B.sc cs-ii -u-1.1 digital logic circuits, digital component floting and fixed point
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This document provides an overview of common number systems and techniques for converting between decimal, binary, octal, and hexadecimal number bases. It discusses how quantities are represented and counted in each base system. Methods for addition, multiplication, and conversion between the different bases are presented for both single and multi-bit numbers. Common powers and fractions are also covered, showing how values are represented in decimal and binary. Exercises provide examples of converting values between the number bases.
B.sc cs-ii -u-1.1 digital logic circuits, digital component floting and fixed point
- 1. Digital Logic Circuits, Digital Component and Data Representation Course: B.Sc-CS-II Subject: Computer Organization And Architecture Unit-1 1
- 2. Common Number Systems 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
- 3. Quantities/Counting (1 of 3) 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
- 4. Quantities/Counting (2 of 3) 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
- 5. Quantities/Counting (3 of 3) 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 Etc.
- 6. Conversion Among Bases • The possibilities: 6 Hexadecimal Decimal Octal Binary pp. 40-46
- 7. Quick Example 7 2510 = 110012 = 318 = 1916 Base
- 8. Decimal to Decimal (just for fun) 8 Hexadecimal Decimal Octal Binary Next slide…
- 9. 12510 => 5 x 100 = 5 2 x 101 = 20 1 x 102 = 100 125 Base Weight 9
- 10. Binary to Decimal 10 Hexadecimal Decimal Octal Binary
- 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 11
- 12. Example 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”
- 13. Octal to Decimal 13 Hexadecimal Decimal Octal Binary
- 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 14
- 15. Example 15 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810
- 16. Hexadecimal to Decimal 16 Hexadecimal Decimal Octal Binary
- 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 17
- 18. Example 18 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 19 Hexadecimal Decimal Octal Binary
- 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. 20
- 21. Example 21 12510 = ?2 2 125 62 12 31 02 15 1 2 7 1 2 3 12 1 12 0 1 12510 = 11111012
- 22. Octal to Binary 22 Hexadecimal Decimal Octal Binary
- 23. Octal to Binary • Technique – Convert each octal digit to a 3-bit equivalent binary representation 23
- 24. Example 24 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 25. Hexadecimal to Binary 25 Hexadecimal Decimal Octal Binary
- 26. Hexadecimal to Binary • Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation 26
- 27. Example 27 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 28. Decimal to Octal 28 Hexadecimal Decimal Octal Binary
- 29. Decimal to Octal • Technique – Divide by 8 – Keep track of the remainder 29
- 30. Example 30 123410 = ?8 8 1234 154 2 8 19 2 8 2 3 8 0 2 123410 = 23228
- 31. Decimal to Hexadecimal 31 Hexadecimal Decimal Octal Binary
- 32. Decimal to Hexadecimal • Technique – Divide by 16 – Keep track of the remainder 32
- 33. Example 33 123410 = ?16 123410 = 4D216 16 1234 77 216 4 13 = D16 0 4
- 34. Binary to Octal 34 Hexadecimal Decimal Octal Binary
- 35. Binary to Octal • Technique – Group bits in threes, starting on right – Convert to octal digits 35
- 36. Example 36 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 37. Binary to Hexadecimal 37 Hexadecimal Decimal Octal Binary
- 38. Binary to Hexadecimal • Technique – Group bits in fours, starting on right – Convert to hexadecimal digits 38
- 39. Example 39 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
- 40. Octal to Hexadecimal 40 Hexadecimal Decimal Octal Binary
- 41. Octal to Hexadecimal • Technique – Use binary as an intermediary 41
- 42. Example 42 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
- 43. Hexadecimal to Octal 43 Hexadecimal Decimal Octal Binary
- 44. Hexadecimal to Octal • Technique – Use binary as an intermediary 44
- 45. Example 45 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148
- 46. Exercise – Convert ... 46 Decimal Binary Octal Hexa- decimal 33 1110101 703 1AF
- 47. Exercise – Convert … 47 Decimal Binary Octal Hexa- decimal 33 100001 41 21 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF Answer
- 48. Common Powers (1 of 2) • Base 10 48 Power Preface Symbol 10-12 pico p 10-9 nano n 10-6 micro µ 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
- 49. Common Powers (2 of 2) • Base 2 49 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
- 50. Review – multiplying powers • For common bases, add powers 50 26 × 210 = 216 = 65,536 or… 26 × 210 = 64 × 210 = 64k ab × ac = ab+c
- 51. Binary Addition (1 of 2) • Two 1-bit values 51 pp. 36-38 A B A + B 0 0 0 0 1 1 1 0 1 1 1 10 “two”
- 52. Binary Addition (2 of 2) • Two n-bit values – Add individual bits – Propagate carries – E.g., 52 10101 21 + 11001 + 25 101110 46 11
- 53. Multiplication (1 of 3) • Decimal (just for fun) 53 pp. 39 35 x 105 175 000 35 3675
- 54. Multiplication (2 of 3) • Binary, two 1-bit values 54 A B A × B 0 0 0 0 1 0 1 0 0 1 1 1
- 55. Multiplication (3 of 3) • Binary, two n-bit values – As with decimal values – E.g., 55 1110 x 1011 1110 1110 0000 1110 10011010
- 56. Fractions • Decimal to decimal (just for fun) 56 pp. 46-50 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14
- 57. Fractions • Binary to decimal 57 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
- 58. Fractions • Decimal to binary 58 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...
- 59. Exercise – Convert ... 59 Decimal Binary Octal Hexa- decimal 29.8 101.1101 3.07 C.82
- 60. Exercise – Convert … 60 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
- 61. Reference Reference Book • Computer Organization & Architecture 7e By Stallings • Computer System Architecture By Mano • Digital Logic & Computer Design By Mano