Number Systems-Number conversion -Arithmetic Operations
- 3. DIGITAL DESIGN What is digital ? • Digital camera, Digital TV, Digital Watch, Digital • Radio, Digital City (e city), Digital Photo Frame …etc ‐ • Which gives the things in countable form • Scene (analog) to Image (digital) Why is digital ? • Countable form, makes easy to manage.accuracy • Easy management makes more useful and versatile
- 4. NUMBER SYSTEM TYPES It is a system for representing numeric values or quantities using different symbols Famous Number System: Decimal , Roman, Binary Decimal System: 0 9 ‐ May evolves: because human have 10 finger Roman System May evolves to make easy to look and feel Pre/Post Concept: (IV, V & VI) is (5 1, 5 & 5+1) ‐ Binary System, Others (Oct, Hex) One can cut an apple in to two
- 5. SIGNIFICANT DIGITS Binary: 11101101 Most significant Bit (MSB) Least significant Bit(LSB) Decimal: 1063079 Most significant digit Least significant digit
- 6. DECIMAL (BASE 10) Uses positional representation Each digit corresponds to a power of 10 based on its position in the number The powers of 10 increment from 0, 1, 2, etc. as you move right to left 1,479 = 1 * 103 + 4 * 102 + 7 * 101 + 9 * 100
- 7. BINARY (BASE 2) Two digits: 0, 1 To make the binary numbers more readable, the digits are often put in groups of 4 1010 = 1 * 23 + 0 * 22 + 1 * 21 + 0 * 20 = 8 + 2 = 10 1100 1001 = 1 * 27 + 1 * 26 + 1 * 23 + 1 * 20 = 128 + 64 + 8 + 1 = 201
- 8. OCTAL (BASE 8) Shorter & easier to read than binary 8 digits: 0, 1, 2, 3, 4, 5, 6, 7, Octal numbers 1368 = 1 * 82 + 3 * 81 + 6 * 80 = 1 * 64 + 3 * 8 + 6 * 1 = 9410
- 9. HEXADECIMAL(BASE 16) Shorter & easier to read than binary 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F “0x” often precedes hexadecimal numbers 0x123 = 1 * 162 + 2 * 161 + 3 * 160 = 1 * 256 + 2 * 16 + 3 * 1 = 256 + 32 + 3 = 291
- 10. DECIMAL(BASE 10) BINARY (BASE 2) OCTAL ( BASE 8) HEXA(BASE 16) 0 0000 0 0 1 0001 1 1 2 0010 2 2 3 0011 3 3 4 0100 4 4 5 0101 5 5 6 0110 6 6 7 0111 7 7 8 1000 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F
- 11. Converting To and From Decimal
- 12. CONVERSION PROCESS DECIMAL ↔ BASE N Successive Division
- 13. CONVERSION PROCESS DECIMAL ↔ BASE N Weighted Multiplication
- 14. DECIMAL ↔ BINARY -EXAMPLE 12510 = ?2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 0 1 12510 = 11111012
- 15. DECIMAL ↔ OCTAL 123410 = ?8 8 1234 154 2 8 19 2 8 2 3 8 0 2 123410 = 23228
- 16. DECIMAL ↔ HEXADECIMAL 123410 = ?16 123410 = 4D216 16 1234 77 2 16 4 13 = D 16 0 4
- 17. BINARY ↔ DECIMAL 1010112 => 1 x 20 = 1 1 x 21 = 2 0 x 22 = 0 1 x 23 = 8 0 x 24 = 0 1 x 25 = 32 4310 Bit “0”
- 18. OCTAL TO DECIMAL 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810 HEXADECIMAL TO DECIMAL ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
- 19. BINARY ↔ OCTAL Technique Group bits in threes, starting on right Convert to octal digits 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 20. OCTAL ↔ BINARY 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 21. BINARY ↔ HEXADECIMAL 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
- 22. HEXADECIMAL ↔ BINARY Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 23. Exercise – Convert ... Decimal Binary Octal Hexa- decimal 33 1110101 703 1AF
- 24. Exercise – Convert ... Decimal Binary Octal Hexa- decimal 33 100001 41 21 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF
- 25. FRACTIONS Decimal to decimal (just for fun) 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14
- 26. FRACTIONS Binary to decimal pp. 46-50 10.1011 => 1 x 2-4 = 0.0625 1 x 2-3 = 0.125 0 x 2-2 = 0.0 1 x 2-1 = 0.5 0 x 20 = 0.0 1 x 21 = 2.0 2.6875
- 27. FRACTIONS Decimal to binary 3.14579 .14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc. 11.001001...
- 28. Exercise – Convert ... Don’t use a calculator! Decimal Binary Octal Hexa- decimal 29.8 101.1101 3.07 C.82
- 29. Exercise – Convert … Decimal Binary Octal Hexa- decimal 29.8 11101.110011… 35.63… 1D.CC… 5.8125 101.1101 5.64 5.D 3.109375 11.000111 3.07 3.1C 12.5078125 1100.10000010 14.404 C.82 Answer
- 30. Binary addition (1 of 2) Two 1-bit values A B A + B 0 0 0 0 1 1 1 0 1 1 1 10 “two”
- 31. Binary addition (2 of 2) Two n-bit values Add individual bits Propagate carries E.g., 1 1 10101 21 + 11001 + 25 101110 46
- 32. Multiplication (1 of 2) Binary, two 1-bit values A B A B 0 0 0 0 1 0 1 0 0 1 1 1
- 33. Multiplication (2 of 2) Binary, two n-bit values 1110 x 1011 1110 1110 0000 1110 10011010
- 34. Common Powers (1 of 2) Base 10 Power Preface Symbol 10-12 pico p 10-9 nano n 10-6 micro 10-3 milli m 103 kilo k 106 mega M 109 giga G 1012 tera T Value .000000000001 .000000001 .000001 .001 1000 1000000 1000000000 1000000000000
- 35. Common Powers (2 of 2) Base 2 Power Preface Symbol 210 kilo k 220 mega M 230 Giga G Value 1024 1048576 1073741824