1. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 4.1 Additive, Multiplicative, and Ciphered Systems of Numeration

2. Copyright 2013, 2010, 2007, Pearson, Education, Inc. What You Will Learn Additive, multiplicative, and ciphered systems of numeration 4.1-2

3. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Systems of Numeration 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). 4.1-3

4. Copyright 2013, 2010, 2007, 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. 4.1-4

5. Copyright 2013, 2010, 2007, 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 4.1-5

6. Copyright 2013, 2010, 2007, 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. 4.1-6

7. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Egyptian Hieroglyphics 4.1-7

8. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 1: From Egyptian to Hindu-Arabic Numerals Write the following numeral as a Hindu- Arabic numeral. 4.1-8 Solution 10,000 + 10,000 + 10,000 + 100 + 100 + 100 + 10 + 1 + 1 + 1 = 30,134

9. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Write the following numeral as a Hindu-Arabic Numberal 4.1-9

10. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 2: From Hindu-Arabic to Egyptian Numerals Write 43,628 as an Egyptian numeral. 4.1-10 Solution 43,628 = 40,000 + 3000 + 600 + 20 + 8

11. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Write 1,234,527 as an Egyptian numeral 4.1-11

12. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Roman Numerals Hindu-Arabic Numerals I1 V5 X10 L50 C100 D500 M1000 4.1-12

13. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Roman Numerals Two advantages over Egyptian system: Uses the subtraction principle as well as addition principle DC = 500 + 100 = 600 CD = 500 – 100 = 400 Uses the multiplication principle for numerals greater than 1000 4.1-13

14. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 4: From Roman to Hindu-Arabic Numerals Write CMLXIV as a Hindu-Arabic numeral. Solution It’s an additive system so, = CM + L + X + IV = (1000 – 100) + 50 + 10 + (5 – 1) = 900 + 50 + 10 + 4 = 964 4.1-14

15. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try THis Write MCMLXXI as a Hindu-Arabic numeral 4.1-15

16. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 5: Writing a Roman Numeral Write 439 as a Roman numeral. Solution 439 = 400 + 30 + 9 = (500 – 100) + 10 + 10 + 10 + (10 – 1) = CDXXXIX 4.1-16

17. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Write 3794 as a Roman Numeral 4.1-17

18. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Multiplicative Systems Multiplicative systems are more similar to the Hindu-Arabic system which we use today. Chinese numerals 4.1-18

19. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Chinese Numerals Written vertically Top numeral from 1 - 9 inclusive Multiply it by the power of 10 below it. 20 = 400 = 4.1-19

20. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 7: A Traditional Chinese Numeral Write 538 as a Chinese numeral. Solution: 4.1-20

21. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Try This Write 3,271 as a Chinese Numeral 4.1-21

22. Copyright 2013, 2010, 2007, 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. 4.1-22

23. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ciphered Systems Ciphered numeration systems require the memorization of many different symbols but have the advantage that numbers can be written in a compact form. 4.1-23

24. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Examples of Ciphered Systems We discuss in detail the Ionic Greek system developed about 3000 B.C. used letters of Greek alphabet as numerals Base is 10 An iota, , placed to the left and above a numeral represents the numeral multiplied by 1000 4.1-24

25. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Examples of Ciphered Systems Hebrew Coptic Hindu Brahmin Syrian Egyptian Hieratic early Arabic 4.1-25

26. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ionic Greek System * Ancient Greek letters not used in the classic or modern Greek language. 4.1-26

27. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Ionic Greek System * Ancient Greek letters not used in the classic or modern Greek language. 4.1-27

28. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 9 Write  as a Hindu-Arabic numeral. Solution The sum is 839. 4.1-28

29. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Example 10 Write 1654 as an Ionic Greek numeral. Solution 1654 = 1000 + 600 + 50 + 4 = (1 × 1000) + 600 + 50 + 4 4.1-29

30. Copyright 2013, 2010, 2007, Pearson, Education, Inc. Homework 4.1-30