Lec 02
- 2. A number system defines how a number can be represented using distinct symbols. A number can be represented differently in different system. Several number systems have been used in the past and can be categorized into two groups: positional and non-positional systems.
- 3. Roman numerals are a good example of a non-positional number system. This number system has a set of symbols S = {I, V, X, L, C, D, M}. The values of each symbol are shown in Table.
- 4. To find the value of a number, we need to add the value of symbols subject to specific rules
- 5. In positional number system, there are only few symbols called digits, and these symbols represent different values depending on the position they occupy in the number. Commonly used number systems are: Decimal Binary Octal Hexadecimal
- 6. Why do we need more number systems? • Humans understand decimal • Digital electronics (computers) understand binary • Since computers have 32, 64, and even 128 bit busses, displaying numbers in binary is cumbersome • Data on a 32 bit data bus would look like the following: 0110 1001 0111 0001 0011 0100 1100 1010 • Hexadecimal (base 16) and octal (base 8) number systems are used to represent binary data in a more compact form.
- 7. The word decimal is a derivative of decem, which is the Latin word for ten. The number system that we use day-to-day life is called the Decimal number system. The most popular & commonly used number system is the Decimal number system as it supports the entire mathematical & accounting concept in the world. The base is equal to ten because there are altogether ten digits (1,2, 3, 4, 5, 6, 7, 8, 9)10
- 8. The binary number system uses two digits to represent numbers, the values are 0 & 1. This numbering system is sometime called the Base 2 numbering system (0,1). Binary digit is often referred to by the common abbreviation BIT. Thus, a “bit” in a computer terminology means either a 0 or a 1. This number system is natural to an electronic machines or devices as their mechanism based on the OFF or ON switching of the circuits. Therefore, 0 represent the OFF & 1 represent ON state of the circuit.
- 9. The octal number system uses eight values to represent numbers. The values are (0, 1, 2, 3, 4, 5, 6, 7)8 and the base of this system is eight.
- 10. The hexadecimal number system has 16- digits or symbols (hexa means six & decimal means 10 so sum is sixteen) are (0, 1, 2, 3, 4, 5,6, 7, 8, 9, A, B, C, D, E, F)16, so it has the base 16. This system uses numerical values from 0 to 9 & alphabets from A to F. Alphabets A to F represent decimal numbers from 10 to 15.
- 11. 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
- 12. 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
- 13. 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
- 14. 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
- 16. 16 Successive Division Weighted Multiplication Octal8 0 1 2 3 4 5 6 7 Successive Division Hexadecimal16 0 1 2 3 4 5 6 7 8 9 A B C D E F Weighted Multiplication Successive Division Weighted Multiplication Binary2 0 1 Decimal10 0 1 2 3 4 5 6 7 8 9
- 17. 2510 = 110012 = 318 = 1916 Base
- 19. 12510 => 5 x 100= 5 2 x 101= 20 1 x 102= 100 125 Base Weight
- 21. 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
- 22. 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”
- 24. 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
- 25. 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810
- 27. 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
- 28. ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
- 30. Technique Divide by two, keep track of the remainder First remainder is bit 0 (LSB, least-significant bit) Second remainder is bit 1 Etc.
- 31. 12510 = ?2 2 125 62 12 31 02 15 1 2 7 1 2 3 12 1 12 0 1 12510 = 11111012
- 33. Technique Convert each octal digit to a 3-bit equivalent binary representation
- 34. 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 36. Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation
- 37. 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 39. Technique Divide by 8 Keep track of the remainder
- 40. 123410 = ?8 8 1234 154 2 8 19 2 8 2 3 8 0 2 123410 = 23228
- 42. Technique Divide by 16 Keep track of the remainder
- 43. 123410 = ?16 123410 = 4D216 16 1234 77 216 4 13 = D16 0 4
- 45. Technique Group bits in threes, starting on right Convert to octal digits
- 46. 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 48. Technique Group bits in fours, starting on right Convert to hexadecimal digits
- 49. 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
- 51. Technique Use binary as an intermediary
- 52. 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
- 54. Technique Use binary as an intermediary
- 55. 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148
- 57. Decimal Binary Octal Hexa- decimal 33 100001 41 21 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF
- 58. 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
- 59. 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
- 60. For common bases, add powers 26 210 = 216 = 65,536 or… 26 210 = 64 210 = 64k ab ac = ab+c
- 61. Two 1-bit values A B A + B 0 0 0 0 1 1 1 0 1 1 1 10 “two”
- 62. Two n-bit values Add individual bits Propagate carries E.g., 10101 21 + 11001 + 25 101110 46 11
- 63. Binary, two 1-bit values A B A B 0 0 0 0 1 0 1 0 0 1 1 1
- 64. Binary, two n-bit values As with decimal values E.g., 1110 x 1011 1110 1110 0000 1110 10011010
- 65. Decimal to decimal 3.14 => 4 x 10-2 = 0.04 1 x 10-1 = 0.1 3 x 100 = 3 3.14
- 66. 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
- 67. 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...
- 69. 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
- 70. Study Project Software Based (Verilog) Hardware Based (using digital ICs) Software & Hardware Based (using FPGA)
- 71. Due on 4th October 2013 One Pager Only Group of two students maximum
- 72. Title Group members Objective/Scope Hardware/Software Requirement Timeline