Number systems
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The document discusses various number systems including decimal, binary, octal, and hexadecimal. It provides techniques for converting between these different number systems, such as dividing by powers of two for binary to decimal conversion, grouping bits in threes for binary to octal, and using binary as an intermediary for conversions between other bases. Arithmetic operations like addition and multiplication are also covered for binary numbers by adding or multiplying individual bits.
Number systems
- 1. Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 2. System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- decimal 16 0, 1, … 9, A, B, … F No No Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 3. Decimal Binary Octal Hexa- decimal 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 4. Decimal Binary Octal Hexa- decimal 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 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 5. Decimal Binary Octal Hexa- decimal 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 20 10100 24 14 21 10101 25 15 22 10110 26 16 23 10111 27 17 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 6. The possibilities: Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 7. 2510 = 110012 = 318 = 1916 Base Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 8. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 9. 12510 => 5 x 100 = 5 2 x 101 = 20 1 x 102 = 100 125 Base Weight Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 10. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 11. Technique Multiply each bit by 2n , where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 12. 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” Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 13. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 14. Technique Multiply each bit by 8n , where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 15. 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 16. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 17. Technique Multiply each bit by 16n , where n is the “weight” of the bit The weight is the position of the bit, starting from 0 on the right Add the results Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 18. ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 19. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 20. Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc. Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 21. 12510 = ?2 2 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1 12510 = 11111012 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 22. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 23. Technique Convert each octal digit to a 3-bit equivalent binary representation Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 24. 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 25. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 26. Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 27. 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 28. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 29. Technique Divide by 8 Keep track of the remainder Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 30. 123410 = ?8 8 1234 154 28 19 28 2 38 0 2 123410 = 23228 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 31. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 32. Technique Divide by 16 Keep track of the remainder Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 33. 123410 = ?16 123410 = 4D216 16 1234 77 216 4 13 = D16 0 4 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 34. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 35. Technique Group bits in threes, starting on right Convert to octal digits Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 36. 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 37. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 38. Technique Group bits in fours, starting on right Convert to hexadecimal digits Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 39. 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 40. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 41. Technique Use binary as an intermediary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 42. 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 43. Hexadecimal Decimal Octal Binary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 44. Technique Use binary as an intermediary Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 45. 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 46. For common bases, add powers 26 × 210 = 216 = 65,536 or… 26 × 210 = 64 × 210 = 64k ab × ac = ab+c Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 47. Two 1-bit values A B A + B 0 0 0 0 1 1 1 0 1 1 1 10 “two” Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 48. Two n-bit values Add individual bits Propagate carries E.g., 10101 21 + 11001 + 25 101110 46 11 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 49. Decimal (just for fun) 35 x 105 175 000 35 3675 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 50. Binary, two 1-bit values A B A × B 0 0 0 0 1 0 1 0 0 1 1 1 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 51. Binary, two n-bit values As with decimal values E.g., 1110 x 1011 1110 1110 0000 1110 10011010 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 52. Decimal to decimal (just for fun) pp. 46-50 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
- 53. 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 Midhunchakkaravarthy Lecturer School of Computer Science Lincoln University College Malaysia
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