Egyptian Fractions
You may find it amazing that fractions, as we know them, barely existed, for the european civilization, until the 17'th century. Even in the 19th century, a method called russian peasant fractions, was the same used by the Europeans since they met the African, and the Egyptians at least since 4000BC in Egypt. As the method was found on several papyrus, we now call this technique egyptian fractions. Let us agree to call a number a unit fraction if it has form 1/n , where n is a positive integer. An egyptian fraction is an expression of the sum of unit fractions
1/a + 1/b + 1/c + ... , where the denominators a,b,c, ... are increasing.
An egyptian number is any number equal which can be expressed as the sum of an integer plus the sum of an Egyptian fraction. Here are some egyptian fractions:1/2 + 1/3 (so 5/6 is an egyptian number), 1/3 + 1/11 + 1/231 (so 3/7 is an egyptian number), 3 + 1/8 + 1/60 + 1/5280 (so 749/5280 is an egyptian number). The egyptians also made note of the fraction 2/3.
1/5 + 1/37 + 1/4070 and 1/6 + 1/22 + 1/66 are regarded as different egyptian fractions even though the sum of each is 5/22.
Other Egyptian fraction pages of note.
The earliest records of egyptian fractions date to nearly 3900 years ago in the papyrus copied by Ahmes (sometimes called Ahmos - ref1, ref2) purportedly from records at least 300 years earlier. It is conjectured that the mysterious, so called, meaningless, egyptian triple 13, 17, 173 actually means
3 + 1/13 + 1/17 + 1/173 = 3.141527
which approximates
to
4 places!!!
(considerably better than the usual 3.16 credited to the
egyptians.)
However, I have not been able to locate a credible
reference for the triple 13, 17, 173 in this context!!!)
However, to the victor goes the spoils; i.e., prior to the "discovery" of the Rhind papyrus, egyptian fractions were thought, by european mathematicians, to come from the Greeks. Even the name pi is Greek.
The rule of Egyptian fractions requires us
to write only unit fractions, integers, and their sums. So if
a duke is awarded 3/7'th of the conquered land, the quanity might
be represented as (1/4 + 1/7 + 1/28)'th of the conquered land,
which is a bit better than
(1/3 + 1/11 + 1/231)'th of the conquered land, but still awkward.
Until the 18'th century, when our present method, from India and
also thousands of years old, of writing any integer in the numerator
became popular in Europe, Egyptian fractions were the primary
method of writing non-integer numbers.
THEOREM. Every rational number is an egyptian number.
The modern proof of the Theorem was discovered in 1880, but European's have known how to compute Egyptian numbers since Fibonacci in the 12'th century. Before exhibiting the rule we make a convention. Given a non-integer r, let [r] denote the smallest integer > r.
Suppose p/q < 1 is written in lowest terms then there is an egyptian fraction with at most p terms and whose sum is p/q (see the proof below). Let r/s = p/q -1/[q/p]. So p/q = 1/[q/p] + r/s. If r=1, we are done; otherwise, repeat the process. Here are some examples:
1. Consider p/q = 4/23. Since 23/4 = 5.65, [23/4] = 6. Compute 4/23 - 1/6 as a fraction, and get 1/138. Thus, 4/23 = 1/6 + 1/138.
2. Consider 5/22. [22/5] = 5. 5/22 - 1/5 =
3/110. Now [110/3] = 37. But
3/110 - 1/37 = 1/4070. So 5/22 = 1/5 + 1/37 + 1/4070.
3. With a little ingenuity, you can determine other egyptian fractions whose sum is 5/22. For example, let's start, for no special reason, with 1/6 instead of 1/5. 5/22 - 1/6 = 2/33. [33/2] = 17 and 2/33 - 1/17 = 1/561. Thus, 5/22 = 1/6 + 1/17 + 1/561.
4. In the last expression for 5/22, keep 1/6 but exchange 1/17 for 1/22. This means we compute 2/33 - 1/22 = 1/66. Thus, 5/22 = 1/6 + 1/22 + 1/66. This expression is more satisfactory since the denominators are not as large as in the proceeding two cases..
Note: There is significant interest in determining which expression is "best" or what the egyptians would have used. We discuss this on the page "The Best Egyptian Fraction."
About the proof of the theorem:
Notice that when 0 < p < q are integers with only 1 as a common divisor (i.e., p/q < 1 is written in lowest terms) the construction in the 2nd paragraph after the theorem gives, via mathematical induction, the proof of the theorem, since for p[q/p]-q < p.
In spite of the Theorem, there is very little
interest in egyptian fractions (or even their modernized version
- continued fractions) today, and only infinite series, a topic
of elementary calculus could indicate their passage (recall from
calculus
e-2 = 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ...). There
are, however, interesting and important problems. We include some
of these in the problem
section below.
