EGYPTIAN FRACTIONS:

FROM AHMES

TO ERDŐS

Keven Hansen Southwestern Illinois College keven.hansen@swic.edu

EGYPTIAN FRACTIONS Ancient Egyptians wrote all proper fractions (except 2/3) as finite sums of distinct unit fractions For example, 3/4 = 1/2 + 1/4 4/9 = 1/3 + 1/9 5/7 = 1/2 + 1/7 + 1/14

AHMES (AHMOSE) The oldest (named) ancient mathematical authority Lived c. 1650 BCE, though the Rhind Papyrus represents work from as

early as 1850 BCE Accomplishments include significant work

with linear equations, arithmetic and geometric progressions, and geometry (areas and volumes)

EGYPTIAN NUMERATION

MULTIPLICATION (DUPLATION) 11 X 257

x 1 257 x 2 514 4 1028 x 8 2056 2827 Essentially, the Egyptians were converting one

multiplicand to binary!

EGYPTIAN DIVISION To divide, Ahmes multiplied: “19 3” was interpreted as “Work with 3 to get 19” 1 3 x 2 6 x 4 12 2/3 2 x 1/3 1 Answer: 19/3 = 6 + 1/3

÷

SCRIBAL TABLES Assisting the scribe were tables of values:

2/n, n/10, and (2/3)n

(#5 RMP): Multiply 2/3 1/10 1/30 by 10: 1 2/3 1/10 1/30 x2 1 1/2 1/10 4 3 1/5 x8 6 1/3 1/15 Total 8

LEONARDO OF PISA (FIBONACCI)

Circa 1170 to 1250 Best known work is Liber Abbaci (1202) Introduced methods for converting between

“vulgar” and Egyptian form Key figure connecting European medieval mathematics to Arabic and African traditions

THE GREEDY ALGORITHM UNTIL a unit fraction results: ONE: form the reciprocal of a/b, round up,

and form the reciprocal again (to identify the largest unit fraction less than a/b).

TWO: subtract this unit fraction and repeat steps one and two on the remainder.

The Egyptian representation is the sum of the unit fractions encountered.

A HEURISTIC METHOD For a/b, find factors of b that sum to a. Rewrite a as this sum and reduce. Or, find an equivalent fraction first. EX: For 5/7, no factors of 7 add up to 5, but 5/7=20/28. Factors of 28: 1,2,4,7,14,28

Since 14 + 4 + 2 = 20, 5/7 = 14/28 + 4/28 + 2/28 = 1/2 + 1/7 + 1/14

J. J. SYLVESTER 1814-1897 Founder of the American Journal of Mathematics Savilian Professor of Geometry (Oxford) Proved the “greedy algorithm” terminates

and brought Egyptian fractions to the attention of “modern” mathematicians

Best known for work in matrix theory, combinatorics, and number theory

EXISTENCE PROOF Assume 0 < a/b < 1 and gcd(a,b) = 1 By division algorithm, b = aq + r, and r ≠ 0 a/b = 1 / (q+1) + (a – r) / [b(q+1)] Let (a – r) / [b(q+1)] = a’/b’ and repeat Since r ≠ 0, each numerator will decrease

and this process will terminate in an Egyptian representation with at most a terms)

UNIQUENESS ? Note that

For any Egyptian representation, you can

“split” the largest denominator to create a new representation:

For example, 3/4 = 1/2 + 1/4 = 1/2 + 1/5 + 1/20 = 1/2 + 1/5 + 1/21 + 1/420 Representations are infinite, not unique!

