+1/4 X 1/7 1 1/14 1/2 X 1/28 1/4 X --------------------------------------- 1/2 + 1/4 + 1/14 + 1/28 6 The number on the left is 1/7 of 6, so it is our answer. By the way, if you try this method on 2/15, you will get 1/8 + 1/120, which is NOT the decomposition listed in the Rhind papyrus, so this must not be exactly the technique used by the author of that document. But it is an interesting method nevertheless. I hope my explanation helped you. - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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