Mathematics in Ancient Egypt. Amber Hedgpeth. June 8, 2017

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1 Mathematics in Ancient Egypt Amber Hedgpeth June 8, 2017 The culture of ancient Egypt is rich and fascinating, with its pharaohs, pyramids, and life around the Nile River. With a rich history of massive constructions and perfectly-timed agriculture, ancient Egyptians logically required some number sense. The mathematical abilities of ancient Egyptians are not as complex as one may think. It turns out that the ancient Egyptians only truly understood addition and subtraction, yet they produced feats of engineering, plentiful crops, and solutions to problems that would not be solved again for thousands of years. Egyptians did not study math for academic purposes but to solve the problems encountered in their everyday lives. Most of the evidence we have of early Egyptian mathematics lies in three surviving math texts of the time: the Moscow Papyrus, the Rhind Papyrus, and the Berlin Papyrus. Every school-aged child knows that the Egyptians used papyri for writing. Unfortunately, the plant-based paper was unlike the clay tablets of other ancient cultures in that it did not survive the elements as readily. Therefore, only a partial record of Egyptian mathematics has survived in papyri. An additional bit of mathematics, particularly one Egyptian method of counting, managed to survive by becoming a part of wall paintings. They Egyptian hieroglyphic counting system only included symbols for powers of 10, and numbers were written additively. These symbols were primarily used in wall paintings. The figure on the left below shows the symbols for the powers of 10 through one million. Egyptians probably did not frequently use higher powers of 10. The figure on the right below gives an example of how a number like 4622 would be written in Egyptian Hieroglyphics.

2 The symbol for 10 is a heel bone, 100 is a coil of rope, 1,000 is a lotus flower, 10,000 is a finger, 100,000 is a tadpole, and 1,000,000 is represented by a man holding his hands in the air. On papyri, Egyptians more frequently used hieratic symbols, which included separate notation for single digits, multiples of 10, multiples of 100, and multiples of Because the symbols for each place value were unique, the symbols did not have to be written in order.

3 and both represent The Moscow Papyrus dates to approximately 1700 B.C. and is 15 feet long, 3 inches wide, and contains about 25 mathematical problems. It is known as the Moscow Papyrus because it is housed at the Moscow Museum of Fine Art to whom it was sold after purchase by V. S. Golenishchev in The papyrus is sometimes known by the buyer s name. The most impressive problem found on the Moscow Papyrus is one describing the formula for finding the volume of a frustum. The derivation of this formula depends on calculus methods, but was obviously known to the ancient Egyptians. The 14 th problem on the papyrus translates: Volume of a frustum. The scribe directs one to square the numbers two and four and to add to the sum of these squares the product of two and four. Multiply this by one third of six. "See, it is 56; your [sic] have found it correctly." What the student has been directed to compute is the number 1 This formula is identical to the general formula for the volume of a frustum. The figure on the next page shows the original problem as shown in the Moscow Papyrus. 1 The Moscow Papyrus. [Online] [Cited: June 8, 2017.]

4 The Rhind Papyrus, sometimes called the Ahmes Papyrus, is named for A. Henry Rhind, a Scottish lawyer who purchased it in Egypt in The papyrus was willed to the British Museum, where it is still housed. In addition to symbols and tables, the papyrus includes 84 mathematical problems with their solutions. Copied in the thirty-third year of the Pharaoh Apophis by a scribe named Ahmose, it presents dozens of math questions and their answers, which lend some insight into the kinds of problems Egyptians could solve. 2 The Rhind Papyrus is larger than the older Moscow papyrus, measuring about 18 feet long and 12 to 13 inches wide. In addition to detailing methods of multiplication and division (accomplished by addition and doubling), the Rhind Papyrus shows how Egyptians dealt with fractions. They did not use fractions with numerators other than 1, except for the special fraction 2/3, and so every fractional number must have an equivalent expression as a sum of fractions with numerator 1. Tables of these equivalents appear on the Rhind Papyrus. Fractions were written in hieroglyphics by placing the hieroglyph meaning part, represented by an open mouth, over the hieroglyph for the number. 2 Brier, B. and Hobbs, H. (2009) Ancient Egypt: Everyday Life in the Land of the Nile. Santa Barbara, CA: Sterling Publishing.

