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Binary Numbers

Binary Numbers. Numbering Systems Counting Symbolic Bases Common Bases (10, 2, 8, 16) Representing Information Binary to Decimal Conversions Bit Groupings Encoding Information. Numbering Systems. There are many different ways to represent numbers Counting Based Systems

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Binary Numbers

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  1. Binary Numbers • Numbering Systems • Counting • Symbolic • Bases • Common Bases (10, 2, 8, 16) • Representing Information • Binary to Decimal Conversions • Bit Groupings • Encoding Information Craig Schock, 2003

  2. Numbering Systems • There are many different ways to represent numbers • Counting Based Systems • The number is represented by the same number of counters • Symbolic • Uses symbols to represent values • 7 = Craig Schock, 2003

  3. Symbolic Numbering Systems • The symbols within the numbering systems are an “abstraction” of the actual number. • The symbol represents a value • Abstraction: A symbolic representation of a thing or idea. • In order for an abstraction to be useful, people must agree on its meaning. • Symbols are combined to represent larger numbers Craig Schock, 2003

  4. Numbering Bases • Each symbolic numbering system has a “base” • Base: The number of symbols in the system • eg. Decimal has 10 symbols (0-9) 0 1 2 3 4 5 6 7 8 9 Craig Schock, 2003

  5. Common Bases • The most common base used in western society is base 10 (decimal) • It is based on 10 symbols • Humans have 10 “digits” • In the computer world, there are other commonly used bases: • Binary (base 2, 0-1) • Octal (base 8, 0-7) • Hexadecimal (base 16, 0-9, A-F) Craig Schock, 2003

  6. Decimal: Representing Numbers • Because decimal has 10 digits, it can easily represent numbers 0-9. • 0 5 • 1 6 • 2 7 • 3 8 • 4 9 • However, representing numbers which are larger than the number of symbols poses a problem. Craig Schock, 2003

  7. Decimal: Positional Notation • Symbolic systems deal with representing large numbers through a positional notation. • The actual value of the symbol is based on its position • Multiplication and addition is employed to increase the value • The factor used in the multiplication is the base First column starts at “1” Note: base 10 x10x10 x10 x1 7 x 10 x 10 + 5 x 10 + 6 x 1 7 5 6 = 2 1 0 10 10 10 Craig Schock, 2003

  8. Information • Information: What is it? • A message received and understood that reduces the recipient’s uncertainty • “Uncertainty” implies that for information to be present, there must be at least 2 possible outcomes. • If there was only 1 outcome, there would be no uncertainty. • If we were to represent information as a numbering system, it would be a system which contained at least 2 symbols. Craig Schock, 2003

  9. Representing Information • We could choose any numbering system to represent information. • Whatever system we choose, it must contain at least 2 symbols. • Electrically, it is easy to represent a system containing 2 symbols using switches. • Off • On • We can also think of philosophical systems based on “truthness” and “falseness” Craig Schock, 2003

  10. Binary – Base 2 • Binary is a numbering system which contains 2 symbols (i.e. base 2) • For easy interoperability with other bases (such as decimal), we choose the digits “0” and “1” to represent the two possible values • A “bit” is a single BInary digiT • A bit is the smallest unit of information • What if the information we need to represent has more than 2 possibilities? Craig Schock, 2003

  11. Binary – Positional Notation • As with decimal, we can represent values which are larger than the number of symbols we have. • This is accomplished through the use of a positional notation. Because the base is 2, the multiplier for each position is x2 Note: base 2 x2x2 x2 x1 1 x 2 x 2 + 1 x 2 + 1 x 1 1 1 1 = 2 1 0 2 2 2 Craig Schock, 2003

  12. Binary to Decimal Conversions • Because humans work well with decimal, it is useful to know how to convert between binary and decimal: • 0000 = 0 1000 = 8 • 0001 = 1 1001 = 9 • 0010 = 2 1010 = 10 • 0011 = 3 1011 = 11 • 0100 = 4 1100 = 12 • 0101 = 5 1101 = 13 • 0110 = 6 1110 = 14 • 0111 = 7 1111 = 15 Craig Schock, 2003

  13. Binary to Decimal Conversions (contd) 8 4 2 1 1x8 + 1x4 + 1x1 = 8 + 4 + 1 = 13 = 1 1 0 1 To convert to decimal, represent the column values in decimal 128 64 32 16 8 4 2 1 1 0 0 1 1 1 0 1 = 1x128 + 1x16 + 1x8 + 1x4 + 1x1 = 128 + 16 + 8 + 4 + 1 = 157 Craig Schock, 2003

  14. How many bits to use? • In the last few slides, we have seen binary numbers which contain differing numbers of digits – 3 bits, 4 bits, and 8 bits • How many bits should one use? • That depends on what information one wishes to represent. • eg. How many bits are necessary to represent the days of the week? Craig Schock, 2003

  15. How Many bits? • Days of the week. Let’s let each bit combination represent one day: • 000 = Sunday • 001 = Monday • 010 = Tuesday • 011 = Wednesday • 100 = Thursday • 101 = Friday • 110 = Saturday • Because there are 7 days of the week, we need enough bits which have at least 7 different combinations. (3 bits) Craig Schock, 2003

