1. © 2010 Pearson Prentice Hall. All rights reserved. CHAPTER 4 Number Representation and Calculation

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 4.4 Looking Back at Early Numeration Systems

3. © 2010 Pearson Prentice Hall. All rights reserved. Objectives 1.Understand and use the Egyptian system. 2.Understand and use the Roman system. 3.Understand and use the traditional Chinese system. 4.Understand and use the Ionic Greek system. 3

4. © 2010 Pearson Prentice Hall. All rights reserved. The Egyptian Numeration System The Egyptians used the oldest numeration system called hieroglyphic notation. 4

5. © 2010 Pearson Prentice Hall. All rights reserved. Write the following numeral as a Hindu-Arabic numeral: Solution: Using the table, find the value of each of the Egyptian numerals. Then add them. 1,000,000 + 10,000 + 10,000 + 10 + 10 + 10 + 1 + 1 + 1 + 1 = 1,020,034 Example 1: Using the Egyptian Numeration System 5

6. © 2010 Pearson Prentice Hall. All rights reserved. Example 2: Using the Egyptian Numeration System Write 1752 as an Egyptian numeral. Solution: First break down the Hindu-Arabic numeral into quantities that match the Egyptian numerals: 1752 = 1000 + 700 + 50 + 2 = 1000 + 100 + 100 + 100 + 100 + 100 + 100 + 100 + 10 + 10 + 10 + 10 + 10 + 1 + 1 Now use the table to find the Egyptian symbol that matches each quantity. Thus, 1752 can be expressed as 6

7. © 2010 Pearson Prentice Hall. All rights reserved. The Roman numerals were used until the eighteenth century and are still commonly used today for outlining, on clocks, and in numbering some pages in books. The Roman Numeration System Roman Numeral IVXLCDM Hindu- Arabic Numeral 1510501005001000 7

8. © 2010 Pearson Prentice Hall. All rights reserved. If the symbols decrease in value from left to right, then add their values to obtain the value of the Roman numeral as a whole. If the symbols increase in value from left to right, then subtract the value of the symbol on the left from the symbol on the right to obtain the value of the Roman numeral as a whole. The Roman Numeration System 8

9. © 2010 Pearson Prentice Hall. All rights reserved. Write CLXVII as a Hindu-Arabic numeral. Solution: Because the numerals decrease in value from left to right, we add their values to find the value of the Roman numeral as a whole. CLXVII = 100 + 50 + 10 + 5 + 1 + 1 = 167 Example 3: Using Roman Numerals 9

10. © 2010 Pearson Prentice Hall. All rights reserved. Write MCMXCVI as a Hindu-Arabic numeral. Solution: Example 4: Using Roman Numerals 10

11. © 2010 Pearson Prentice Hall. All rights reserved. The Traditional Chinese Numeration System 11

12. © 2010 Pearson Prentice Hall. All rights reserved. Example 6: Using the Traditional Chinese Numeration System Write 3264 as a Chinese numeral. 12

13. © 2010 Pearson Prentice Hall. All rights reserved. 1 The Traditional Japanese Numeration Traditional Japanese 一二三四五六七八九十百千万 Hindu-Arabic 123456789101001,00010,000 (10 4 ) Traditional Japanese 億 (oku) 兆 (chou) 京 (kei) Hindu-Arabic 100,000,000 (10 8 )1,000,000,000,000 (10 12 )10 (10 16 ) E.g., World population: 7.2 10 9 = 72 億 Honda Civic Hybrid: ¥2,500,000 = ¥250 万 ≈ $25,000

14. © 2010 Pearson Prentice Hall. All rights reserved. 1 Japanese Currency ¥1000 ¥5000 ¥10,000 ¥1¥10¥50

15. © 2010 Pearson Prentice Hall. All rights reserved. Unit Prefix 1,00010 3 kilo- 1,000,00010 6 mega- 1,000,000,00010 9 giga- 1,000,000,000,00010 12 tera- 1,000,000,000,000,00010 15 peta- 1,000,000,000,000,000,00010 18 exa- 1

16. © 2010 Pearson Prentice Hall. All rights reserved. The Ionic Greek Numeration System The ancient Greeks used letters from their alphabet for numerals. The symbols are written right next to one another. 16

17. © 2010 Pearson Prentice Hall. All rights reserved. Example 7: Using the Ionic Greek Numeration System Write ψλδ as a Hindu-Arabic numeral. Solution: Retrieving what each Greek numeral represents, ψ =700, λ=30, δ=4, next we add the digits left to right according to their positions. ψλδ = 700 + 30 + 4 = 734 Thus, ψλδ represents 734 in Hindu-Arabic numerals. 17