Why did in ancient cultures, Ancient Egypt, Babylonians, and Chinese, primarily write out their mathematical texts in words?
Answers
The introduction of writing in Egypt in the period (c. 3000 BCE) brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.
One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century BCE) was a satiric letter in which one scribe, taunts his rival, Amen-em-, for his incompetence as an adviser and manager. “You are the clever scribe at the head of the troops,” chides at one point,
a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments—it is 60 cubits high, 30 cubits in the middle…and the generals and the scribes turn to you and say, “You are a clever scribe, your name is famous. Is there anything you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7 cubits.
This problem, and three others like it in the same letter, cannot be solved without further data. But the point of the humour is clear, as challenges his rival with these hard, but typical, tasks.
What is known of Egyptian mathematics tallies well with the tests posed by the scribe Horizon. The information comes primarily from two long papyrus documents that once served as textbooks within scribal schools. The Rhonda papyrus (in the British Museum) is a copy made in the 17th century BCE of a text two centuries older still. In it is found a long table of fractional parts to help with division, followed by the solutions of 84 specific problems in arithmetic and geometry. The papyrus (in the Moscow Museum of Fine Arts), dating from the 19th century BCE, presents 25 problems of a similar type. These problems reflect well the functions the scribes would perform, for they deal with how to distribute beer and bread as wages, for example, and how to measure the areas of fields as well as the volumes of pyramids and other solids.
The numeral system and arithmetic operations
The Egyptians, like the Romans after them, expressed numbers according to a decimal scheme, using separate symbols for 1, 10, 100, 1,000, and so on; each symbol appeared in the expression for a number as many times as the value it represented occurred in the number itself. For example, stood for 24. This rather cumbersome notation was used within the hieroglyphic writing found in stone inscriptions and other formal texts, but in the papyrus documents the scribes employed a more convenient abbreviated script, called hieratic writing, where, for example, 24 was written .
Ancient Egyptians customarily wrote from right to left. Because they did not have a positional system, they needed separate symbols for each power of 10.
Ancient Egyptians customarily wrote from right to left. Because they did not have a positional system, they needed separate symbols for each power of 10.
Encyclopedia Britannica, Inc.
Egyptian hieratic numerals
Encyclopædia Britannica, Inc.
In such a system, addition and subtraction amount to counting how many symbols of each kind there are in the numerical expressions and then rewriting with the resulting number of symbols. The texts that survive do not reveal what, if any, special procedures the scribes used to assist in this. But for multiplication they introduced a method of successive doubling. For example, to multiply 28 by 11, one constructs a table of multiples of 28 like the following:
Table of multiples of 28.
The several entries in the first column that together sum to 11 (i.e., 8, 2, and 1) are checked off. The product is then found by adding up the multiples corresponding to these entries; thus, 224 + 56 + 28 = 308, the desired product.
To divide 308 by 28, the Egyptians applied the same procedure in reverse. Using the same table as in the multiplication problem, one can see that 8 produces the largest multiple of 28 that is less then 308 (for the entry at 16 is already 448), and 8 is checked off. The process is then repeated, this time for the remainder (84) obtained by subtracting the entry at 8 (224) from the original number (308). This, however, is already smaller than the entry at 4, which consequently is ignored, but it is greater than the entry at 2 (56), which is then checked off. The process is repeated again for the remainder obtained by subtracting 56 from the previous remainder of 84, or 28, which also happens to exactly equal the entry at 1 and which is then checked off. The entries that have been checked off are added up, yielding the quotient: 8 + 2 + 1 = 11. (In most cases, of course, there is a remainder that is less than the divisor.)
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Answer:
brought with it the formation of a special class of literate professionals, the scribes. By virtue of their writing skills, the scribes took on all the duties of a civil service: record keeping, tax accounting, the management of public works (building projects and the like), even the prosecution of war through overseeing military supplies and payrolls. Young men enrolled in scribal schools to learn the essentials of the trade, which included not only reading and writing but also the basics of mathematics.
One of the texts popular as a copy exercise in the schools of the New Kingdom (13th century BCE) was a satiric letter in which one scribe, taunts his rival, Amen-em-, for his incompetence as an adviser and manager. “You are the clever scribe at the head of the troops,” chides at one point,
a ramp is to be built, 730 cubits long, 55 cubits wide, with 120 compartments—it is 60 cubits high, 30 cubits in the middle…and the generals and the scribes turn to you and say, “You are a clever scribe, your name is famous. Is there anything you don’t know? Answer us, how many bricks are needed?” Let each compartment be 30 cubits by 7 cubits.
This problem, and three others like it in the same letter, cannot be solved without further data. But the point of the humour is clear, as challenges his rival with these hard, but typical, tasks.
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