1. Introduction to Computer Engineering by Richard E. Haskell Number Systems Module M3.1 Sections 2.1-2.2

2. Introduction to Computer Engineering by Richard E. Haskell Number Systems Counting in Binary Hexadecimal and Octal Numbers Positional Notation Fractional Numbers

3. Introduction to Computer Engineering by Richard E. Haskell Counting in Binary Position:8421 0000000000 0001100011 0010200102 0011300113 0100401004 0101501015 0110601106 0111701117 1000810008 BINARY HEX

4. Introduction to Computer Engineering by Richard E. Haskell Counting in Binary Position:8421 1000810008 1001910019 1010A1010A 1011B1011B 1100C1100C 1101D1101D 1110E1110E 1111F1111F BINARY HEX

5. Introduction to Computer Engineering by Richard E. Haskell Counting in Binary 128 64 32 16 8 4 2 1 0 0 1 1 0 1 0 0 52 1 0 1 0 0 0 1 1163 1 1 1 1 1 1 1 1 255 BINARY DEC

6. Introduction to Computer Engineering by Richard E. Haskell

7. 87C9: 32,768 1,792 192 9 87C9 16 = 34,761 10

8. Introduction to Computer Engineering by Richard E. Haskell Positional Notation N = P 4 P 3 P 2 P 1 P 0 = P 4 b 4 + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 584 10 = 5 x 10 2 + 8 x 10 1 + 4 x 10 0 = 500 + 80 + 4 = 584

9. Introduction to Computer Engineering by Richard E. Haskell Positional Notation N = P 4 P 3 P 2 P 1 P 0 = P 4 b 4 + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 10110 2 = 1 x 2 4 + 0 x 2 3 + 1 x 2 2 + 1 x 2 1 + 0 x 2 0 = 16 + 0 + 4 + 2 + 0 = 22 10 Binary

10. Introduction to Computer Engineering by Richard E. Haskell Positional Notation N = P 4 P 3 P 2 P 1 P 0 = P 4 b 4 + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 3AF 16 = 3 x 16 2 + A x 16 1 + F x 16 0 = 3 x 256 + 10 x 16 + 15 x 1 = 768 + 160 + 15 = 943 10 Hex

11. Introduction to Computer Engineering by Richard E. Haskell Positional Notation N = P 4 P 3 P 2 P 1 P 0 = P 4 b 4 + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 457 8 = 4 x 8 2 + 5 x 8 1 + 7 x 8 0 = 4 x 64 + 5 x 8 + 7 x 1 = 256 + 40 + 7 = 303 10 Octal

12. Introduction to Computer Engineering by Richard E. Haskell Fractional Numbers N =...P 3 P 2 P 1 P 0. P -1 P -2 P -3... =... + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 + P -1 b -1 + P -2 b -2 + P -3 b -3 +... 375.17 10 = 3 x 10 2 + 7 x 10 1 + 5 x 10 0 + 1 x 10 -1 + 7 x 10 -2 = 300 + 70 + 5 + 0.1 + 0.07 = 375.17

13. Introduction to Computer Engineering by Richard E. Haskell Fractional Numbers N =...P 3 P 2 P 1 P 0. P -1 P -2 P -3... =... + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 + P -1 b -1 + P -2 b -2 + P -3 b -3 +... 1101.11 2 = 1 x 2 3 + 1 x 2 2 + 0 x 2 1 + 1 x 2 0 + 1 x 2 -1 + 1 x 2 -2 = 8 + 2 + 0 + 1 + 1/2 + 1/4 = 11.75 10 Binary

14. Introduction to Computer Engineering by Richard E. Haskell Fractional Numbers N =...P 3 P 2 P 1 P 0. P -1 P -2 P -3... =... + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 + P -1 b -1 + P -2 b -2 + P -3 b -3 +... 1AB.6 16 = 1 x 16 2 + A x 16 1 + B x 16 0 + 6 x 16 -1 = 1 x 256 + 10 x 16 + 11 x 1 + 6/16 = 256 + 160 + 11 + 0.375 = 427.375 10 Hex

15. Introduction to Computer Engineering by Richard E. Haskell Fractional Numbers N =...P 3 P 2 P 1 P 0. P -1 P -2 P -3... =... + P 3 b 3 + P 2 b 2 + P 1 b 1 + P 0 b 0 + P -1 b -1 + P -2 b -2 + P -3 b -3 +... 173.25 8 = 1 x 8 2 + 7 x 8 1 + 3 x 8 0 + 2 x 8 -1 + 5 x 8 -2 = 1 x 64 + 7 x 8 + 3 x 1 + 2/8 + 5/64 = 64 + 56 + 3 + 0.25 + 0.078125 = 123.328125 10 Octal

16. Introduction to Computer Engineering by Richard E. Haskell Questions What is the decimal value of 243 5 ? What is the decimal value of 21.3 4 ?