Non User
Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z	Business Non User Industries Finance Tax

Home > Egyptian fraction

Refer to the harmonic series for the proof that you can express all numbers, even very large, in this way. One algorithm to produce an Egyptian fraction representation for a given rational number r=a/b between 0 and 1 is the following greedy algorithm due to James Joseph Sylvester:

1. Find the largest unit fraction just less than r. The denominator can be found by dividing b by a, discarding the remainder, and adding one. (If there's no remainder, we're done right away because r is itself a unit fraction.)
2. Subtract the found unit fraction from r. Then continue with step 1, using this new smaller value for r.

Example: convert 19/20 into an Egyptian fraction.

• 20/19 = 1 with some remainder, so our first unit fraction is 1/2.
• 19/20 - 1/2 = 9/20.
• 20/9 = 2 with some remainder, so our second unit fraction is 1/3.
• 9/20 - 1/3 = 7/60
• 60/7 = 8 with some remainder, so our third unit fraction is 1/9.
• 7/60 - 1/9 = 1/180 which is itself a unit fraction.

So our result is

Note that the representation of a given rational number as an Egyptian fraction is not unique, and that the above algorithm does not always yield the shortest such representation:

1 Fractions in Egypt

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 know from ancient Egyptian hieroglyphs 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 ¹/₇ added together become 19. What is the quantity?

"Assume 7. 7 and ¹/₇ 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 ¹/₂, ¹/₃, ¹/₄, ¹/₁₀, ....

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".

2 See also

• Egyptian numerals

Politics	Business	Finance

Topics: Egyptian Fraction Unit Fractions Number Ancient Remainder...

---

Egyptian LoverEgyptian Lover (born Greg Broussard is an American electro hop artist, one of the best known performers of the underground West Coast dance scene. He began recording around Los Angeles in 1983 (see 1983 in music) and released his debut single and club hit Egyptian languagesThe Egyptian languages are a subfamily of the Afro-Asiatic languages. The only members are the ancient Egyptian language and its descendant, Coptic. They are extinct in regular usage, though Coptic is preserved as a liturgical language in the Coptic Churc Egyptian mythologyEgyptian mythology (or Egyptian religion is the name for the succession of beliefs held by the people of Egypt until the coming of Christianity and Islam. The timespan involved is nearly three thousand years, and beliefs varied considerably over time, so Egyptian languageRecords of the Ancient Egyptian language have been dated to 2600 BC. It is part of the Afro-Asiatic group of languages and is related to Hamitic (North African languages) and Semitic (languages such as Arabic and Hebrew). The language survived until about Egyptian hieroglyphHieroglyphs are a system of writing used by the Ancient Egyptians, using a combination of logographic, syllabic, and alphabetic elements. Etymology The word hieroglyph comes from the Greek words (hierogluphos) hiero , meaning "sacred", and glyph , meaning Egyptian Islamic JihadThe Egyptian Islamic Jihad EIJ , also called the Islamic Jihad and the Jihad Group is an Egyptian Islamic group active since the late 1970s with origins in the Muslim Brotherhood. It suffered setbacks as a result of numerous arrests of operatives worldwid Egyptian soul Egyptian chronology Egyptian Vulture Egyptian Museum Egyptian calendar Egyptian Sand Sea Egyptian Goose Egyptian fraction Egyptian Nightjar