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Math 300 Exploration Egyptian Fractions A unit fraction is a fraction whose numerator is 1 and whose denominator is a natural number. A decomposition of a positive rational number into distinct unit fractions is called an Egyp- tian fraction decomposition. For example, + is an Egyptian fraction decomposition for , but ++ is not. I may write EFD in place of Egyptian fraction decomposition. The ancient Egyptians had no notation for fractions other than unit fractions (except for so they did in fact make use of such decompositions, even though pizza had not been invented yet. Much of what we know about Egyptian mathematics comes from the Rhind Papyrus, which dates to approximately 1550 BC. Computational techniques are demonstrated to solve problems such as this: Divide 100 hekats of barley among 5 men so that the common difference is the same and so that the sum of the two smallest is 1/7 the sum of the three largest. (Keep in mind this is at least 1000 years before Algebra is invented.) The Egyptian fraction decomposition for a rational number is not unique, as you may have already seen. Computational questions. 1. Find four different EFDs for 3/20. 2. Use the greedy algorithm to find an EFD for 3/7, and for 8/11. Can you find a different EFD for 8/11 with the same number of terms, but smaller denomi- nators? (Try considering only even denominators.) 3. The start of the Rhind papyrus consists of a table for duplicating unit fractions. For example, if you wanted to write 2, you could not write the table gives the EFD. Double ++ without using any fractions with numerators greater than one. (Hint: 1-+1) The Main Question. How do we know that any positive rational number has an EFD? Related questions. 1. Does it follow that if one can find an EFD for a given rational number, then one can find infinitely many? 2. Does the greedy algorithm always provide the shortest possible EFD for a given rational number?
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