The Origins of Mathematics in Mesopotamia
The ancient Egyptians made their number system more efficient by introducing more symbols.
The inhabitants of Mesopotamia (our book calls them Babylonians) achieved a similar result by introducing positional notation.
The Babylonians used 60 as their base. How do we write a number in base 60?
The Babylonians used 60 as their base. How do we write a number in base 60? Write the following base 10 numbers in base 60: 78 10 = 38,921 10 = 216,061 10 =
The Babylonians used 60 as their base. How do we write a number in base 60? Write the following base 10 numbers in base 60: 78 10 = 1,18 60 38,921 10 = 216,061 10 =
The Babylonians used 60 as their base. How do we write a number in base 60? Write the following base 10 numbers in base 60: 78 10 = 1,18 60 38,921 10 = 10,48,41 60 216,061 10 =
The Babylonians used 60 as their base. How do we write a number in base 60? Write the following base 10 numbers in base 60: 78 10 = 1,18 60 38,921 10 = 10,48,41 60 216,061 10 = 1,0,1,1 60
How did the Babylonians represent fractions with their base 60 positional system?
How did the Babylonians represent fractions with their base 60 positional system? Write the following base 10 fractions in base 60: 1/5 10 = 1/6 10 = 1/12 10 = 1/30 10 =
How did the Babylonians represent fractions with their base 60 positional system? Write the following base 10 fractions in base 60: 1/5 10 = 0;12 60 1/6 10 = 1/12 10 = 1/30 10 =
How did the Babylonians represent fractions with their base 60 positional system? Write the following base 10 fractions in base 60: 1/5 10 = 0;12 60 1/6 10 = 0;10 60 1/12 10 = 1/30 10 =
How did the Babylonians represent fractions with their base 60 positional system? Write the following base 10 fractions in base 60: 1/5 10 = 0;12 60 1/6 10 = 0;10 60 1/12 10 = 0;5 60 1/30 10 =
How did the Babylonians represent fractions with their base 60 positional system? Write the following base 10 fractions in base 60: 1/5 10 = 0;12 60 1/6 10 = 0;10 60 1/12 10 = 0;5 60 1/30 10 = 0;2 60
How did the Babylonians do division?
How did the Babylonians do division? Perform the following divisions in base 60: (a) 18,5 divided by 6. (b) 42,7,15 divided by 4.
How did the Babylonians do division? Perform the following divisions in base 60: (a) 18,5 divided by 6 is 3,0;50. (b) 42,7,15 divided by 4.
How did the Babylonians do division? Perform the following divisions in base 60: (a) 18,5 divided by 6 is 3,0;50. (b) 42,7,15 divided by 4 is 10,31,48;45.
When answering the reading guide and discussion questions, try to write your responses in the historical style. In particular, the Babylonians would not have used symbols for +,, /, or =. For example: 18,5 divided by 6 is 18,5 multiplied by 0;10. 18,0 multiplied by 0;10 is 180;0 3,0. 5 multiplied by 0;10 is 0;50. The result is 3,0;50. It is a good idea to check your work using in base 10.
What are the disadvantages of using position notation as opposed to additive notation? What are the advantages? Why is positional notation conceptually more difficult than additive notation? How might the need for positional notation have arisen in ancient civilizations?
Where can you find evidence of the use of bases other than 10 in modern life? (Hint: thank about time, measurement, and number words in English and other languages.) Can you see an advantage of base 60 over base 10?
n n 60 n 10 1/6 1/12 1/15 1/30
n n 60 n 10 1/6 0;10 0.1666 1/12 0;5 0.8333 1/15 0;4 0.0666 1/30 0;2 0.0333
n (20 / n) 60 (20 / n) 10 6 20 0;10 20 0.1666 12 20 0;5 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333 Because reciprocals written in sexagesimal notation are often simpler than they are written in decimal notation, the process of division (multiplication by a reciprocal) is also simpler.
n (20 / n) 60 (20 / n) 10 6 20 0;10 20 0.1666 12 20 0;5 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 20 0;10 = 0;200 20 0.1666 12 20 0;5 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 20 0;10 = 3;20 20 0.1666 12 20 0;5 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 20 0;5 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 20 0;5 = 0;100 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 20 0;5 = 1;40 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 1;40 20 0.8333 15 20 0;4 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 1;40 20 0.8333 15 20 0;4 = 0;80 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 1;40 20 0.8333 15 20 0;4 = 1;20 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 1;40 20 0.8333 15 1;20 20 0.0666 30 20 0;2 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 1;40 20 0.8333 15 1;20 20 0.0666 30 20 0;2 = 0;40 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 1;40 20 0.8333 15 1;20 20 0.0666 30 0;40 20 0.0333
n (20 / n) 60 (20 / n) 10 6 3;20 20 0.1666 12 1;40 20 0.8333 15 1;20 20 0.0666 30 0;40 20 0.0333 We can often do division base 60 in our heads more easily than division base 10.
How would the Babylonians have represented the value 1/8?
How would the Babylonians have represented the value 1/8? How would the Babylonians have represented the value 1/9?
Why wouldn t the Babylonians have had a nice representation for the value 1/7? What are the first three numbers in the sexagesimal approximation to this number?
Do discussion questions 5, 6, and 7.
The second section in this chapter highlights some of the problems ancient Babylonians could solve.
The solution to Problem 2.1 on page 28 seems to contain several errors. If you use the numbers given, what should you get for the volume, the number of workers, and the total cost? Give your answers both in base 10 and in base 60. Where might these errors have come from? Problem 2.1: A canal 5 GAR long, 1 1/2 GAR wide, and 1/2 GAR deep is to be dug. Each worker is assigned to dig 10 GIN, and is paid 6 SE. Find the area, volume, number of workers, and total cost. Solution: Multiply length and width to get 7;30 SAR, the area. Multiply 7;30 by depth to get 45 SAR, the volume. Multiply the reciprocal of the assignment, 6, by 4,30, which is the number of workers. Multiply 4,30 by the wages to get 9 GIN, the total expenses.
The solution to Problem 2.2 on page 20 also seems to contain an error. To find it, translate the procedure for finding the length L and the width W into formulas in terms of the sum L + W and the area LW. Where might this error have come from? Show geometrically why this procedure works. Problem 2.2: The length and width of a canal are together 6;30 GAR; the area of the canal is 7;30 SAR. What are the length and width? Solution: Take half the sum of the length and width, which is 3;15. Square 3;15 to get 10;33,45. Subtract the product of length and width, 7;30, from 10;33,45 to get 3;3,45. Take its square root, which is 1;45. Add it to the sum of the length and width, to get 5 GAR, the length, and subtract it from the sum to get 1;30 GAR, the width.
Rewrite the procedure described for the solution of Problem 2.3 using modern notation. Show geometrically why this procedure works. Problem 2.3: A canal s area is 7;30 SAR, and its length exceeded its width by 3;30 GAR. Solution: Take half of the amount by which the length exceeded the width, which is 1;45. Square 1;45 to get 3;3,45. Add 7;30 to get 10;33,45. Take its square root, to get 3;15. Add 1;45 to the square root to get 5 GAR, the length. Subtract 1;45 from the square root to get 1;30 GAR, the width.
The third section in this chapter discusses ancient Babylonian geometry.
The tablet YBC 7289 is shown together with a redrawing and a translation of the cuneiform numbers. How would we write the base 60 number 1;24,51,10 in base 10? What is the significance of this number? How might it have been obtained?
What is the significance of the tablet Plimpton 322? How might the ancient Babylonians have used such a table?