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Number systems
- 1. 1. Number Systems Chapt. 2 Location in course textbook
- 2. Common Number Systems 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
- 4. Quantities/Counting (1 of 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 p. 33
- 5. Quantities/Counting (2 of 3) 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
- 6. Quantities/Counting (3 of 3) 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 Etc.
- 7. Conversion Among Bases The possibilities: Hexadecimal Decimal Octal Binary pp. 40-46
- 8. Quick Example 2510 = 110012 = 318 = 1916 Base
- 9. Decimal to Decimal (just for fun) Hexadecimal Decimal Octal Binary Next slide
- 10. 12510 => 5 x 100 = 5 2 x 101 = 20 1 x 102 = 100 125 Base Weight
- 11. Binary to Decimal Hexadecimal Decimal Octal Binary
- 12. Binary to Decimal 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
- 13. Example 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
- 14. Octal to Decimal Hexadecimal Decimal Octal Binary
- 15. Octal to Decimal 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
- 16. Example 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810
- 17. Hexadecimal to Decimal Hexadecimal Decimal Octal Binary
- 18. Hexadecimal to Decimal 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
- 19. Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
- 20. Decimal to Binary Hexadecimal Decimal Octal Binary
- 21. Decimal to Binary Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.
- 22. Example 12510 = ?2 2 125 62 12 31 02 15 12 7 12 3 12 1 12 0 1 12510 = 11111012
- 23. Octal to Binary Hexadecimal Decimal Octal Binary
- 24. Octal to Binary Technique Convert each octal digit to a 3-bit equivalent binary representation
- 25. Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 26. Hexadecimal to Binary Hexadecimal Decimal Octal Binary
- 27. Hexadecimal to Binary Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation
- 28. Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 29. Decimal to Octal Hexadecimal Decimal Octal Binary
- 30. Decimal to Octal Technique Divide by 8 Keep track of the remainder
- 31. Example 123410 = ?8 8 1234 154 28 19 28 2 38 0 2 123410 = 23228
- 32. Decimal to Hexadecimal Hexadecimal Decimal Octal Binary
- 33. Decimal to Hexadecimal Technique Divide by 16 Keep track of the remainder
- 34. Example 123410 = ?16 123410 = 4D216 16 1234 77 216 4 13 = D16 0 4
- 35. Binary to Octal Hexadecimal Decimal Octal Binary
- 36. Binary to Octal Technique Group bits in threes, starting on right Convert to octal digits
- 37. Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 38. Binary to Hexadecimal Hexadecimal Decimal Octal Binary
- 39. Binary to Hexadecimal Technique Group bits in fours, starting on right Convert to hexadecimal digits
- 40. Example 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
- 41. Octal to Hexadecimal Hexadecimal Decimal Octal Binary
- 42. Octal to Hexadecimal Technique Use binary as an intermediary
- 43. Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
- 44. Hexadecimal to Octal Hexadecimal Decimal Octal Binary
- 45. Hexadecimal to Octal Technique Use binary as an intermediary
- 46. Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148
- 47. Exercise Convert ... Dont use a calculator! Decimal Binary Octal Hexa- decimal 33 1110101 703 1AF Skip answer Answer
- 48. Exercise Convert Decimal Binary Octal Hexa- decimal 33 100001 41 21 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF Answer
- 49. Common Powers (1 of 2) Base 10 Power Preface Symbol 10-12 pico p 10-9 nano n 10-6 micro 袖 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
- 50. Common Powers (2 of 2) Base 2 Power Preface Symbol 210 kilo k 220 mega M 230 Giga G Value 1024 1048576 1073741824 What is the value of k, M, and G? In computing, particularly w.r.t. memory, the base-2 interpretation generally applies
- 51. Example / 230 = In the lab 1. Double click on My Computer 2. Right click on C: 3. Click on Properties
- 52. Exercise Free Space Determine the free space on all drives on a machine in the lab Drive Free space Bytes GB A: C: D: E: etc.
- 53. Review multiplying powers For common bases, add powers 26 210 = 216 = 65,536 or 26 210 = 64 210 = 64k ab ac = ab+c
- 54. Binary Addition (1 of 2) Two 1-bit values pp. 36-38 A B A + B 0 0 0 0 1 1 1 0 1 1 1 10 two
- 55. Binary Addition (2 of 2) Two n-bit values Add individual bits Propagate carries E.g., 10101 21 + 11001 + 25 101110 46 11
- 56. Multiplication (1 of 3) Decimal (just for fun) pp. 39 35 x 105 175 000 35 3675
- 57. Multiplication (2 of 3) Binary, two 1-bit values A B A B 0 0 0 0 1 0 1 0 0 1 1 1
- 58. Multiplication (3 of 3) Binary, two n-bit values As with decimal values E.g., 1110 x 1011 1110 1110 0000 1110 10011010
- 59. Fractions 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
- 60. 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
- 61. Fractions Decimal to binary p. 50 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...
- 62. Exercise Convert ... Dont use a calculator! Decimal Binary Octal Hexa- decimal 29.8 101.1101 3.07 C.82 Skip answer Answer
- 63. 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