1 1 11 ( 1)n n n n

= ++ +

PAUL ERDŐS 1913-1996 Legendary, prolific, eccentric mathematician who made significant

contributions to Egyptian fractions (and left many open problems)

Traveled widely, worked collaboratively Contributed to combinatorics, graph

theory, probability, among other branches

ALGORITHMS

Scores of algorithms exist for converting to Egyptian form Greedy, Binary, Continued Fraction, Farey,

Golomb, Splitting, and many more, each worth a classroom exploration…

“Best” algorithms are those that: Minimize the number of unit fractions OR Keep the denominators small

THE FAREY ALGORITHM F1 0/1 1/1

F2 0/1 1/2 1/1

F3 0/1 1/3 1/2 2/3 1/1

F4 0/1 1/4 1/3 1/2 2/3 3/4 1/1

F5 0/1 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 1/1

F6 0/1 1/6 1/5 1/4 1/3 2/5 1/2 3/5 2/3 3/4 4/5 5/6 1/1

F7 0/1 1/7 1/6 1/5 1/4 2/7 1/3 2/5 3/7 1/2 4/7 3/5 2/3 5/7 3/4 4/5 5/6 6/7 1/1 Find p/q in Fq & the entry preceding it; repeat. The pairwise products of denominators yield an Egyptian form: 3/7 = 1/3 + 1/15 + 1/35

THE SPLITTING ALGORITHM Not proven as an algorithm until 1993 (L. Beeckmans) Relies on the splitting formula given earlier 3/5 = 1/5 + 1/5 + 1/5 = 1/5 + (1/6 + 1/30) + (1/6 + 1/30)

= 1/5 + 1/6 + 1/30 + 1/7 + 1/42 + 1/31 + 1/930

OPEN QUESTIONS The Erdős-Straus Conjecture: for all n, 4/n

has a 3-term Egyptian representation. The Sierpinski Conjecture: The same is

true of 5/n. Does the “odd greedy algorithm”

(restricting to odd denominators) always terminate?

Does any “best” algorithm run in polynomial time?

MAKE YOUR OWN CONJECTURE

THE TELESCOPING SERIES Since , we have:

1 1 1( 1) 1n n n n

= ++ +

1

1 1 1 1 1 11 2 2 2 6 3

1 1 1 1 1 1 1 1 12 6 12 4 2 6 12 20 5

1 1( 1) 1

k

n n n k=

= + = + +

= + + + = + + + +

= ++ +∑

1

11( 1)n n n

∞

=

=+∑

As k goes to infinity,

THE HARMONIC SERIES Proven divergent by Oresme, c. 1320-

1385 For rational numbers greater than one:

Sum the harmonic series until adding the next term would exceed the given rational.

Apply any algorithm above to the remainder 10/1 requires more than 20 000 t !

THE HARMONIC TRIANGLE

THANKS !!!! keven.hansen@swic.edu

IMAGE SOURCES

Page of Rhind Papyrus: www. Aldokkan.com/science/mathematics.htm Seated Scribe: ritournelleblog.com/2010/12/23/ancient-egyptian-treasure-at-the-louvre/ Hieratic Numerals: www.skypoint.com/members/waltzman/mathematics.html Hieroglyphic Numerals: www.eyelid.co.uk.numbers.htm Scribal Tools: www.ancientegyptonline.co.uk/scribe.html Noseless Scribe: jameswwatts.net (Syracuse University) Fibonacci: www.history.mcs.st-and.ac.uk/history/PictDisplay/Fibonacci.html Sylvester: www.history.mcs.st-and.ac.uk/history/PictDisplay/Sylvester.html Erdos: garydavis.blogs.umassd.edu/erdos-number/paulerdos Bleicher: experts.news.wisc.edu/experts/712 Farey: www.history.mcs.st-and.ac.uk/history/PictDisplay/Farey.html Fraction Rainbow Triangle: original work by Keven Hansen Telescope: www.clker.com/clipart-telescope-5.html Oresme: wikipedia.org/wiki/Nicole_Oresme

Oresme’s Traite du l’espere, Bibliotheque Nationale, Paris, fonds francais 565, fol. 1r Leibniz:

wp.clipart.com/famous/philosophy/philosophy_2/Gottfried_Wilhelm_von_Leibniz.png.html Original Braunschweig, Herzog Anton Ulrich Museum

Two Scribes: astromic.blogspot.com/2012/09/literacy-in-ancient-egypt.html Horus Eye: mathsisgoodforyou.com/topicsPages/egyptianmaths/horusfractions.html