  16. Bits and Combinations • How can we tell how many combinations a given number of bits will provide? • The answer is based on a simple formula: n n bits provides 2 combinations 1 bit = 2 combinations 2 bits = 4 combinations 3 bits = 8 combinations 4 bits = 16 combinations 5 bits = 32 combinations 6 bits = 64 combinations 7 bits = 128 combinations 8 bits = 256 combinations 9 bits = 512 combinations 10 bits = 1024 combinations 11 bits = 2048 combinations 12 bits = 4096 combinations 13 bits = 8192 combinations 14 bits = 16384 combinations 15 bits = 32768 combinations 16 bits = 65536 combinations Craig Schock, 2003

  17. More combinations • How many bits are necessary to represent • The days of the month? • Your age • The year of your birth? • The number of cars you have owned? • Your salary? • The salary of a CEO of a multinational corporation? • The Canadian national debt? • The US national debt? • The number of atoms in the universe? Craig Schock, 2003

  18. Bit groupings • Relatively speaking, there are few cases where the information content only requires 1 bit. • Bits are usually grouped into larger units. • Common groupings include: • 4 bits = 1 nybble (not commonly used) • 8 bits = 1 byte • 1024 bytes = 1 kilobyte (or 1K) • 1024 * 1024 bits = 1048576 bits = 1 megabits = 131072 bytes • 1024 * 1024 bytes = 1048576 bytes = 1 Megabyte • 1024 * 1024 * 1024 bytes = 1073741824 bytes = 1 Gigabyte Craig Schock, 2003

  19. What does it mean? • Exercise: What do the following numbers mean? • 102, 102, 102, 126, 126, 102, 102, 102 • 0, 24, 24, 0, 24, 24, 24, 24 • 24, 24, 24, 24, 24, 0, 24, 24 Craig Schock, 2003

  20. What does it mean? 102 = 01100110 0 = 00000000 24 = 00011000 102 = 01100110 24 = 00011000 24 = 00011000 102 = 01100110 24 = 00011000 24 = 00011000 126 = 01111110 0 = 00000000 24 = 00011000 126 = 01111110 24 = 00011000 24 = 00011000 102 = 01100110 24 = 00011000 0 = 00000000 102 = 01100110 24 = 00011000 24 = 00011000 102 = 01100110 24 = 00011000 24 = 00011000 Craig Schock, 2003

  21. What does it mean? 11 11 11 11 11 11 11 11 11 11 11 111111 11 111111 11 11 11 11 11 11 11 11 11 11 11 11 11 Craig Schock, 2003

  22. Information Encoding • The exercise illustrates and important concept in computer science. • We have learned that binary numbers can be used to represent information. Information, as a series of bits, can be represented as decimal numbers. The example provided a series of decimal numbers. But what do they mean? • What the numbers mean is a matter of definition • Information is encoded using bits • Encoding is an abstraction. Craig Schock, 2003

  23. ASCII Codes • ASCII – American Standard Code for Information Interchange • ASCII is an 7 bit encoding which is used to represent the Roman alphabet, numbers, standard symbols and printer control characters. • Some companies expanded ASCII to 8 bits. They used the extra combinations to encode special characters Craig Schock, 2003

  24. The ASCII Standard Dec Char 0 NUL 1 SOH 2 STX 3 ETX 4 EOT 5 ENQ 6 ACK 7 BEL 8 BS 9 HT 10 LF 11 VT 12 FF 13 CR 14 SO 15 SI 16 DLE 17 DC1 18 DC2 19 DC3 20 DC4 21 NAK 22 SYN Dec Char 23 ETB 24 CAN 25 EM 26 SUB 27 ESC 28 FS 29 GS 30 RS 31 US 32 SPACE 33 ! 34 " 35 # 36 $ 37 % 38 & 39 ' 40 ( 41 ) 42 * 43 + 44 , 45 - Dec Char 46 . 47 / 48 0 49 1 50 2 51 3 52 4 53 5 54 6 55 7 56 8 57 9 58 : 59 ; 60 < 61 = 62 > 63 ? 64 @ 65 A 66 B 67 C 68 D Dec Char 69 E 70 F 71 G 72 H 73 I 74 J 75 K 76 L 77 M 78 N 79 O 80 P 81 Q 82 R 83 S 84 T 85 U 86 V 87 W 88 X 89 Y 90 Z 91 [ Dec Char 92 \ 93 ] 94 ^ 95 _ 96 ` 97 a 98 b 99 c 100 d 101 e 102 f 103 g 104 h 105 i 106 j 107 k 108 l 109 m 110 n 111 o 112 p 113 q 114 r Dec Char 115 s 116 t 117 u 118 v 119 w 120 x 121 y 122 z 123 { 124 | 125 } 126 ~ 127 DEL Craig Schock, 2003

  25. Using ASCII • Using the ASCII standard, we can encode an English sentence as a series of ASCII values • I would like to buy some cheese. • 73, 32, 119, 111, 117, 108, 100, 32, 108, 105, 107, 101, 32, 116, 111, 32, 98, 117, 121, 32, 115, 111, 109, 101, 32, 99, 104, 101, 101, 115, 101, 46 • How many bytes are needed to encode the above sentence using ASCII? • 32 byes Craig Schock, 2003

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