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The world of mathmatics and astronomy owes alot to the Babylonians. The sexagesimal system is used for the calculation of time and angles, and is still considered because of the multiple divisibility of the number 60. We also get the Greek day of 12 “doublehours”, and the zodiac and its signs. The mathematicians of the Old Babylonian period went way beyond the challenges of their official accounting duties. They introduced a numeral system which took advantage of the idea of place value. They developed methods that took advantage of the means of expressing numbers; they solved linear and quadratic problems by methods like the ones used now in algebra their success with the study of Pythagorean number triples was a great achievement in number theory. The older Sumerian system of numbers followed adding the decimal (base10) principle like the type from the Egyptians. But the Old Babylonian system changed it into a placevalue system with the base of 60 (sexagesimal). The reasons for the choice of 60 are unknown, but one good mathematical reason may have been that there were already so many divisors (2, 3, 4, and 5, and some multiples) of the base, which would have greatly simplified the operation of division. For numbers from 1 to 59, the symbols for 1 and for 10 were combined in the simple way of adding (e.g., represented 32). But, to express larger importance,
the Babylonians used the understanding of place value. For example, 60 was written as , 70 as , 80 as , and so on. In fact, could represent any power of 60. The context influenced which power was meant in the equation. The Babylonians seems to be the ones who developed a placeholder symbol that took the place of a zero by the 3rd century BC, but its exact meaning and use are still not know. They had no mark to separate numbers into integral and fractional parts (same as with the modern decimal point). The threeplace numeral 3 7 30 could represent 3 1/8 (i.e., 3 + 7/60 + 30/602), 187 1/2 (i.e., 3 ´ 60 + 7 + 30/60), 11,250 (i.e., 3 ´ 602 + 7 ´ 60 + 30), or a multiple of these numbers by any power of 60.
The four arithmetic operation, was accomplished in the same way as in the modern decimal system, except that carrying happened when a sum reached 60 rather than 10. Multiplication was simplified by using tables. One most common tablet lists the multiples of a number by 1, 2, 3, 19, 20, 30, 40, and 50. To multiply two numbers various places long, the writer first broke the problem down into several multiplications, each by a one number place, and then they looked up the value of each product in the right tables. He found the answer to the problem by adding up these results. These tables also helped in division, because the values that head them were all reciprocals of regular numbers.
Regular numbers are numbers that their prime factors divide the base; the reciprocals of these numbers have only a limited number of places. The reciprocals of no regular numbers produce an unlimited amount of a repeating numeral. In base 10, only numbers with factors of 2 and 5 (e.g., 8 or 50) are regular, and the reciprocals (1/8 = 0.125, 1/50 = 0.02) have limited expressions, but the reciprocals of other numbers (such as 3 and 7) repeat forever.
0.3 with the repeating bar over the 3 only and
.142857 with the repeating bar over the whole decimal
The bar indicates the digits that continually repeat. In base 60, only numbers with factors of 2, 3, and 5 are regular. For example 6 and 54 are regular, so that their reciprocals (10 and 1, 6, 40) are limited. The entries in the multiplication table for 1, 6, 40 are at the same time multiples of its reciprocal 1/54. To divide a number by any regular number, consult the table of multiples for its reciprocal.
Babylonian Mathematical tablet. ( Yale Babylonian Collection )
Back side of the Babylonian mathmatical tablet (Yale Babylonian Collection)
A tablet in the collection of Yale University shows a square with its diagonals. On one side it is written “30,” under one diagonal “42, 25, 35” and right along the same diagonal “1, 24, 51, 10” for example, (1 + 24/60 + 51/602 + 10/603). The third number is the correct value of Ö2 to four sexagesimal places (equal in the decimal system to 1.414213…, which is too low by 1 in the seventh place), the second number is the product of the third number and the first and which gives the length of the diagonal. Example: when the side is 30. The scribe thus appears to have known an equivalent of the familiar long method of finding square roots. An additional element of sophistication is that, by choosing 30 (that is, 1/2) for the side, the scribe obtained as the diagonal the reciprocal of the value of Ö2 (since Ö2/2 = 1/Ö2), a result useful for purposes of division.
the Babylonians used the understanding of place value. For example, 60 was written as , 70 as , 80 as , and so on. In fact, could represent any power of 60. The context influenced which power was meant in the equation. The Babylonians seems to be the ones who developed a placeholder symbol that took the place of a zero by the 3rd century BC, but its exact meaning and use are still not know. They had no mark to separate numbers into integral and fractional parts (same as with the modern decimal point). The threeplace numeral 3 7 30 could represent 3 1/8 (i.e., 3 + 7/60 + 30/602), 187 1/2 (i.e., 3 ´ 60 + 7 + 30/60), 11,250 (i.e., 3 ´ 602 + 7 ´ 60 + 30), or a multiple of these numbers by any power of 60.
The four arithmetic operation, was accomplished in the same way as in the modern decimal system, except that carrying happened when a sum reached 60 rather than 10. Multiplication was simplified by using tables. One most common tablet lists the multiples of a number by 1, 2, 3, 19, 20, 30, 40, and 50. To multiply two numbers various places long, the writer first broke the problem down into several multiplications, each by a one number place, and then they looked up the value of each product in the right tables. He found the answer to the problem by adding up these results. These tables also helped in division, because the values that head them were all reciprocals of regular numbers.
Regular numbers are numbers that their prime factors divide the base; the reciprocals of these numbers have only a limited number of places. The reciprocals of no regular numbers produce an unlimited amount of a repeating numeral. In base 10, only numbers with factors of 2 and 5 (e.g., 8 or 50) are regular, and the reciprocals (1/8 = 0.125, 1/50 = 0.02) have limited expressions, but the reciprocals of other numbers (such as 3 and 7) repeat forever.
0.3 with the repeating bar over the 3 only and
.142857 with the repeating bar over the whole decimal
The bar indicates the digits that continually repeat. In base 60, only numbers with factors of 2, 3, and 5 are regular. For example 6 and 54 are regular, so that their reciprocals (10 and 1, 6, 40) are limited. The entries in the multiplication table for 1, 6, 40 are at the same time multiples of its reciprocal 1/54. To divide a number by any regular number, consult the table of multiples for its reciprocal.
Babylonian Mathematical tablet. ( Yale Babylonian Collection )
Back side of the Babylonian mathmatical tablet (Yale Babylonian Collection)
A tablet in the collection of Yale University shows a square with its diagonals. On one side it is written “30,” under one diagonal “42, 25, 35” and right along the same diagonal “1, 24, 51, 10” for example, (1 + 24/60 + 51/602 + 10/603). The third number is the correct value of Ö2 to four sexagesimal places (equal in the decimal system to 1.414213…, which is too low by 1 in the seventh place), the second number is the product of the third number and the first and which gives the length of the diagonal. Example: when the side is 30. The scribe thus appears to have known an equivalent of the familiar long method of finding square roots. An additional element of sophistication is that, by choosing 30 (that is, 1/2) for the side, the scribe obtained as the diagonal the reciprocal of the value of Ö2 (since Ö2/2 = 1/Ö2), a result useful for purposes of division.
jtj2008 
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