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Cosmic Game © Douglass A. White, 2012 v151207 224
more accessible to our modern view. It turns out the Egyptians also sometimes used
"vulgar" fractions (with greater than unity numerators) during their calculations but then
put the result in unit fractions.
Using the same process they could find the square root of 3.16227766, and then the
square root of that root, and so on until they had their list of the roots of 10.
Appendix B: Methods for Calculating Egyptian Unit Fractions
(from Wikipedia, "Egyptian fractions")
Modern historians of mathematics have studied the Rhind papyrus and other ancient
sources in an attempt to discover the methods the Egyptians used in calculating with
Egyptian fractions. In particular, study in this area has concentrated on understanding the
tables of expansions for numbers of the form 2/n in the Rhind papyrus. Although these
expansions can generally be described as algebraic identities, the methods used by the
Egyptians may not correspond directly to these identities. Additionally, the expansions in
the table do not match any single identity; rather, different identities match the
expansions for prime and for composite denominators, and more than one identity fits
the numbers of each type:
• For small odd prime denominators p, the expansion 2/p = 2/(p + 1) + 2/p(p + 1)
was used.
• For larger prime denominators, an expansion of the form 2/p = 1/A + (2A − p)/Ap
was used, where A is a number with many divisors (such as a practical number)
between p/2 and p. The remaining term (2A − p)/Ap was expanded by
representing the number (2A − p)/Ap as a sum of divisors of A and forming a
fraction d/Ap for each such divisor d in this sum (Hultsch 1895, Bruins 1957). As
an example, Ahmes' expansion 1/24 + 1/111 + 1/296 for 2/37 fits this pattern with
A = 24 and (2A − p)/Ap = 11 = 3 + 8, as 1/24 + 1/111 + 1/296 = 1/24 + 3/(24 × 37)
+ 8/(24 × 37). There may be many different expansions of this type for a given p;
however, as K. S. Brown observed, the expansion chosen by the Egyptians was
often the one that caused the largest denominator to be as small as possible,
among all expansions fitting this pattern.
• For composite denominators, factored as p×q, one can expand 2/pq using the
identity 2/pq = 1/aq + 1/apq, where a = (p+1)/2. For instance, applying this
method for pq = 21 gives p = 3, q = 7, and a = (3+1)/2 = 2, producing the
expansion 2/21 = 1/14 + 1/42 from the Rhind papyrus. Some authors have
preferred to write this expansion as 2/A × A/pq, where A = p+1 (Gardner 2002);
replacing the second term of this product by p/pq + 1/pq, applying the distributive
law to the product, and simplifying leads to an expression equivalent to the first
expansion described here. This method appears to have been used for many of the
composite numbers in the Rhind papyrus (Gillings 1982, Gardner 2002), but there
are exceptions, notably 2/35, 2/91, and 2/95 (Knorr 1982).
• One can also expand 2/pq as 1/pr + 1/qr, where r = (p+q)/2. For instance, Ahmes
expands 2/35 = 1/30 + 1/42, where p = 5, q = 7, and r = (5+7)/2 = 6. Later scribes