# Mathematics, Very Old

# Mathematics, Very Old

From the dawn of civilization, humankind has needed to count and measure. Even the earliest civilizations developed effective and efficient number systems. Ancient mathematics is surprisingly sophisticated and, in many cases, quite similar to the mathematics used today.

## Tally Bones

An early evidence of a "mathematical system" was found on a bone discovered in the Czech Republic. It dates from about 30,000 b.c.e. The bone contains fifty-five individual tally marks, divided into eleven groups of five marks each, just as tally marks might be grouped today. There is a dividing line, separating the first twenty-five marks from the remaining twenty, that makes totaling the tally marks even easier. No one knows just what the bone's owner was counting, but it may have been domestic animals such as sheep or some type of game animal.

Another ancient tally bone is the Ishango Bone, found in the Democratic Republic of the Congo. The Ishango Bone dates from 9000 b.c.e. to 6500 b.c.e. It is a bone tool handle, and it also contains tally marks that were probably used to keep a record of domestic items, perhaps sheep or cattle.

## Early Number Systems

It is a short conceptual leap from a tally system to a number system. Early humans depended on body parts to keep an accurate count. Many civilizations developed a **base-10** system, which used ten fingers as a basis for counting. A base-10 system is still used today. Some ancient cultures included fingers and toes in a **base-20** system. The word *score* (twenty) is derived from the base-20 number system.

The first concrete evidence of a numerical system comes from a ceremonial Egyptian weapon dating from King Menes (3000 b.c.e.). It contains hieroglyphics that described plunder taken by the king. One of the hieroglyphic figures is probably exaggerated, but it lists 1,422,000 oxen as part of the spoils of victory. Whether the mace head hieroglyphics give an accurate account or not is unimportant. What is significant about the hieroglyphic numbers is that more than 5,000 years ago the Egyptians could comprehend and represent extremely large numbers.

The Egyptian number system was additive. That is, each multiple of ten had its own symbol. The number of oxen, 1,422,000, was represented by a single symbol for 1,000,000, four symbols for 100,000, two symbols for 10,000, and two symbols for 1,000. In order to understand the hieroglyphic number, it was necessary to find the sum of all the symbols: (1 × 1,000,000) + (4 × 100,000) + (2 × 10,000) + (2 × 1,000). Today's number system is positional. A number such as 345 is understood as (3 × 100) + (4 × 10) + (5 × 1). In the number 635, the number 3 now has the value of 3 × 10. There are only ten digits in today's system, and no number up to 9,999,999 requires more than seven digits. The Egyptian system required many more symbols to represent numbers. In the Egyptian system, a number like 45 would require four 10-symbols and five 1-symbols. Still, the Egyptian number system worked so well that there were essentially no changes for 3,000 years after its invention.

Many other number systems that followed the Egyptian number system were also additive, including the Roman numerals that are still used today. Several other early number systems, such as those developed by the Mayan, Incan, and Babylonian civilizations, rival, and in some cases surpass, the number system of today.

The Babylonians refined the number system that had been used by the Sumerians. The Sumerians left no written record to describe their number system, but it appears that the Babylonians adopted it without much alteration. By approximately 2500 b.c.e., the Babylonians were using a positional number system that is similar to the number system used today. The Baby-lonians used a **base-60** system, with the value of a symbol dependent on its position in the number. For example, using modern symbols in the Babylonian system, the number 632 would represent (6 × 60^{2}) + (3 × 60^{1}) + (2 × 60^{0}), or (6 × 3600) + (3 × 60) + (2 × 1) = 21,782 in our base-10 system. The Babylonian system continued to develop, and by about 500 b.c.e., clay tablets show numbers written with a symbol that is used as a zero, the first use of zero in a positional system.

The Babylonians also used base-60 for their fractions. By using base-60 for fractions and whole numbers, the Babylonians avoided much of the computation problems associated with fractions in today's number system. The following fractions may be represented with the same denominator of 60 . Thus, the task of finding a common denominator was essentially eliminated in the Babylonian system. The Babylonians also extended the base-60 to units of weight, distance, and time. Today's sixty-minute hour and sixty-second minute are inherited from the Babylonians.

About 100 c.e. the Maya of Central America developed a number system that was positional like the Babylonian system. The Maya used base-20 instead of base-60. The Maya were also the first people to make full use of zero in their number system.

About 1,300 years later, the Incas of South America invented a recordkeeping system that used a base-10 positional system and used zero. The Incas kept records on a series of knotted cords called *quipu.* The knots on the quipu cord indicated the digit, and the position of the cord indicated the value of the digit, much like today's number system. A missing cord was used to show zero.

## Early Texts

The Babylonians left behind records about their mathematics. The Babylonians wrote on clay tablets in a simple script called cuneiform. One of the

earliest clay tablets, called Plimpton 322, shows a table of whole number values that fit the Pythagorean theorem. Plimpton 322 dates from 1900 b.c.e. to 1600 b.c.e.

Other clay tablets contain what appear to be mathematics lessons. Among the problems found on the clay tablets are **quadratic equations** in the form (in modern notation) *x* ^{2} + 6*x* = 16, and problems that may be represented as cubic equations. The clay tablets show that the Babylonians could solve systems of equations such as .

Other tablets reveal that the Babylonians used 3.125 for the value of π, quite close to the present approximation of 3.1416. Some tablets contain tables of cube roots and square roots with great accuracy. The value of , for example, is correct to six places after the decimal.

