Page:The Rhind Mathematical Papyrus, Volume I.pdf/29

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DIVISION OF 2 BY ODD NUMBERS

13

namely, by 1+²⁄₃. The true smallest term will then be 1+²⁄₃ and we shall have the true division of loaves

1+²⁄₃10+²⁄₃ ¹⁄₆2029+¹⁄₆38+¹⁄₃.

This problem is notable because, while the Egyptian mathematician did not have such a thing as simultaneous equations, yet by methods which were within his knowledge he could sometimes obtain the result when there were two unknown quantities, as illustrated here.

The process of false position was employed by Diophantus and by Arabic writers, and has continued in use even down to our own day, being found in older arithmetics;^[1] it was probably dropped from use about the time that algebra began to be generally taught in our schools.

3. The third of the special processes was a process of completion, used for determining the amount to be added when we have very nearly a given number. It was used especially in the second kind of multiplication as explained on page 5. Problems 21-23 are given as problems in completion and show the method of solving such problems. Thus in Problem 21 we have to complete ²⁄₃ ¹⁄₁₅ to 1. To determine the answer, these fractions are applied to 15. ²⁄₃ ¹⁄₁₅ of 15 make 11 and require 4 more to make the whole of 15. 4 is the same as ¹⁄₅ ¹⁄₁₅ of 15, and therefore ¹⁄₅ ¹⁄₁₅ is what is required to complete the given fractions to 1. The three problems are all solved in this way.^[2]

I may add that there is another group of problems, 7–20, before which the author puts the words, “Example of making complete,” but probably by mistake as these problems are all simple multiplications. See page 23, footnote 2.

Table of the Division of 2 by Odd Numbers

Inasmuch as the Egyptian mathematician performed his multiplications mostly by doubling or halving it was necessary that he should be able to double any numerical quantity, a reciprocal as well as a whole number. This could easily be done with the reciprocal of an even number, but for odd numbers it was convenient to have a special table. To determine the double of a reciprocal number was the same as

↑ See, for example, Benjamin Greenleaf, The National Arithmetic, revised, Boston, 1853, page 286. He calls it, “Single Position.”
↑ In the formation of the table for the division of 2 by odd numbers completion is one of the processes for all but three of the numbers. In simple cases the completion may not have required elaborate calculations, but whenever the author did not know at once what were the fractions required, the procedure illustrated in Problems 21-23 would be used. In six cases the word “remainder” is put in, indicating the result of a completion process. These are for 2 divided by 17, 19, 23, 37, 41, and S3. Completion is also used in Problems 30–34.

[1] See, for example, Benjamin Greenleaf, The National Arithmetic, revised, Boston, 1853, page 286. He calls it, “Single Position.”

[2] In the formation of the table for the division of 2 by odd numbers completion is one of the processes for all but three of the numbers. In simple cases the completion may not have required elaborate calculations, but whenever the author did not know at once what were the fractions required, the procedure illustrated in Problems 21-23 would be used. In six cases the word “remainder” is put in, indicating the result of a completion process. These are for 2 divided by 17, 19, 23, 37, 41, and S3. Completion is also used in Problems 30–34.

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