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The First Mathematicians
Introduction
The first mathematics can be traced to the ancient country of Babylon and to Egypt during the 3rd millennium BC. A number system with a base of 60 had developed in Babylon over time. Large numbers and fractions could be represented and formed the basis of advanced mathematical evolution. From at least 1700 BC, Pythagorean triples were studied. The study of linear and quadratic equations led to form of primitive numerical algebra. Meanwhile, similar figures, areas, and volumes were studied as well as the primitive values for pi obtained. The Greeks inherited the Babylonian principles and developed mathematics from 450 BC. They discovered that all real numbers could not accurately express all values, such as relationships between sides. Irrational numbers were born. The Greeks progressed rapidly in mathematics from 300 BC. Progress also sped in the Islamic countries of Syria, India, and Iran. Their work had a different focus from that of the Greeks, but all Greek principles held true. This basis was later brought to Europe and developed further there.
Babylonians: Writing and Base 60 System
The Babylonian system of writing was called cuneiform and was based on a series of straight lined symbols. These symbols were wet and baked in the hot sun to preserve. Curved lines could not be drawn. These cuneiform symbols led to many tables used to aid calculation. As stated previously, they used a base 60 system, which has ten proper divisors, instead of our current system, base 10 with only two proper divisors. In this respect, their system may have been more advanced since many more numbers have a finite form. Two examples of these tables are the tables found at Senkerah on the Euphrates River in 1854, which date from 2000 BC. This table was used to figure the squares of numbers to 59 and cubes of numbers up to 32. However, a drawback of this system is the lack of a proper 0. Also, context was required to determine if 1 meant 1, 61, or 361, etc.
Babylonians: Multiplication and Division
The Babylonians used the fact that
ab = ((a + b)² - a ² - b²)/2
and therefore
ab = (a + b)² / 4 - (a - b)² / 4
to make multiplication possible. If the user of this formula wished to multiply two numbers, all he would need is a chart of squares. However, their process of division was a more difficult task. They used the fact that
a.b = a.(1 / b)
to solve division problems. If the user of this formula wished to divide number, all he would need is a table of reciprocals. Consider the following reciprocal chart translated into Arabic numbers.
2 30
3 20
4 15
5 12
6 10
8 7 30
9 6 40
10 6
12 5
15 4
16 3 45
18 3 20
20 3
24 2 30
25 2 24
27 2 13 20
Not all reciprocals were possible because certain numbers have no base 60 finite fractions. To compute 1/13, for instance, the Babylonians could write
1/13 = 7/91 = 7.(1/91) = approximately 7.(1/90)
Babylonians: Pythagorean Triples and Problems to Solve Them
Another Babylonian tablet, dated between 1900 BC and 1600 BC contains certain Pythagorean triples where
a² + b² = c²
Many believe that it is the oldest number theory document ever recorded. Another Babylonian tablet contains the problem
4 is the length and 5 the diagonal. What is the breadth? Its size is not known. 4 times 4 is 16. 5 times 5 is 25. You take 16 from 25 and there remains 9. What times shall I take in order to get 9? 3 times 3 is 9. 3 is the breadth.
This is a good example of using Pythagorean triples to solve certain problems by the Babylonians.
Egyptians and Romans: Number System
The Roman and Egyptian systems did not make Arithmetic calculations easy. Multiplication of Roman numerals is nearly impossible and exceedingly complex. Unlike the Babylonians, the Egyptians did not develop fully their understanding of mathematics. Instead, they concerned themselves with practical applications of mathematics.
Egyptians and Romans: Multiplication
In 1650 BC, the scribe Ahmes wrote the Rhind Papyrus, named for its Scottish Egyptologist author A. Henry Rhind. The scroll is 6 meters long and 1/3 of a meter wide. The scribe Ahmes was copying a document predated 200 years before him. Consider, for instance, the multiplication of 41 and 59.
41 59
1 59
2 118
4 236
8 472
16 944
32 1888
64 3776
Since 41 falls between 32 and 64, they carry out the following simple subtraction problem
41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0
Hence,
41 = 32 + 8 + 1
Then add the corresponding totals
59
1 59 X
2 118
4 236
8 472 X
16 944
32 1888 X
2419
Hence the answer, 2419. If the factors were reversed and the factors of 41 used, then the same answer could be reached. These are two good examples of different historical cultures solving the same types of problems totally differently before modern mathematics.