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Study in Egyptian Fractions 1 STUDY IN EGYPTIAN FRACTIONS Robert P. Schneider Whether operating with fingers, stones or hieroglyphic symbols, the ancient mathematician would have been in many respects as familiar with the integers and rational numbers as is the modern researcher. If we regard both written symbols and unwritten arrangements of objects to be forms of notation for numbers, then through the ancient notations and their related number systems flowed the same arithmetic properties—associativity, commutativity, distributivity—that we know so intimately today, having probed and extended these empirical laws using modern tools such as set theory and Peano’s axioms. It is fascinating to contemplate that we share our base-ten number system and its arithmetic, almost as natural to us as sunlight and the air we breathe, with the scribes of ancient Egypt [1, p. 85]. We use different symbols to denote the numbers and operations, but our arithmetic was theirs as well. Our notation for fractions even mimics theirs with its two-tiered arrangement foreshadowing our numerator and denominator; however, the Egyptian notation for fractions leaves no space for numbers in the numerator, the position being filled by a hieroglyphic symbol meaning “part” [1, p. 93]. Aside from having special symbols to denote 2/3 and 3/4, the Egyptians always understood the numerator to be equal to unity in their fractions [4]. The representation of an improper fraction as the sum of distinct unit fractions, or fractions having numerators equal to one, is referred to as an “Egyptian fraction” [3]. I found the Egyptians’ use of fractions, as well as their multiplication and division algorithms [1, pp. 88-101], to be surprisingly subtle; in fact, certain questions that would 2 have been of interest to Egyptian mathematicians remain unanswered still. T - Straus conjecture, for example, concerns whether the fraction 4/ can be writtenhe Erdősas the sum of three unit fractions for every positive integer [3]. While푛 the conjecture is suggested by ancient mathematical tables, my own cursory푛 investigation into the problem had me delving into polynomials, the law of quadratic reciprocity, -cyclic groups, hypergeometric series, continued fractions and other topics largely beyondℤ the scope of ancient thinkers—yet in the end, nearly all of my own findings could have been arrived at using Egyptian methods and notations. After a false start searching for integer roots to quadratic polynomials, I began my investigation into the -Straus conjecture with the observation that Erdős 1 + 1 1 1 = = + . ( + 1) + 1 ( + 1) 푛 푛 푛 푛 푛 푛 푛 Then we have the general formula for decomposing the fraction / having numerator equal to a positive integer 퐴 푛 퐴 = + . + 1 ( + 1) 퐴 퐴 퐴 푛 푛 푛 푛 Using this identity for different cases, we find that 2/ can always be written as the sum of either one or two unit fractions, depending on whether푛 is even or odd; 3/ can always be written as the sum of either one or two unit fractions,푛 except when푛 1 (mod 3); and 4/ can always be written as the sum of one, two, or three unit fractions,푛 ≡ except when 푛1 (mod 4). In the noted cases 1 (mod 3) and 1 (mod 4), the above formula푛 fails≡ to yield unit fractions. 푛 ≡ 푛 ≡ 3 Another obvious tactic may be used to improve this result; if we partition into distinct positive integers, say = + + + + , then we might rewrite퐴 the problem in terms of smaller fractions퐴 that푎1 may푎2 be푎3 easier⋯ to푎푘 solve or look up in a table = + + + + . 퐴 푎1 푎2 푎3 푎푘 ⋯ 푛 푛 푛 푛 푛 The simplest such partition yields the identity 1 1 = + . 퐴 퐴 − 푛 푛 푛 Applying this identity in the case = 4, 1 (mod 4), we find 퐴 푛 ≡ 4 1 3 = + . 푛 푛 푛 By the result cited above for = 3, the fraction 3/ on the right-hand side may be written as the sum of two unit fractions퐴 except when 푛1 (mod 3). It follows then that 4/ may be written as the sum of three or fewer unit fractions푛 ≡ unless both 1 (mod 3) and푛 1 (mod 4) hold true simultaneously, leaving only the case 1푛 (≡mod 12) as requiring푛 ≡ more than three unit fractions to represent 4/ using the above푛 ≡ decomposition procedure. 푛 Therefore, 4/ can be written as the sum of three or fewer unit fractions in all cases when 1 (mod 12푛), and the number of terms may be increased by decomposition to satisfy푛 the≢ statement of the -Straus conjecture [3] if fewer than three are produced. Furthermore, if there exists Erdősat least one divisor | such that 1 (mod 12), we may 1 1 write 4/ as the sum of three unit fractions; then푑 the푛 product 푑 ≢ 푑1 4 4 4 1 4 = = � � 푛 푑1푑2 푑2 푑1 is also the sum of three unit fractions. I was not able to address whether it is generally possible for 4/ to be written as three unit fractions when 1 (mod 12), only to show that such a representation푛 cannot always be obtained by the methods푛 ≡ outlined above. One part of my investigation into Egyptian fractions certainly lay beyond the reach of Egyptian mathematics; I used Euler’s formula for continued fractions [5] and a recursive argument to extend the above methods for decomposing fractions, the proof of which I omit here, leading to the identity when , = 1, = , + 푎푖 푥 ∈ ℝ 푎0 푏푖 푥+푎푖+1 1 = = ∞ 푘 + 푎푖 1 푏0 � �� 푖+1� 1 + 1 푥 푘=0 푖=0 푥 푎 푏 − 1 + 2 1 푏1 + 푏 − 3 2 푏 푏 − 3 푏 − ⋯ which holds when the infinite series converges. Yet only slight abuses of Egyptian notation are necessary to allow for the expression of such modern mathematical ideas. If he were to allow the introduction of infinite processes, the Egyptian scribe would not be far from the study of infinite geometric series, the Riemann zeta function, and other sums of unit fractions central to modern mathematics; in addition, the scribe might adjoin a miniature fraction to the whole number in the denominator of his unit fraction, giving birth to regular continued fractions [2] and brushing up against the theory of irrational numbers. Recognizing the Egyptians’ ingenuity in working with multiplication, division, and operations involving fractions, it seems that many of the arithmetic patterns studied in 5 number theory and other branches of modern mathematics—topics such as finite geometric series, quadratic reciprocity, distribution of primes in arithmetic progressions and Goldbach’s conjecture—were in principle available for investigation by ancient minds. Moreover, certain patterns appear more naturally in one number system or notation than in another; perhaps talented scribes noticed and experimented with such patterns, rediscovering them again and again over the millennia, their unrecorded knowledge evaporating in history. References 1. Joseph, G. G. The Crest of the Peacock: Non-European Roots of Mathematics. Princeton & Oxford: Princeton University Press, 2011. 2. “Continued fraction.” Wikipedia: The Free Encyclopedia. Jan. 31, 2012 <http://en.wikipedia.org/wiki/Continued_fraction>. 3. “Egyptian fraction.” Wikipedia: The Free Encyclopedia. Jan. 31, 2012 <http://en.wikipedia.org/wiki/Egyptian_fraction>. 4. “Egyptian numerals.” Wikipedia: The Free Encyclopedia. Jan. 31, 2012 <http://en.wikipedia.org/wiki/Egyptian_numerals>. 5. “Euler’s continued fraction formula.” Wikipedia: The Free Encyclopedia. Jan. 31, 2012 <http://en.wikipedia.org/wiki/Euler's_continued_fraction_formula>. .
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