The earliest records of advanced, organized mathematics (beyond just counting) date back to the ancient Mesopotamian country of Babylonia and to Egypt of the fourth millennium b.c.. The Egyptian civilization, one of the world's oldest, developed in the valley of the Nile. Its study and use of mathematics developed only in response to the practical needs within agriculture, business, and industry. Specifically, Egyptian mathematics was primarily arithmetic, with an emphasis on measurement, surveying, and calculation in geometry. Its use was elementary, with no trace of the necessity of later abstract mathematical concepts such as proofs and axioms. This mathematics was generally arrived at by trial and error as the way to obtain desired results. However, no records exist that show how the Egyptians reached their conclusions. It is true, though, that they possessed advanced mathematical knowledge, for without it their accomplishments in engineering, astronomy, and administration would not have been possible.
Some of our knowledge of ancient Egyptian mathematics first appeared from the sacred carvings in the temples called hieroglyphics. Occurring as far back as 3400 b.c., these inscriptions of a simple grouping system were made primarily on stone. Later a more complex number system called hieratic evolved from hieroglyphics. Hieratic developed on papyrus, wood, and pottery as a method to more rapidly write. From hieratic developed the demotic numeral system that was eventually adopted for general use. Another important source of mathematical knowledge about the Egyptians came from texts that were composed about 1800 b.c. and written by a class of literate professionals, called scribes. Because of their reading, writing, and mathematical skills, the scribes performed the duties of modern-day civil servants: that of tax accounting, record keeping, management of public works, and overseeing military payrolls and supplies. However, only a few of these original documents survived. Two papyrus documents that once served as scribal textbooks were discovered intact and contain collections of mathematical problems along with their solutions and a decimal numeration system with separate symbols for the successive powers of 10 (1, 10, 100, etc.). The first text, the Rhind Mathematical Papyrus (also called the Ahmes Papyrus), was named after Scottish Egyptologist A. Henry Rhind who purchased it in Luxor in 1858. It was written around 1650 b.c. by the scribe Ahmes from another text that was two centuries older. The Rhind discussed unit fractions, presented solutions of 84 specific problems in arithmetic and geometry, and contained a table of representations of "2/n" as a sum of distinct unit fractions for odd "n" between 5 and 101. The second text was called the Moscow Mathematical Papyrus, also known as the Golenishchev Papyrus, and was purchased by V. S. Golenishchev. Its origin was from the nineteenth century b.c. and presented 25 problems involving the distribution of beer and bread as wages, and how to calculate the areas of fields and the volumes of pyramids and other solids.
In Egyptian mathematics the symbol for 1 "vertical stroke" was written five times to represent the number 5; the symbol for 10 "heal bone (or 'hobble' for cattle)" was written seven times to represent the number 70; and the symbol for 100 "snare (or coil of rope)" was written four times to represent the number 400. Together, these three groups of symbols (smallest symbols on the right and largest symbols on the left) were brought together to represent the number 475. Some of the remaining symbols, and what it is believed the symbols represent, are 1,000 "lotus flower"; 10,000 "bent finger"=; 100,000 "burbot fish (or frog)"; and 1,000,000 "kneeling god with arms raised above his head".
As indicated above, the Egyptians possessed an early (hieroglyphics) grouping system that separated the units (10s, 100s, etc.) in the numbers to be added. Thus, the number 475 is represented as (4 snares) + (7 lotus flowers) + (5 vertical strokes). The number 263 is represented as (2 snares) + (6 lotus flowers) + (3 vertical strokes), and 475 + 263 is represented as (6 snares) + (13 lotus flowers, or 1 snare and 3 lotus flowers) + (8 vertical strokes) = (7 snares) + (3 lotus flower) + (8 vertical strokes).
Multiplication was basically binary arithmetic, based on successive doublings. For instance, multiplying 47 by 27 is solved by associating 1 = 20 with 47, 2 = 21 with 94 (47 x 2), 4 = 22 with 188 (47 x 4), 8 = 23 with 376 (47 x 8), and 16 = 24 with 752 (47 x 16). Since 32 > 27, there is no need to go beyond the 16 entry. Now, go through a number of subtractions (27 - 16 = 11, 11 - 8 = 3, 3 - 2 = 1, 1 - 1 = 0) to show that 27 = 16 + 8 + 2 + 1. Next, add together the numbers associated with the right-hand numbers. Therefore, 47 x 27 = 47 x (16 + 8 + 2 + 1) = 47 x (24 + 23 + 21 + 20). Substitute into this equation the previously calculated values for the powers of 2 multiplied by 47 to obtain: 47 x 27 = 752 + 376 + 94+ 47 = 1,269.
Division is based on the reverse process of multiplication. Using the previous example, dividing 1,269 by 47 is solved by consecutive subtractions of "1,269 - 752 = 517; 517 - 376 = 141; 141 - 94 = 47; 47 - 47 = 0". Therefore, 1,269 /47 = 16 + 8 + 2 + 1 = 27.
The Egyptians used sums of unit fractions (1/n), supplemented by the fraction 2/3 that was not represented by a sum of unit fractions, to express all other fractions. For example, if 5 loaves of bread needed to be divided among 12 men, the fraction 5/12 became the sum of the unit fractions 1/3 and 1/12 where 1/3 + 1/12 = 4/12 + 1/12 = 5/12. Therefore, each man received one-third and one-twelfth of a loaf. Using this system, the Egyptians were able to solve all problems of arithmetic that involved fractions, as well as some elementary problems in algebra. However, the procedure can sometimes become quite complicated for large numbers and several combinations of unit fractions are possible for the solution.
Historians view the Egyptian achievement in mathematics as modest. But, their methods persisted for nearly three thousand years. Egyptian mathematics was most important in its effect on the emerging Greek mathematics between the sixth and fourth centuries b.c..
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