1. Chapter 4 Section 1- Slide 1 Copyright © 2009 Pearson Education, Inc. AND

2. Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 2 Chapter 4 Systems of Numeration

3. Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 3 WHAT YOU WILL LEARN Additive, multiplicative, and ciphered systems of numeration Place-value systems of numeration Egyptian, Hindu-Arabic, Roman, Chinese, Ionic Greek, Babylonian, and Mayan numerals

4. Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 4 Section 1 Additive, Multiplicative, and Ciphered Systems of Numeration

5. Chapter 4 Section 1- Slide 5 Copyright © 2009 Pearson Education, Inc. Systems of Numeration A system of numeration consists of a set of numerals and a scheme or rule for combining the numerals to represent numbers A number is a quantity. It answers the question “How many?” A numeral is a symbol such as , 10 or  used to represent the number (amount).

6. Chapter 4 Section 1- Slide 6 Copyright © 2009 Pearson Education, Inc. Types Of Numeration Systems Four types of systems used by different cultures will be discussed. They are:  Additive (or repetitive)  Multiplicative  Ciphered  Place-value

7. Chapter 4 Section 1- Slide 7 Copyright © 2009 Pearson Education, Inc. Additive Systems An additive system is one in which the number represented by a set of numerals is simply the sum of the values of the numerals. It is one of the oldest and most primitive types of systems. Examples: Egyptian hieroglyphics and Roman numerals.

8. Chapter 4 Section 1- Slide 8 Copyright © 2009 Pearson Education, Inc. Egyptian Hieroglyphics

9. Chapter 4 Section 1- Slide 9 Copyright © 2009 Pearson Education, Inc. Roman Numerals Hindu-Arabic Numerals I1 V5 X10 L50 C100 D500 M1000

10. Chapter 4 Section 1- Slide 10 Copyright © 2009 Pearson Education, Inc. Example At the end of a movie the credits indicated a copyright date of MCMXCVII. How would you write this as a Hindu-Arabic numeral? Solution: It’s an additive system so, = M + CM + XC + V + I + I = 1000 + (1000 – 100) + (100 – 10) + 5 + 1 + 1 = 1000 + 900 + 90 + 7 = 1997

11. Chapter 4 Section 1- Slide 11 Copyright © 2009 Pearson Education, Inc. Multiplicative Systems Multiplicative systems are more similar to the Hindu-Arabic system which we use today. Example: Chinese numerals.

12. Chapter 4 Section 1- Slide 12 Copyright © 2009 Pearson Education, Inc. Chinese Numerals Written vertically Top numeral from 1 - 9 inclusive Multiply it by the power of 10 below it. 20 = 400 =

13. Chapter 4 Section 1- Slide 13 Copyright © 2009 Pearson Education, Inc. Example Write 538 as a Chinese numeral.

14. Chapter 4 Section 1- Slide 14 Copyright © 2009 Pearson Education, Inc. Ciphered Systems In this system, there are numerals for numbers up to and including the base and for multiples of the base. The number (amount) represented by a specific set of numerals is the sum of the values of the numerals.

15. Chapter 4 Section 1- Slide 15 Copyright © 2009 Pearson Education, Inc. Examples of Ciphered Systems: Ionic Greek system (developed about 3000 B.C. and used letters of Greek alphabet as numerals). Hebrew system Coptic system Hindu system Early Arabic systems

16. Chapter 4 Section 1- Slide 16 Copyright © 2009 Pearson Education, Inc. Ionic Greek System * Ancient Greek letters not used in the classic or modern Greek language.

17. Chapter 4 Section 1- Slide 17 Copyright © 2009 Pearson Education, Inc. Ionic Greek System * Ancient Greek letters not used in the classic or modern Greek language.

18. Chapter 4 Section 1- Slide 18 Copyright © 2009 Pearson Education, Inc. Example

19. Copyright © 2009 Pearson Education, Inc. Chapter 4 Section 1- Slide 19 Section 2 Place-Value or Positional-Value Numeration Systems

20. Chapter 4 Section 1- Slide 20 Copyright © 2009 Pearson Education, Inc. Place-Value System (or Positional Value System) The value of the symbol depends on its position in the representation of the number. It is the most common type of numeration system in the world today. The most common place-value system is the Hindu-Arabic numeration system. This is used in the United States.

21. Chapter 4 Section 1- Slide 21 Copyright © 2009 Pearson Education, Inc. Place-Value System A true positional-value system requires a base and a set of symbols, including a symbol for zero and one for each counting number less than the base. The most common place-value system is the base 10 system.  It is called the decimal number system.

22. Chapter 4 Section 1- Slide 22 Copyright © 2009 Pearson Education, Inc. Hindu-Arabic System Digits: In the Hindu-Arabic system, the digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Positions: In the Hindu-Arabic system, the positional values or place values are: … 10 5, 10 4, 10 3, 10 2, 10, 1.

23. Chapter 4 Section 1- Slide 23 Copyright © 2009 Pearson Education, Inc. Expanded Form To evaluate a number in this system, begin with the rightmost digit and multiply it by 1. Multiply the second digit from the right by base 10. Continue by taking the next digit to the left and multiplying by the next power of 10. In general, we multiply the digit n places from the right by 10 n–1 in order to show expanded form.

24. Chapter 4 Section 1- Slide 24 Copyright © 2009 Pearson Education, Inc. Example: Expanded Form Write the Hindu-Arabic numeral in expanded form. a) 63 b) 3769 Solution: 63 = (6  10 1 ) + (3  1 ) or (6  10) + 3 3769 = (3  1000) + (7  100) + (6  10) + 9 or (3  10 3 ) + (7  10 2 ) + (6  10 1 ) + (9  1 )