To expand search, see Arithmetic. Laterally related topics: Number Systems, Numerology, Magic Squares, Bookkeeping, Modular Arithmetic, Logarithms, The Number Concept, The Abacus, Exponentials, Interpolation, Zero, Fractions, The Real Number System, Irrationals, The Extraction of Roots, Mental Arithmetic, The Negative Numbers, and Imaginary and Complex Numbers.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Anagnostakis, Christopher and Goldstein, Bernard R. On an error in the Babylonian table of Pythagorean triples. Centaurus 18 (1973/74), 64--66. (Reviewer: E. M. Bruins.) SC: 01A15, MR: 58 #20994.
The authors explain a well-known mistake in the Babylonian tablet Plimpton 322 (column I, entry 10) as a consequence of a certain method of computation and of the neglect of a medial zero. It is a very appealing theory, and could give us some insight the way Babylonians did their mathematics. Other solutions have also been proposed. A good example of how we can learn from mistakes! Closely related topics: Sumerians and Babylonians and Pythagorean Triangles and Triples.Modify notes on this entry Modify bibliography entry Make comment on this entry
Bruins, Evert M. Egyptian arithmetic. Janus 68 (1981), no. 1-3, 33--52. (Reviewer: Paul Ernest.) SC: 01A15, MR: 83a:01003.
Discusses the construction of the 2/n table in the Rhind papyrus, using an extensive computer search. Fairly technical. Doesn't give a magical answer, but does apparently discredit some other theories. Might be a topic suitable for some independent study projects. Closely related topic: The Rhind/Ahmes Papyrus.Modify notes on this entry Modify bibliography entry Make comment on this entry
Knuth, Donald E. Ancient Babylonian algorithms. Twenty-fifth anniversary of the Association for Computing Machinery. Comm. ACM 15 (1972), no. 7, 671--677; errata, ibid. 19 (1976), no. 2, 108; MR: 52#13133. SC: 01A15, MR: 52 #13132.
Were there computer scientists among the ancient Babylonians? Probably not. However, some of the ideas in computer science occurred to the ancient Babylonians as well. The author here discusses Babylonian algorithms in particular. Most algorithms are of course given as examples, but Knuth notes one text that is an exception: "Length and width is to be equal to the area. You should proceed as follows. Make two copies of one parameter. Subtract 1. Form the reciprocal. Multiply by the parameter you copied. This gives the width." Knuth explains, "In other words, if x+y=xy, it is possible to compute y by the procedure y=(x-1)-1x. The fact that no numbers are given made this passage particularly hard to decipher, and it was not properly understood for many years; hence we can see the advantages of numerical examples. The above procedure reads surprisingly like a program for a 'stack' machine like the Burroughs B5500!". Knuth finds a table involving compound interest where he finds evidence of a "DO I = 1 TO N" loop and something like a "WHILE" clause. He also discusses how one tablet may have been obtained by sorting a large set of numbers. "Thus, Inakibit seems to have the distinction of being the first man in history to solve a computational problem that takes longer than one second of time on a modern electronic computer!" [However, note that this statement was made in 1972.] Some tablets cited are available here in English for the first time (Knuth translated them using German and French translations, and at times Akkadian and Sumerian vocabularies as well). See errata in Knuth, Donald E., Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Closely related topics: Sumerians and Babylonians, Computation, and Logarithms.Modify notes on this entry Modify bibliography entry Make comment on this entry
Knuth, Donald E. Errata: "Ancient Babylonian algorithms" (Comm. ACM 15 (1972), no. 7, 671--677). Comm. ACM 19 (1976), no. 2, 108. SC: 01A15, MR: 52 #13133.
An errata to Knuth, Donald E., Ancient Babylonian algorithms. The table that was sorted was not as extensive as Knuth previously believed, and involved a "file" of about 500 instead of about 800. As Knuth notes "My italicized statement on p. 676 that 'this table contains every one' of the 231 regular sexagesimal numbers of six digits or less, is false; the table contains only 136 of those 231." The misunderstanding was due to a failure "to read the accompanying German commentary carefully enough, since [Neugebauer] departed from his usual custom in this particular case. Many of the lines in his rendition of the table were not on the original clay tablet at all, they were interpolated to show what the tablet would have looked like if it had been complete." Closely related topics: Sumerians and Babylonians, Computation, and Logarithms.Modify notes on this entry Modify bibliography entry Make comment on this entry
Lambert, Joseph B.; Ownbey-McLaughlin, Barbara and McLaughlin, Charles D. Maya arithmetic. Amer. Sci. 68 (1980), no. 3, 249--255. (Reviewer: M. P. Closs.) SC: 01A10, MR: 82f:01002.
As the authors explain, there is no real evidence that the Mayas used either multiplication or division. However, the authors discuss how the Mayan notation would suit itself naturally to both of these operations. The authors suggest that in the Mayan calendrical system, where multiples of 360 are used in the third place instead of multiples of 202=400, it might be easier to "multiply by converting calendrical figures to vigesimal, perform the operations, and reconvert the answer to calendrical notation." The authors don't discuss a simple way to do this conversion. Closely related topic: The Maya.Modify notes on this entry Modify bibliography entry Make comment on this entry
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