Numbers and Number Systems

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Numbers and Number Systems

• Instructor Eng. Haya Sammaneh

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Introduction
Input device
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Number Systems

• Decimal -- 10 symbols (0,1,2,3,4,5,6,7,8,9)
• Binary -- 2 symbols (0,1)
• Octal -- 8 symbols (0,1,2,3,4,5,6,7)
• Hexadecimal -- 16 symbols (0,1,2,3,4,5,6,7,8,9,A,B
  ,C,D,E,F) , A 10 , B 11 .F 15

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Converting with fraction, examples

• Binary to Decimal
• Ex (1101.1)2 ? 123 122 021 120 12-1
  (13.5)10
• Octal to Decimal
• Ex (673)8 ? 6 82 781 380 (443)10
• Hexadecimal to Decimal
• Ex. (A9C)16 ? A 162 9161 C 160
• 10 162 9161 12
  160 (2788)10

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Converting, example

• Binary to Octal
• Ex (011011.101100)2?(33.54)8
• Binary to Hexadecimal
• Ex (11111011.11011000)2?(FB.D8)16
• Hexadecimal to Binary
• Ex. (A3.B)16 ? (10100011.1011)2

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Converting, example

• Decimal to Binary
• EX (12.3)10 ? (1100.01001)2
• 12 / 2 6 ( Remainder 0 (right) )
• 6 / 2 3 ( Remainder 0 )
• 3 / 2 1 ( Remainder 1 )
• 1 / 2 0 ( Remainder 1 (left))
• 0.3 2 0.6 (left)
• 0.6 2 1.2
• 0.2 2 0.4
• 0.4 2 0.8
• 0.8 2 1.6 (right)
• Using the same method to convert the Decimal
  number to any base.

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Signed Numbers

• In general a N-bit integer can store numbers in
  the range of -2N-1 -1 to 2N -1

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Ways to represent negatives

• Negative integers are stored via twos complement
  representation
• 1s complement
• reverse the bits to get the negative
• Ex 1101 ? 1s complement ? 0010
• 2s complement
• It happens by reversing the bits (1s complement)
  then adding 1.
• Ex 1101 ? 2s complement ? 0011
• Get 1s complement ? 0010
• then add 1 ? 0011

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Binary arithmetic addition and multiplication

• 0 0 0
• 0 1 1
• Overflow If there is not enough room to hold
  the result correctly.
• If the two numbers are of opposite signs, no
  overflow can occur. (Why not?)
• multiplication
• 0 0 0
• 0 1 0
• 1 0 0
• 1 1 1

• 1 0 1
• 1 1 0 and carry 1

(Result is smaller than one of them)
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Concepts of bit, byte and word

• Bit is the smallest data item in computers (short
  for "binary digit" . Each data item, or bit, can
  assume either the value 0 or the value 1.
• Computer circuitry performs various simple bit
  manipulations, such as examining the value of a
  bit, setting the value of a bit and reversing a
  bit (from 1 to 0 or from 0 to 1).
• Bytes are composed of eight bits which is the
  smallest grouping of numbers . Large amounts of
  memory are indicated in terms of kilobytes (1,024
  bytes), megabytes (1,048,576 bytes), and
  gigabytes (1,073,741,824 bytes).
• Words The size of a word varies from one
  computer to another, depending on the CPU. For
  computers with a 16-bit CPU, a word is 16 bits (2
  bytes). On large mainframes, a word can be as
  long as 64 bits (8 bytes) or 128 bit (16 byte).
• Some computers and programming languages
  distinguish between short-words and long-words. A
  short-word is usually 2 bytes long, while a
  long-word is 4 bytes.

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Standard Alphanumeric Formats

• Problem Representing text strings, such as
  Hello, world, in a computer
• The standards for representing letters (alpha)
  and numbers
• ASCII American standard code for information
  interchange
• Unicode

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Character Code ASCII and Unicode

• Why do computers use numbers for the names of
  letters?
• Because they store all information in number
  form.
• Technical detail they store information as
  bytes each byte consists of 8 bits each
  bit is either the number 0 or 1

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Character Code ASCII and Unicode

• ASCII and Unicode are two computer languages
  for naming letters
• The ASCII name for a is 61
• The Unicode name for a is U0061

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ASCII Reference TableControl , Numeric,
Alphabetic, Punctuations Codes
7416 111 0100
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ASCII

• Most widely used coding scheme
• Computer systems can represent up to 256 letters
• Technical detail with one 8-bit byte (28 256)
• ASCII only uses 7 bits (27 128)
• The first 32-127 are called ASCII letters
  (characters)
• 1-32 Thats for control characters like the
  option key.

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ASCII Problem

• No one agrees on what letters the numbers 128-256
  stand for.

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Unicode fixing the ASCII problem

• Most common 16-bit form represents 65,536
  characters.
• Multilingual These characters cover the
  principal written languages of the Americas,
  Europe, the Middle East, Africa, India, Asia, and
  Pacifica.