**Rhind Papyrus.** While Babylonian mathematicians were inscribing cuneiform mathematics onto their clay tablets, Egyptian mathematicians were writing hieroglyphic mathematics on papyrus. One of the best sources of Egyptian mathematics is the Rhind Papyrus, named for archeologist Henry Rhind. It is 18 feet long and 13 inches wide and dates from 1650 b.c.e. It was written by copyist Ahmes, who claimed to be copying problems that had been known to Egyptians for at least 200 years. Thus, the eighty-five problems of the Rhind Papyrus date from about 2000 b.c.e. The Rhind Papyrus is the earliest arithmetic text ever written, and it is essentially a handbook of mathematics exercises.

The problems and solutions do not break any theoretical ground. The solutions are given because they work, and not because they are justified or logically shown to be correct. Some problems in the Rhind Papyrus, written in modern notation, are 100 (7 + ½ + ¼ + ⅛) = (*x, x* + 8*x* = 45, *x* + , and ⅓(1 + ⅓ + ¼) = *x*.

Although the problems are based on practical mathematics, the specific numbers are unlikely to be found in the everyday life of ancient Egypt, as this problem shows: Divide twenty-three loaves of bread among seventeen men. The Rhind Papyrus also contains formulas for the volumes of cylinders and prisms. Problems involving circles give the value of *π* as 3.1604, a value that is astonishingly close to the modern approximation of 3.1416.

**Moscow Papyrus.** Another source of Egyptian mathematics is the Moscow Papyrus, named for the Museum of Fine Arts in Moscow, where it is on display. It is much smaller than the Rhind Papyrus, at 18 feet long and 3 inches wide. The Moscow Papyrus dates from about 1850 b.c.e. and contains thirty exercise problems, written like those in the Rhind Papyrus. The problems were "already old" according to the copyist. One problem shows the formula for the area of a trapezoid, A = ½(*b* _{1} + *b* _{2}) *h*, which is identical to the formula used today. Another problem gives the formula for the area of a parallelogram in terms of its four side lengths: A = ¼(*a* + *c* )(*b* + *d* ). This formula, however, works only if the parallelogram is very close in shape to a rectangle. It is easy to show that the formula is incorrect for extreme parallelogram shapes, but the goal for this papyrus is the same as for the Rhind Papyrus: to provide exercises, not theoretical mathematics.

The Moscow Papyrus contains one problem that asks for the volume of the frustum of a pyramid. A frustum is the shape that is left if the top of a pyramid is cut by a plane parallel to the base. Math historians cannot explain how the Egyptians found this formula. The mathematics needed to derive the formula were not in existence until about 300 b.c.e., when Greek mathematicians rediscovered it. The only possible explanation is that, through a series of many problems, the Egyptians discovered an equation that gave the correct solution. Although the formula holds for all frustums, there was no attempt by the copyist to generalize it.

**Other Papyrus Texts.** Other mathematics papyrus texts include the Egyptian Mathematical Leather Roll, which contains twenty-six sums of unit fractions; the Kahun Papyrus, which contains six problems not fully deciphered; the Berlin Papyrus, which contains two problems that employ simultaneous equations; and the Reisner Papyrus, which contains volume calculations.

## Practical Mathematics

The Egyptians developed practical **geometry** , necessitated by the Nile river floods every spring. The floodwaters erased all boundary markers, and when the waters receded, boundary lines had to be re-established. From at least 2500 b.c.e., the Egyptians used a special case of the **Pythagorean theorem** to set out the boundary lines. They used cords marked off by knots into twelve equal lengths. Then, the rope could be shaped into a right triangle with sides of the proportion 3-4-5. Two such triangles form a rectangle, and by replicating triangles and rectangles along the shores of the Nile, the Egyptians could reset the property lines.

In 2750 b.c.e. the Egyptians built a pyramid at Saqqara, Egypt. Inscriptions on the pyramid indicate that the builders used rectangular coordinates to erect the foundation of the pyramid. The Egyptian coordinate system is fundamentally the same system that is used today. The modern coordinate system was discovered in the mid-seventeenth century, some 4,000 years after the Egyptian system.

Another pyramid, the Great Pyramid, dates from about the same time period. The Great Pyramid demonstrates the practical mathematics of the Egyptians. The four base edges each measure within 4.5 inches of the average length of 751 feet. Each triangular face of the pyramid faces one of the cardinal compass points to within a tenth of a degree. Such precision is difficult to comprehend but nevertheless testifies to the practical development

of mathematics that the Egyptians achieved.

see also Number System, Real.

*Arthur V. Johnson II*

## Bibliography

Boyer, Carl B. *History of Mathematics.* New York: John Wiley & Sons, Inc., 1968.

Bunt, L., P. Jones, and J. Bedient. *The Historical Roots of Elementary Mathematics.* New York: Dover Publications, Inc., 1976.

Burton, David. *The History of Mathematics: An Introduction.* New York: Allyn and Bacon, 1985.

Chace, A. *The Rhind Mathematical Papyrus.* Reston, VA: National Council of Teachers of Mathematics, 1979.

Gillings, R. *Mathematics in the Time of the Pharaohs.* New York: Dover Publications, Inc., 1972.

McLeish, J. *The Story of Numbers: How Mathematics Shaped Civilization.* New York: Fawcett Columbine, 1991.

Resnikoff, H. and R. O. Wells, Jr. *Mathematics in Civilization.* New York: Dover Publications, Inc., 1976.

Robbins, G., and C. Shute. *The Rhind Mathematical Papyrus: An Ancient Egyptian Text.* New York: Dover Publications, Inc., 1987.

## HOW OLD IS THE OLDEST?

The oldest "mathematical artifact" currently known is a piece of baboon fibula with twenty-nine notches in it. It was discovered in the mountains between South Africa and Swaziland and dates from 35,000 b.c.e.

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