5 Examples are shown in the figure below. Special symbols for some fractions are given, as well. The symbols are all parts of a larger hieroglyph, the symbol for well-being, which is represented by the falcon eye of Horus. The figure below shows the full hieroglyph, highlighting its parts, and it also gives the Egyptian meaning behind each part of the symbol. The 84 problems of the Rhind Papyrus deal with methods for dividing loaves of bread among men, relative values of precious metals, and the amount of bread needed to force-feed geese, and so we see

6 why the treatment of fractions in the same text would have been important. Also, though we have already mentioned that Egyptian mathematics was focused on the practical, the Rhind Papyrus includes some problems that are recreational in nature, including one akin to the Kits, cats, sacks, and wives, how many were going to St. Ives? trick problem known to many. Evidently, a few playful scribes enjoyed solving math problems for fun! The third mathematical papyrus is Berlin Papyrus 6619, or simply the Berlin Papyrus. This papyrus dates to around B.C. Part of a larger collection of papyri including literature and medical discoveries, the Berlin Papyrus gives us the only evidence that the ancient Egyptians may have understood the Pythagorean relation, long before the Pythagoreans. The possibly coincidental evidence comes from a problem that amounts to finding the side of a square whose area, along with that of a square with ¾ its side-length, sums to a square of area This problem gives solution (6,8,10), a multiple of the most basic Pythagorean triple (3,4,5). The Berlin Papyrus also shows that ancient Egyptians were capable of solving some quadratic equations. A piece of the Berlin Papyrus is shown below. 3 Kaplan, R. and Kaplan E. (2011) Hidden Harmonies: The Lives and Times of the Pythagorean Theorem. New York, NY: Bloomsbury Press.

7 While the Greeks are typically given the most credit for developing modern mathematics, a study of Egyptian mathematics reveals their importance in the development of number sense, written mathematical communication, a base-ten number system, and standardized measurement. Some wall paintings and three important papyri, now housed in museums, are all we have left of the mathematics of a culture of ancient engineers and scientists. Students use Egyptian methods of calculation for fun, and the internal workings of modern day computers use the same methods of multiplication and division as the Egyptians. Thus, we see that the basic principles of mathematics have been understood by men (and women) for thousands of years, and we are left to wonder what other discoveries the Egyptians made long before those to whom they are credited.

8 Bibliography Ancient Egyptian Multiplication, Division, Root Extraction. [Online] [Cited: June 8, 2017.] Ancient Library: (Society for the Promotion of the Egyptian Museum Berlin). [Online] [Cited: June 8, 2017.] A piece of the Berlin Paprus THE MATHEMATICAL TOURIST. [Online] [Cited: June 8, 2017.] Brier, B. and Hobbs, H. (2009) Ancient Egypt: Everyday Life in the Land of the Nile. Santa Barbara, CA: Sterling Publishing. Egyptian Mathematics. [Online] [Cited: June 8, 2017.] Egyptian Numerals. [Online] [Cited: June 8, 2017.] Hieroglyphics: Egyptian, Mayan, and Chinese Characters. [Online] [Cited: June 8, 2017.] Kaplan, R. and Kaplan E. (2011) Hidden Harmonies: The Lives and Times of the Pythagorean Theorem. New York, NY: Bloomsbury Press. mothnrust. (2008, September 5). Egyptian Maths [Video File]. Retrieved from Reimer, D. (2014) Count Like an Egyptian: A Hands-On Introduction to Ancient Mathematics. Princeton, NJ: Princeton University Press. Rudman, P.S. (2007) How Mathematics Happened: The First 50,000 Years. Amherst, NY: Prometheus Books. The Moscow Papyrus. [Online] [Cited: June 8, 2017.] The RHIND PAPYRUS. [Online] [Cited: June 8, 2017.]

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