The Cosmic Game ( PDFDrive.com )

Cosmic Game © Douglass A. White, 2012 v151207 225
used a more general form of this expansion, n/pq = 1/pr + 1/qr, where r =(p +
q)/n, which works when p + q is a multiple of n (Eves 1953).
• For some other composite denominators, the expansion for 2/pq has the form of
an expansion for 2/q with each denominator multiplied by p. For instance,
95=5×19, and 2/19 = 1/12 + 1/76 + 1/114 (as can be found using the method for
primes with A = 12), so 2/95 = 1/(5×12) + 1/(5×76) + 1/(5×114) = 1/60 + 1/380 +
1/570 (Eves 1953). This expression can be simplified as 1/380 + 1/570 = 1/228
but the Rhind papyrus uses the unsimplified form.
• The final (prime) expansion in the Rhind papyrus, 2/101, does not fit any of these
forms, but instead uses an expansion 2/p = 1/p + 1/2p + 1/3p + 1/6p that may be
applied regardless of the value of p. That is, 2/101 = 1/101 + 1/202 + 1/303 +
1/606. A related expansion was also used in the Egyptian Mathematical Leather
Roll for several cases.
Appendix C: Finding Cube Roots With an Electronic Calculator
(From Wikipedia, "Cube root")
"There is a simple method to compute cube roots using a non-scientific calculator, using
only the multiplication and square root buttons, after the number is on the display. No
memory is required.
• Press the square root button once. (Note that the last step will take care of this
strange start.)
• Press the multiplication button.
• Press the square root button twice.
• Press the multiplication button.
• Press the square root button four times.
• Press the multiplication button.
• Press the square root button eight times.
• Press the multiplication button...
This process continues until the number does not change after pressing the multiplication
button because the repeated square root gives 1 (this means that the solution has been
figured to as many significant digits as the calculator can handle). Then:
• Press the square root button one last time.
At this point an approximation of the cube root of the original number will be shown in
the display.
If the first multiplication is replaced by division, instead of the cube root, the fifth root
will be shown on the display.
Why this method works
After raising x to the power in both sides of the above identity, one obtains: