Understanding Place Value Systems: Babylonian and Hindu-Arabic Methods

1. Section 4.2 Place Value System

2. Objectives: • Understand and use the Babylonian System. • Understand and use the Hindu-Arabic Expanded Notation with addition and subtraction. • Use the Galley Method for multiplication. • Use Napier’s Rods for multiplication.

3. Key Terms: • Place Value System – the placement of the symbols in a numeral determines the value of the symbols, also called a positional system. • NOTE: In order to have a true place value system, you must have a symbol for zero.

4. Babylonian Number System • The Babylonians developed an early example of a place value system. • This system was based on powers of 60, called a sexagesimal system. • There are only 2 symbols in the Babylonian system: • Represents 1 - • Represents 10 -

5. For Example: • The number 23 can be written as: ,however, for larger numbers, they used several symbols separated by spaces, and multiplied the value of these groups, of symbols, by increasing powers of 60.

6. Example 1: • Convert to Hindu-Arabic

7. Example 2: • Convert to Hindu-Arabic

8. Example 3: • Convert to Hindu-Arabic

9. Example 4: 7,717 • Convert to Babylonian • In order to convert, we need to divide by 60, similar to converting seconds to hours and minutes.

10. Example 5: 7,573 • Convert to Babylonian • In order to convert, we need to divide by 60, similar to converting seconds to hours and minutes.

11. Example 6: 128,485 • Convert to Babylonian • In order to convert, we need to divide by 60, similar to converting seconds to hours and minutes.

12. Section 4.2 Assignment I • Class work: • TB pg. 216/1 – 16 All • Remember you must write the problem and show ALL work to receive credit for this assignment. • NO work, NO grade!

13. Hindu-Arabic Numeration System Place Value • Based on Powers of 10. • Writing numbers in expanded notation. • 6,582 = (6x103)+(5x102)+(8x101)+(2x100)

14. Example 7: 5,389 • Write the number using expanded notation.

15. Example 8: 31,157 • Write the number using expanded notation.

16. Example 9: 2,100,405 • Write the number using expanded notation.

17. Section 4.2 Continued Addition and Subtraction using the Hindu-Arabic Expanded Notation

18. Example 10: 4,625 + 814 • Add/Subtract using Expanded Notation

19. Example 11: 5,264 + 583 • Add/Subtract using Expanded Notation

20. Example 12: 728 – 243 • Add/Subtract using Expanded Notation

21. Example 13: 4,317 – 2,561 • Add/Subtract using Expanded Notation

22. Section 4.2 Assignment II • Class work: • TB pg. 216/33 – 40 All • Remember you must write the problem and show ALL work to receive credit for this assignment. • NO work, NO grade!

23. Galley Method: 685 x 49 • Begin by constructing a rectangle.

24. Galley Method: 685 x 49 • Divide into triangles called a galley.

25. Galley Method: 685 x 49 • Compute partial products in each box

26. Galley Method: 685 x 49 • Add numbers along the diagonals.

27. Example 14: 7 x 364 • Multiply using the Galley Method.

28. Example 15: 22 x 867 • Multiply using the Galley Method.

29. Example 16: 239 x 456 • Multiply using the Galley Method.

30. Napier’s Rods/Bones • Developed by John Napier in the 16th Century, for doing multiplication. • TB pg. 215 The Napier's rods consist of strips of wood, metal or heavy cardboardand are three dimensional.

31. Example 17: 8 x 346 • Using Napier’s Rods

32. Example 18: 21 x 768 • Using Napier’s Rods

33. Example 19: 241 x 365 • Using Napier’s Rods

34. Section 4.2 Assignment III • Class work: • TB pg. 216/41 – 52 All • Remember you must write the problem and show ALL work to receive credit for this assignment. • NO work, NO grade!