1. Historical Numeration Systems
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
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
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
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
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
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
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
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
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 simple grouping 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) =
4 (10) =
5 (1) =
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 or 1,000,000.

20. Ancient Roman Numeration

21. Example: Roman Numeral
Write the number below in our system.
MCMXLVII
Solution
M= 1000
CM=
XL =
V= 5
I= 1
Answer:
= 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
This art will need to be changed to make the number on the table “table 3”

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, ,

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.

51. Greek Numeration The classical Greeks used a ciphered counting system.
They had 27 individual symbols for numbers, based on the 24 letters of the Greek alphabet, with 3 Phoenician letters added.
The Greek number symbols are shown on the next slide.

52. Greek Numeration
Table 2
Table 2
(cont.)

53. Example: Greek Numerals
Interpret each Greek numeral.
a) ma
b) cpq

54. Example: Greek Numerals
Solution
a) ma
b) cpq
Answer: 41
Answer: 689

55. Example: Greek Numerals