Number systems
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The document discusses various number systems including decimal, binary, octal, and hexadecimal. It provides examples of converting between these different bases using techniques like dividing by the base, tracking remainders, and grouping bits. Common powers are also defined for bases 10 and 2. The key concepts covered are representation of quantities in different number systems, conversion between number systems, addition and multiplication in binary, and representing fractions in binary.
Number systems
- 1. Number Systems Eng. Mustafa H. Salah Mechatronics Engineering University for Electronic Technology
- 2. Common Number Systems Used by Used in System Base Symbols humans? computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- 16 0, 1, … 9, No No decimal A, B, … F
- 3. Quantities/Counting (1 of 3) Hexa- Decimal Binary Octal 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
- 4. Quantities/Counting (2 of 3) Hexa- Decimal Binary Octal 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
- 5. Quantities/Counting (3 of 3) Hexa- Decimal Binary Octal 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
- 6. Conversion Among Bases • The possibilities: Decimal Octal Binary Hexadecimal
- 7. Quick Example 2510 = 110012 = 318 = 1916 Base
- 8. Decimal to Decimal (just for fun) Decimal Octal Binary Hexadecimal
- 9. Weight 12510 => 5 x 100 = 5 2 x 101 = 20 1 x 102 = 100 125 Base
- 10. Binary to Decimal Decimal Octal Binary Hexadecimal
- 11. 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
- 12. Example Bit “0” 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
- 13. Octal to Decimal Decimal Octal Binary Hexadecimal
- 14. 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
- 15. Example 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810
- 16. Hexadecimal to Decimal Decimal Octal Binary Hexadecimal
- 17. 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
- 18. Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
- 19. Decimal to Binary Decimal Octal Binary Hexadecimal
- 20. 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.
- 21. Example 12510 = ?2 2 125 2 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 0 1 12510 = 11111012
- 22. Octal to Binary Decimal Octal Binary Hexadecimal
- 23. Octal to Binary • Technique • Convert each octal digit to a 3-bit equivalent binary representation
- 24. Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 25. Hexadecimal to Binary Decimal Octal Binary Hexadecimal
- 26. Hexadecimal to Binary • Technique • Convert each hexadecimal digit to a 4-bit equivalent binary representation
- 27. Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 28. Decimal to Octal Decimal Octal Binary Hexadecimal
- 29. Decimal to Octal • Technique • Divide by 8 • Keep track of the remainder
- 30. Example 123410 = ?8 8 1234 8 154 2 8 19 2 8 2 3 0 2 123410 = 23228
- 31. Decimal to Hexadecimal Decimal Octal Binary Hexadecimal
- 32. Decimal to Hexadecimal • Technique • Divide by 16 • Keep track of the remainder
- 33. Example 123410 = ?16 16 1234 16 77 2 16 4 13 = D 0 4 123410 = 4D216
- 34. Binary to Octal Decimal Octal Binary Hexadecimal
- 35. Binary to Octal • Technique • Group bits in threes, starting on right • Convert to octal digits
- 36. Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 37. Binary to Hexadecimal Decimal Octal Binary Hexadecimal
- 38. Binary to Hexadecimal • Technique • Group bits in fours, starting on right • Convert to hexadecimal digits
- 39. Example 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
- 40. Octal to Hexadecimal Decimal Octal Binary Hexadecimal
- 41. Octal to Hexadecimal • Technique • Use binary as an intermediary
- 42. Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
- 43. Hexadecimal to Octal Decimal Octal Binary Hexadecimal
- 44. Hexadecimal to Octal • Technique • Use binary as an intermediary
- 45. Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148
- 46. Exercise – Convert ... Hexa- Decimal Binary Octal decimal 33 1110101 703 1AF Answer
- 47. Exercise – Convert… Hexa- Decimal Binary Octal decimal 33 100001 41 21 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF
- 48. Common Powers (1 of 2) • Base 10 Power Preface Symbol Value 10-12 pico p .000000000001 10-9 nano n .000000001 10-6 micro .000001 10-3 milli m .001 103 kilo k 1000 106 mega M 1000000 109 giga G 1000000000 1012 tera T 1000000000000
- 49. Common Powers (2 of 2) • Base 2 Power Preface Symbol Value 210 kilo k 1024 220 mega M 1048576 230 Giga G 1073741824 • What is the value of “k”, “M”, and “G”?
- 50. Example In the lab… 1. Double click on My Computer 2. Right click on C: 3. Click on Properties / 230 =
- 51. Review – multiplying powers • For common bases, add powers ab ac = ab+c 26 210 = 216 = 65,536 or… 26 210 = 64 210 = 64k
- 52. 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”
- 53. Binary Addition (2 of 2) • Two n-bit values • Add individual bits • Propagate carries • E.g., 1 1 10101 21 + 11001 + 25 101110 46
- 54. Multiplication (1 of 3) • Decimal (just for fun) 35 x 105 175 000 35 3675
- 55. 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
- 56. Multiplication (3 of 3) • Binary, two n-bit values • As with decimal values • E.g., 1110 x 1011 1110 1110 0000 1110 10011010
- 57. 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
- 58. Fractions • Binary to decimal 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
- 59. Fractions • Decimal to binary .14579 x 2 3.14579 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 11.001001... etc.
- 60. Exercise – Convert ... Hexa- Decimal Binary Octal decimal 29.8 101.1101 3.07 C.82 Answer
- 61. Exercise – Convert … Hexa- Decimal Binary Octal 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
- 62. Thank you