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

2. © 2010 Pearson Prentice Hall. All rights reserved. 2 4.2 Number Bases in Positional Systems

3. © 2010 Pearson Prentice Hall. All rights reserved. Objectives 1.Change numerals in bases other than ten to base ten. 2.Change base ten numerals to numerals in other bases. 3

4. © 2010 Pearson Prentice Hall. All rights reserved. Your Turn What is the decimal notation for the following Mayan numeral? 1

5. © 2010 Pearson Prentice Hall. All rights reserved. Binary (base 2) System Computer signals in terms of ‘yes’/’no’, ‘on’/’off’, ‘high voltage’/’low voltage’, 1/0. Computer data storage also in terms of ‘on’/’off’, ‘north pole’/’south pole’, 1/0. Binary system uses 2 symbols: 0, 1 E.g., 1000001 two = 65 ten 1

6. © 2010 Pearson Prentice Hall. All rights reserved. Binary (base 2) System 64+0 +1 1 6432168421 2 16 2828 2424 23232 2121 2020 1000001 = 65

7. © 2010 Pearson Prentice Hall. All rights reserved. What Is the Binary Numeral in Decimal? 64+0+16+8+0+2+0 1 6432168421 2 16 2828 2424 23232 2121 2020 1011010 = 90

8. © 2010 Pearson Prentice Hall. All rights reserved. Binary (base 2) System DecimalBinary 00 11 210 311 4100 5101 6110 7111 81000 91001 101010 111011 121100 131101 1

9. © 2010 Pearson Prentice Hall. All rights reserved. Your Turn Find the decimal notation for the following binary notation. 1.0110 binary 2.1111 binary 1

10. © 2010 Pearson Prentice Hall. All rights reserved. Hexadecimal (base 16) Notation Uses 16 symbols: E.g., 2B hexadecimal = 43 ten A3 hexadecimal = 163 ten 1 Decimal0123456789101112131415 Hexadec0123456789ABCDEF

11. © 2010 Pearson Prentice Hall. All rights reserved. Hexadecimal (base 16) Notation 256161 16 2 16 1 16 0 2B 1 2x16+11x = 43 ten What is 2B hex in decimal notation?

12. © 2010 Pearson Prentice Hall. All rights reserved. Hexadecimal (base 16) Notation 256161 16 2 16 1 16 0 A3 1 10x16+3x1 = 163 ten What is A3 hex in decimal notation?

13. © 2010 Pearson Prentice Hall. All rights reserved. Your Turn Find the decimal numeral for the following hex numeral. 1.12A 2.FFF 1 Decimal0123456789101112131415 Hexadec0123456789ABCDEF

14. © 2010 Pearson Prentice Hall. All rights reserved. How Is Binary Notation Related to Hex Notation? 1286432168421 2727 2626 2525 2424 23232 2121 2020 01011010 1 1 16 1 16 0 1286432168421 2727 2626 2525 2424 23232 2121 2020 01011010 5x16 +10x1 = 90 ten 64+16+8+2= 90 ten 5A

15. © 2010 Pearson Prentice Hall. All rights reserved. Adding Numbers in Binary Notation 1011 +0101 10000 1

16. © 2010 Pearson Prentice Hall. All rights reserved. Your Turn 1001 +0101 1 Find the sum of the following binaries. 2.e 1010 +0111

17. © 2010 Pearson Prentice Hall. All rights reserved. Changing Numerals in Bases Other Than Ten to Base Ten The base of a positional numeration system refers to the number whose powers define the place values. The place values in a base five system are powers of five: …,5 4, 5 3, 5 2, 5 1, 1 =…,5 × 5 × 5 × 5, 5 × 5 × 5, 5 × 5, 5, 1 =…,625, 125, 25, 5, 1 17

18. © 2010 Pearson Prentice Hall. All rights reserved. Changing to Base Ten To change a numeral in a base other than ten to a base ten numeral, 1.Find the place value for each digit in the numeral. 2.Multiply each digit in the numeral by its respective place value. 3.Find the sum of the products in step 2. 18

19. © 2010 Pearson Prentice Hall. All rights reserved. Example 1: Converting to Base Ten Convert 4726 eight to base ten. Solution: The given base eight numeral has four places. From left to right, the place values are 8 3, 8 2, 8 1, and 1 Find the place value for each digit in the numeral: 19

20. © 2010 Pearson Prentice Hall. All rights reserved. Example 1 continued Multiply each digit in the numeral by its respective place value. 20

21. © 2010 Pearson Prentice Hall. All rights reserved. Changing Base Ten Numerals to Numerals in Other Bases We use division to determine how many groups of each place value are contained in a base ten numeral. 21

22. © 2010 Pearson Prentice Hall. All rights reserved. Example 5: Using Division to Convert from Base Ten to Base Eight Convert the base ten numeral 299 to a base eight numeral. Solution: The place values in base eight are …,8 3, 8 2,8 1,1, or …,512, 64, 8, 1. The place values that are less than 299 are 64, 8, and 1. We use division to show how many groups of each of these place values are contained in 288. 22

23. © 2010 Pearson Prentice Hall. All rights reserved. Example 5 continued These divisions show that 299 can be expressed as 4 groups of 64, 5 groups of 8 and 3 ones: 23

24. © 2010 Pearson Prentice Hall. All rights reserved. Example 7: Using Divisions to Convert from Base Ten to Base Six Convert the base ten numeral 3444 to a base six numeral. Solution: The place values in base six are …6 5, 6 4, 6 3,6 2, 6 1, 1, or …7776, 1296, 216, 36, 6, 1. We use the powers of 6 that are less than 3444 and perform successive divisions by these powers. 24

25. © 2010 Pearson Prentice Hall. All rights reserved. Example 7 continued Using these four quotients and the final remainder, we can immediately write the answer. 25