Egyptian fraction

An Egyptian fraction is a sum of fractions whose numerators are equal to 1, whose denominators are positive integers, and all of whose denominators differ from each other. It can be shown that every positive rational number can be written in that form.

Mathematical historians sometimes describe algebra as having developed in three primary stages:

1. rhetorical algebra, wherein the problem was stated in words of the language of the ancient mathematician;
2. syncopated algebra, wherein some words of the problem were abbreviated, for easier comprehension;
3. symbolic algebra, where in symbols for operators and operands made comprehension still easier.

Typical of symbolism is denoting "the unknown" by "x". We known from ancient Egyptian hieroglypics on clay or papyrus that ancient Egyptian priests, in their rhetorical algebra, used the word "aha" meaning "heap" or "set" for the unknown.

This is shown in the Rhind Papyrus (circa 1650 B.C.) in The British Museum in London in a translation of one of its "aha" problems:

"Problem 24: A quantity and its added together become 19. What is the quantity?

"Assume 7. 1 and of 7 is 8. As many times as 8 must be multiplied to give 19, so many times 7 must be multiplied to give the required number."

In modern symbolic form, x + x/7 = 8x/7 = 19, or x = 133/8. Proof: 133/8 + 133/(7 · 8) = 133/8 + 19/8 = 152/8 = 19.

Note the fractions in this problem. Ancient Egyptians calculated by unit fractions, such as , , , , ....

The hieroglyph for an open mouth denoted the fractional solidus, with a number hieroglyph written below this "open mouth" icon to denote the denominator of the fraction.

Any fraction we write with a non-unit numerator was written by ancient Egyptians as a sum of unit fractions no two of whose denominators are the same.

These sums of unit fractions have, therefore, become known as "Egyptian fractions".

We use them every day in making change for a dollar:
1 penny ;
1 nickel;
1 dime ;
1 quarter ;
1 half-dollar .

The great British mathematician, James Joseph Sylvester, developed an algorithm for converting any non-unit fraction into the sum of unit or Egyptian fractions:

1. Given a fraction, such as , find the greatest Egyptian fraction just less than .
2. Find this by performing division: 24 ÷ 7 = 3 + remainder 3. So is the trial number, just less than .
3. Perform subtraction: .
4. Hence, the conversion, .\n