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Egyptian mathematics

Egyptian mathematics refers to the style and methods of mathematics performed by scribes in Ancient Egypt, deriving in large part from the rare discoveries of ancient papyri: in particular, the Rhind Mathematical Papyrus, dating from the Second Intermediate Period or around 1650 BC (though it is a copy of a now lost Middle Kingdom papyrus), and the Moscow Mathematical Papyrus from around 1850 BC, both of which appear to be ancient mathematics textbooks.

As early as 2450 BC ancient Egyptian mathematicians derived the earliest known systematic method for the approximative calculation of the circle on the basis of the "sacred" 3-4-5 triangle. Additionally, the Rhind Mathematical Papyrus preserves the first known aproximate value of π (at 3.16), the earliest known attempt at squaring the circle, and the earliest known use of arithemetic and geometric series to arrive at mathematical solutions. The Rhind Mathematical Papyrus, written by the scribe Ahmes, is allegedly a copy of a lost scroll dating around 1850 BC. Apparently, Egyptian mathematicians were millennia ahead of the rest of the world.

Furthermore, quoting Mathpages.com,

... the 2/n table of the Rhind Papyrus, which dates from more than a thousand years before Pythagoras, seems to show an awareness of prime and composite numbers, a crude version of the 'Sieve of Eratosthenes,' a knowledge of the arithmetic, geometric, and harmonic means, and of the 'perfectness' of the number 6. This all seems to suggest a greater number-theoretic sophistication than is generally credited to the ancient Egyptians. (The Rhind Papyrus 2/N Table)

Moreover, the Berlin Papyrus shows that the ancient Egyptians knew how to solve 2nd order algebraic equations around 1800 BC [1].

In the few documents that have survived over the millennia, the mathematical problems we see are mainly confined to applications of arithmetic in practical contexts. For example, many problems address how a number of loaves can be divided equally between a number of men. Based upon surviving records, most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. The problems in the Moscow and Rhind Mathematical Papyri are expressed in a practical context and seem to be exercises to practice the techniques for the manipulation of numbers.

Egyptian methods demonstrated in the few documents we have principally employed the method of doubling and halving a known number to approach the solution, together with the method of false position. Allied with their decimal number systems, unit fractions, and tables of common results (such as the decomposition of non-unit fractions, such as 2/n) into unit fractions, these methods allowed complex manipulation of problems.


Contents

Numerals

Main article: Egyptian numerals

The numeral system employed in the documents handed down to us was a decimal system, written in hieroglyphs and hieratic. Both systems existed from at least the Early Dynastic Period. (It should be noted that the hieratic system does not differ from the hieroglyphic system beyond a use of simplifying litagures for rapid writing.)

The Rhind Mathematical Papyrus is written in hieratic, and contains many examples of how ancient Egyptians did their mathematical calculations.

Multiplication

Egyptian multiplication employed in these few documents was done by repeated doubling of the number to be multiplied (the multiplicand), and choosing which of the doublings to add together (essentially a form of binary arithmetic). The multiplicand would be written out next to the figure 1, then the multiplicand would be added to itself (i.e. doubled) and would be written out next to the number 2, and so on, until the doublings gave a number greater than the number to be multiplied by (the multiplier). Then, the doubled numbers (1, 2, etc.) would be repeatedly subtracted from the multiplier to select which of the results of the existing calculations should be added together to create the answer.

So, for example, Problem 69 on the Rhind Papyrus provides the following illustration:

To multiply 80 × 14
Egyptian calculation Modern calculation
Result Multiplier Result Multiplier
<hiero>V20*V20*V20*V20:V20*V20*V20*V20</hiero> <hiero>Z1</hiero> 80 1
<hiero>V1*V1*V1*V1:V1*V1*V1*V1</hiero> <hiero>V20</hiero> / 800 10
<hiero>V20*V20*V20:V20*V20*V20-V1</hiero> <hiero>Z1*Z1</hiero> 160 2
<hiero>V20:V20-V1*V1:V1</hiero> <hiero>Z1*Z1*Z1*Z1</hiero> / 320 4
<hiero>V20:V20-V1-M12</hiero> [= hiero] 1120 14

The tick mark (/) denotes the intermediate results that are added together to produce the final answer.

Fractions

Main article: Egyptian fraction

Rational numbers could also be expressed, but only as sums of unit fractions, i.e. sums of reciprocals of positive integers, except for 2/3 and (rarely) 3/4. The hieroglyph indicating a fraction looked like a mouth, which meant "part", and fractions were written with this fractional solidus, i.e. the numerator 1, and the positive denominator below. Special symbols were used for for 1/2 and for two non-unit fractions, 2/3 (used frequently) and 3/4 (used less frequently).

Problem 25 on the Rhind Papyrus uses the method of false position to solve the problem "a quantity and its half added together become 16; what is the quantity?" (i.e., in modern algebraic notation, what is x if xx=16).

Assume 2

       1 2 /
       ½ 1 /
Total 1½ 3

As many times as 3 must be to give 16, so many times must 2 be multiplied to give the answer.

     1      3 /
     2      6
     4     12 /
     2/3    2
     1/3    1 /

Total 5 1/3 16

So:

 1   5 1/3 (1 + 4 + 1/3)
 2  10 2/3

The answer is 10 2/3.

Check -

     1   10 2/3
     ½    5 1/3

Total 1½ 16

Problem 31 sets the problem "q quantity, its 1/3, its 1/2 and its 1/7, added together, become 33; what is the quantity?" (i.e. what is x if x + 1/3 x + 1/2 x + 1/7 x =33), giving the answer 14 1/4 1/56 1/97 1/194 1/388 1/679 1/776 (i.e. 14 28/97).

Geometry

Problem 50 uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter (so 1/9 is subtracted from the diameter, and the resulting figure is multiplied by itself, using the doubling method). In essence, this assumes that π is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent) and was not surpassed until Archimedes (211875/67441 = 3.14163, an error of just over 1 in 10,000).

See also

External links

  • http://www-gap.dcs.st-and.ac.uk/~history/Indexes/Egyptians.html
  • http://scitsc.wlv.ac.uk/university/scit/modules/mm2217/en.htm

Further reading

  • Chace, Arnold Buffum. 1927–1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. 2 vols. Classics in Mathematics Education 8. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0873531337
  • Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0871692325
  • Couchoud, Sylvia. 1993. Mathématiques égyptiennes: Recherches sur les connaissances mathématiques de l'Égypte pharaonique. Paris: Éditions Le Léopard d'Or
  • Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited
  • Robins, R. Gay. 1995. "Mathematics, Astronomy, and Calendars in Pharaonic Egypt". In Civilizations of the Ancient Near East, edited by Jack M. Sasson, John R. Baines, Gary Beckman, and Karen S. Rubinson. Vol. 3 of 4 vols. New York: Charles Schribner's Sons. (Reprinted Peabody: Hendrickson Publishers, 2000). 1799–1813
  • Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0714109444
  • Struve, Vasilij Vasil'evič, and Boris Aleksandrovič Turaev. 1930. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer
Last updated: 08-12-2005 10:45:41
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