Divide 1077 by 25:
|
1 |
25* |
|
|
2 |
50* |
|
|
4 |
100 |
|
|
8 |
200* |
|
|
16 |
400 |
|
|
32 |
800* |
|
|
2/5 |
2 |
|
|
1+2+8+32+ 1/3 + 1/15 =43 |
25+50+200+800+2=1077 |
As the egyptians wrote 1/3 + 1/15 for our 2/5. Infact, the Ahmos scroll contains a table of decompositions of each odd fractions of the form 2/n where n ranges from 3 to 101. Here are a few:
|
2/5 |
1/3 + 1/5 |
|
2/7 |
1/4 + 1/28 |
|
2/9 |
1/6 + 1/18 |
|
2/15 |
1/10 + 1/30 |
|
2/17 |
1/12 + 1/51 + 1/68 |
|
2/101 |
1/101 + 1/202 + 1/303 + 1/606 |
| for more see Rhind Papyrus 2/n table | |
Multiplication of egyptian fractions
Multiply 383/15 by 130/3 or 25 + 8/15 = 25 + 1/3 +1/5 by 43 + 1/3
|
1* |
25 + 1/3 +1/5 |
|
2* |
50 + 2/3 + 2/5 = 50 + 2/3 + (1/3 + 1/15) = 51 + 1/15 |
|
4 |
102 + 2/15 = 102 + 1/10 + 1/30 |
|
8* |
204 + 1/5 + 1/15 |
|
16 |
408 + 2/5 + 2/15 = 408 + (1/3 + 1/15) + (1/10 + 1/30) |
|
32* |
816 + (2/3 + 2/15 )+ (1/5 + 1/15) = 816 + 2/3 + 2/5 = 816 + 2/3 +1/10 + 1/30 |
|
2/3 |
544 + 4/9 + 1/15 + 1/45 |
|
1/3* |
272 + 2/9 + 1/30 + 1/90 |
|
1 + 2 + 8 + 32 + 1/3 = 43 |
25 + 1/3 +1/5 + 51 + 1/15 + 204 +
1/5 + 1/15 + 816 + 2/3 +1/10 + 1/30 + 272+ 2/9 + 1/30 + 1/90 = 1369 + 1/3 + 1/5 + 1/6 + 1/18 + 1/30 + 1/90 |
PROBLEMS USING EGYPTIAN FRACTIONS
1. The Mullah's horse: The former Grand Wizier, Mullah Nasrudin was approached by three men with 19 horses. The men asked him to adjudicate the will of their recently dead father which required that his horses be divided among his three sons so that the oldest son receives 1/2, the middle son gets 1/3, and the youngest son would get 1/7. With little hesitation Nasrudin added his own horse to the herd and said, "What is half of 20, 1/4 of 20, and 1/5 of 20" After some time the men replied, "10, 5, and 4". The eldest son then took 10 of the horses, the middle son took 5 of the horses, and the youngest son took 4 of the horses. The Mullah Nasrudin, then took the remaining horse and rode home. Can you explain what occured?
2. Find all the solutions (there are less than 10) to the problem
(n-1)/n = 1/a + 1/b + 1/c, where a < b,
b < c, a, b, and c are positive integers with least common
multiple n. Note. a = 2, b = 4, c = 6, and n = 12 gives one solution.
3. How many different egyptian fractions can be used to describe 2/3? Two of them are 1/2 + 1/3 + 1/6 and 1/3 + 1/10 + 1/15.
5. Write a program (in BASIC or any other language) for computing egyptian fractions representing all fractions p/q, p < q with q at most 50 (hint: look at the proof above to check your answer see the Rhind Papyrus 2/n table.
6. The following is problem 33 from the Ahmes Papyrus. Solve it using egyptian fractions only: The sum of a certain quantity together with its two-thirds and its one-seventh becomes 37. What is the quantity?
7. Want to solve an unsolved problem? One of the most famous problems on Egyptian Fractions asks, "Can every proper fraction of the form 4/q be expressed with an egyptian fraction with less than 4 terms?" Can every proper fraction of the form 5/q be expressed with an egyptian fraction with less than 4 terms?
8. The sailor, coconut, and monkey problem: Five sailors were abandoned on an island. To provide food, they collected all the coconuts they could find. During the night one of the sailors awoke and decided to take his share of the coconuts. He divided the nuts into five equal piles and discovered that one was left over, so he threw the extra coconut to the monkies. He then hid his share and went back to sleep. A little later a second sailor awoke and had the same idea as the first. He divided the remainder of the nuts into five equal piles, discovered also that one was left over, and through it to the monkies before hiding his share. In turn each of the other three sailors did the same - dividing the observable amount into five equal piles, hiding one, throwing one left over to the monkies. The next morning the sailors, looking innocent, divided the remaining nuts into five piles with none left over. Find the smallest number of nuts in the original pile.
4. 355/113 approximates to 6 places. (355/113) - 3
= 16/113. Find an egyptian fraction whose sum is 16/113.
9. Above we noted the triple 13, 17, 173
used to approximate- The numbers 3 and 7 were
very important to egyotian mythology.
URL: http://www.math.buffalo.edu/mad/Ancient-Africa/egyptian-fractions.html
GOTO The Best Egyptian Fraction
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