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# Number Systems

Number Systems. Module M3.1 Sections 2.1-2.2. Number Systems. Counting in Binary Hexadecimal and Octal Numbers Positional Notation Fractional Numbers. Counting in Binary. BINARY HEX. Position: 8 4 2 1. 0 0 0 0 0 0 0 0 1 1 0 0 1 0 2 0 0 1 1 3 0 1 0 0 4 0 1 0 1 5 0 1 1 0 6

## Number Systems

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1. Number Systems Module M3.1 Sections 2.1-2.2

2. Number Systems • Counting in Binary • Hexadecimal and Octal Numbers • Positional Notation • Fractional Numbers

3. Counting in Binary BINARY HEX Position: 8 4 2 1 0 0 0 0 0 0 0 0 1 1 0 0 1 0 2 0 0 1 1 3 0 1 0 0 4 0 1 0 1 5 0 1 1 0 6 0 1 1 1 7 1 0 0 0 8

4. Counting in Binary BINARY HEX Position: 8 4 2 1 1 0 0 0 8 1 0 0 1 9 1 0 1 0 A 1 0 1 1 B 1 1 0 0 C 1 1 0 1 D 1 1 1 0 E 1 1 1 1 F

5. Counting in Binary BINARY DEC 128 64 32 16 8 4 2 1 0 0 1 1 0 1 0 0 52 1 0 1 0 0 0 1 1 163 1 1 1 1 1 1 1 1 255

6. 87C9: 32,768 1,792 192 9 87C916 = 34,76110

7. Positional Notation N = P4P3P2P1P0 = P4b4 + P3b3 + P2b2 + P1b1 + P0b0 58410 = 5 x 102 + 8 x 101 + 4 x 100 = 500 + 80 + 4 = 584

8. Positional Notation N = P4P3P2P1P0 = P4b4 + P3b3 + P2b2 + P1b1 + P0b0 Binary 101102 = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 0 x 20 = 16 + 0 + 4 + 2 + 0 = 2210

9. Positional Notation N = P4P3P2P1P0 = P4b4 + P3b3 + P2b2 + P1b1 + P0b0 Hex 3AF16 = 3 x 162 + A x 161 + F x 160 = 3 x 256 + 10 x 16 + 15 x 1 = 768 + 160 + 15 = 94310

10. Positional Notation N = P4P3P2P1P0 = P4b4 + P3b3 + P2b2 + P1b1 + P0b0 Octal 4578 = 4 x 82 + 5 x 81 + 7 x 80 = 4 x 64 + 5 x 8 + 7 x 1 = 256 + 40 + 7 = 30310

11. Fractional Numbers N = ...P3P2P1P0. P-1P-2P-3... = ... + P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b-3 + ... 375.1710 = 3 x 102 + 7 x 101 + 5 x 100 + 1 x 10-1 + 7 x 10-2 = 300 + 70 + 5 + 0.1 + 0.07 = 375.17

12. Fractional Numbers N = ...P3P2P1P0. P-1P-2P-3... = ... + P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b-3 + ... Binary 1101.112 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 + 1 x 2-1 + 1 x 2-2 = 8 + 2 + 0 + 1 + 1/2 + 1/4 = 11.7510

13. Fractional Numbers N = ...P3P2P1P0. P-1P-2P-3... = ... + P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b-3 + ... Hex 1AB.616 = 1 x 162 + A x 161 + B x 160 + 6 x 16-1 = 1 x 256 + 10 x 16 + 11 x 1 + 6/16 = 256 + 160 + 11 + 0.375 = 427.37510

14. Fractional Numbers N = ...P3P2P1P0. P-1P-2P-3... = ... + P3b3 + P2b2 + P1b1 + P0b0 + P-1b-1 + P-2b-2 + P-3b-3 + ... Octal 173.258 = 1 x 82 + 7 x 81 + 3 x 80 + 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 = 123.32812510

15. Questions What is the decimal value of 2435? What is the decimal value of 21.34?

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