Section 4-1

1. Section 4-1 • Historical Numeration Systems

2. Chapter 4: Numeration Systems Hindus- Arabic 670 AD It is very important moment in development of Mathematics 1-Relatived easy ways to express the numbers using 10 symbols 2-Relatived easy rules for arithmetic operations. 3- It allows several methods and devices to compute arithmetic operations, even use of computer and calculators.

3. Chapter 4: Numeration Systems Ancient Civilization European Hindu- Arabic 670 AD time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

4. Chapter 4: Numeration Systems European Hindu- Arabic 670 AD time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

5. Chapter 4: Numeration Systems European Hindu- Arabic 670 AD time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

6. Chapter 4: Numeration Systems European Hindu- Arabic 670 AD time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

7. Chapter 4: Numeration Systems European Hindu- Arabic 670 AD time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

8. Chapter 4: Numeration Systems European Hindu- Arabic 670 AD time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

9. Chapter 4: Numeration Systems Mayan 2000 BC-1546 AD European time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

10. Chapter 4: Numeration Systems European Hindu- Arabic 670 AD time now 4000 BC 3000 BC 2000 BC 1000 BC 1000 AD 0 2000 AD Babylonian Egypt Indian Greece---Rome

11. Historical Numeration Systems • Basics of Numeration • Ancient Egyptian Numeration • Ancient Roman Numeration • Classical Chinese Numeration

12. Numeration Systems The various ways of symbolizing and working with the counting numbers are called numeration systems. The symbols of a numeration system are called numerals. Two question are, How many symbols we need to represent numbers and what is the optimal way for grouping these symbols.

13. Example: Counting by Tallying Tally sticks and tally marks have been used for a long time. Each mark represents one item. For example, eight items are tallied by writing the following:

14. Counting by Grouping Counting by grouping allows for less repetition of symbols and makes numerals easier to interpret. The size of the group is called the base (usually ten) of the number system.

15. Ancient Egyptian Numeration – Simple Grouping The ancient Egyptian system is an example of a simplegrouping system. It uses ten as its base and the various symbols are shown on the next slide.

16. Ancient Egyptian Numeration

17. Example: Egyptian Numeral Write the number below in our system. Solution 2 (100,000) = 200,000 3 (1,000) = 3,000 1 (100) = 100 4 (10) = 40 5 (1) = 5 Answer: 203,145

18. Ancient Roman Numeration The ancient Roman method of counting is a modified grouping system. It uses ten as its base, but also has symbols for 5, 50, and 500. The Roman system also has a subtractive feature which allows a number to be written using subtraction. A smaller-valued symbol placed immediately to the left of the larger value indicated subtraction.

19. Ancient Roman Numeration The ancient Roman numeration system also has a multiplicative feature to allow for bigger numbers to be written. A bar over a number means multiply the number by 1000. A double bar over the number means multiply by 10002 or 1,000,000.

20. Ancient Roman Numeration

21. Example: Roman Numeral Write the number below in our system. MCMXLVII Solution M= 1000 CM= -100 + 1000 XL = -10 + 50 V= 5 I= 1 I= 1 Answer: 1000 + 900 + 40 + 5 + 1 + 1= 1947

22. Example: Roman Numeral

23. Traditional Chinese Numeration – Multiplicative Grouping A multiplicative grouping system involves pairs of symbols, each pair containing a multiplier and then a power of the base. The symbols for a Chinese version are shown on the next slide.

24. Chinese Numeration

25. Example: Chinese Numeral Interpret each Chinese numeral. a) b)

26. Example: Chinese Numeral Solution 7000 200 400 0 (tens) 1 80 Answer: 201 2 Answer: 7482

27. Example: Chinese Numeral A single symbol rather than a pair denotes as 1 multiplier an when a particular power is missing the omission is denoted with zero symbol.

28. Example: Chinese Numeral

29. Section 4-2 • More Historical Numeration Systems

30. More Historical Numeration Systems • Basics of Positional Numeration • Hindu-Arabic Numeration • Babylonian Numeration • Mayan Numeration • Greek Numeration

31. Positional Numeration A positional system is one where the various powers of the base require no separate symbols. The power associated with each multiplier can be understood by the position that the multiplier occupies in the numeral.

32. Positional Numeration In a positional numeral, each symbol (called a digit) conveys two things: 1. Face value – the inherent value of the symbol. 2. Place value – the power of the base which is associated with the position that the digit occupies in the numeral.

33. Positional Numeration To work successfully, a positional system must have a symbol for zero to serve as a placeholder in case one or more powers of the base is not needed.

34. Hindu-Arabic Numeration – Positional One such system that uses positional form is our system, the Hindu-Arabic system. The place values in a Hindu-Arabic numeral, from right to left, are 1, 10, 100, 1000, and so on. The three 4s in the number 45,414 all have the same face value but different place values.

35. Hindu-Arabic Numeration Hundred thousands Millions Ten thousands Thousands Decimal point Hundreds Tens Units 7, 5 4 1, 7 2 5 .

36. Babylonian Numeration The ancient Babylonians used a modified base 60 numeration system. The digits in a base 60 system represent the number of 1s, the number of 60s, the number of 3600s, and so on. The Babylonians used only two symbols to create all the numbers between 1 and 59. ▼ = 1 and ‹ =10

37. Example: Babylonian Numeral Interpret each Babylonian numeral. a) ‹‹‹▼▼▼▼ b) ▼▼‹‹‹▼▼▼▼ ▼

38. Example: Babylonian Numeral Solution ‹‹‹▼▼▼▼ Answer: 34 ▼▼‹‹‹▼▼▼▼ ▼ Answer: 155

39. Example: Babylonian Numeral

40. Example: Babylonian Numeral

41. Example: Babylonian Numeral

42. Example: Babylonian Numeral

43. Example: Babylonian Numeral

44. Mayan Numeration The ancient Mayans used a base 20 numeration system, but with a twist. Normally the place values in a base 20 system would be 1s, 20s, 400s, 8000s, etc. Instead, the Mayans used 360s as their third place value. Mayan numerals are written from top to bottom. Table 1

45. Mayan Numeration

46. Example: Mayan Numeral Write the number below in our system. Solution Answer: 3619

47. Example: Mayan Numeral Write the number below in our system.

48. Example: Mayan Numeral Write the number below in our system.

49. Example: Mayan Numeral Write the number below in Mayan Numeral.

50. Example: Mayan Numeral Write the number below in Mayan Numeral.