Egyptian Fractions - Mathematicians of the African Diaspora
Egyptian Fractions
You may find it amazing that fractions, as<br>we know them, barely existed, for the european civilization, until<br>the 17'th century. Even in the 19th century, a method called russian<br>peasant fractions, was the same used by the Europeans since they<br>met the African, and the Egyptians at least since 4000BC in Egypt.<br>As the method was found on several papyrus, we now call this technique<br>egyptian fractions . Let us agree to call a number a unit<br>fraction if it has form 1/n , where n is a positive integer.<br>An egyptian fraction is an expression of the sum of unit<br>fractions
1/a + 1/b + 1/c + ... , where the denominators<br>a,b,c, ... are increasing.
An egyptian number is any number equal<br>which can be expressed as the sum of an integer plus the sum of<br>an Egyptian fraction. Here are some egyptian fractions:1/2 + 1/3<br>(so 5/6 is an egyptian number), 1/3 + 1/11 + 1/231 (so 3/7 is<br>an egyptian number), 3 + 1/8 + 1/60 + 1/5280 (so 749/5280 is an<br>egyptian number). The egyptians also made note of the fraction<br>2/3.
1/5 + 1/37 + 1/4070 and 1/6 + 1/22 + 1/66 are<br>regarded as different egyptian fractions even though the sum of<br>each is 5/22.
Other<br>Egyptian fraction pages of note .
The earliest records of<br>egyptian fractions date to nearly 3900 years ago in the papyrus<br>copied by Ahmes (sometimes called Ahmos - ref1,<br>ref2) purportedly from<br>records at least 300 years earlier. It is conjectured that the<br>mysterious, so called, meaningless, egyptian triple 13, 17, 173<br>actually means
3 + 1/13 + 1/17 + 1/173 = 3.141527<br>which approximates to<br>4 places !!!
(considerably better than the usual 3.16 credited to the<br>egyptians.)
However, I have not been able to locate a credible<br>reference for the triple 13, 17, 173 in this context!!! )
However, to the victor goes the spoils; i.e.,<br>prior to the "discovery" of the Rhind papyrus, egyptian<br>fractions were thought, by european mathematicians, to come from<br>the Greeks. Even the name pi is Greek.
The rule of Egyptian fractions requires us<br>to write only unit fractions, integers, and their sums. So if<br>a duke is awarded 3/7'th of the conquered land, the quanity might<br>be represented as (1/4 + 1/7 + 1/28)'th of the conquered land,<br>which is a bit better than
(1/3 + 1/11 + 1/231)'th of the conquered land, but still awkward.<br>Until the 18'th century, when our present method, from India and<br>also thousands of years old, of writing any integer in the numerator<br>became popular in Europe, Egyptian fractions were the primary<br>method of writing non-integer numbers.
THEOREM. Every rational number<br>is an egyptian number.
The modern proof of the Theorem was discovered<br>in 1880, but European's have known how to compute Egyptian numbers<br>since Fibonacci in the 12'th century. Before exhibiting the rule<br>we make a convention. Given a non-integer r, let [r] denote the<br>smallest integer > r.
Suppose p/q proof<br>below). Let r/s = p/q -1/[q/p]. So p/q = 1/[q/p] + r/s. If r=1,<br>we are done; otherwise, repeat the process. Here are some examples:
1. Consider p/q = 4/23. Since 23/4 = 5.65,<br>[23/4] = 6. Compute 4/23 - 1/6 as a fraction, and get 1/138. Thus,<br>4/23 = 1/6 + 1/138.
2. Consider 5/22. [22/5] = 5. 5/22 - 1/5 =<br>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<br>other egyptian fractions whose sum is 5/22. For example, let's<br>start, for no special reason, with 1/6 instead of 1/5. 5/22 -<br>1/6 = 2/33. [33/2] = 17 and 2/33 - 1/17 = 1/561. Thus, 5/22 =<br>1/6 + 1/17 + 1/561.
4. In the last expression for 5/22, keep 1/6<br>but exchange 1/17 for 1/22. This means we compute 2/33 - 1/22<br>= 1/66. Thus, 5/22 = 1/6 + 1/22 + 1/66. This expression is more<br>satisfactory since the denominators are not as large as in the<br>proceeding two cases..
Note: There is significant interest in determining<br>which expression is "best" or what the egyptians would<br>have used. We discuss this on the page "The<br>Best Egyptian Fraction ."
About the proof<br>of the theorem:
Notice that when 0
In spite of the Theorem, there is very little<br>interest in egyptian fractions (or even their modernized version<br>- continued fractions) today, and only infinite series, a topic<br>of elementary calculus could indicate their passage (recall from<br>calculus
e-2 = 1/2 + 1/6 + 1/24 + 1/120 + 1/720 + ...). There<br>are, however, interesting and important problems. We include some<br>of these in the problem<br>section below.
Divide 1077 by 25:
25*
50*
100
200*
16
400
32
800*
2/5
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.<br>Infact, the Ahmos scroll contains a table of decompositions of<br>each odd fractions of the form 2/n where n ranges from 3 to 101.<br>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<br>Papyrus 2/n table
Multiplication of egyptian fractions
Multiply 383/15 by 130/3...