COMPUTER ORGANIZATION AND ARCHITECTURE

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COMPUTER ORGANIZATION ANDARCHITECTURE

• Chapter 2Number System and Data Representation

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Chapter II Number System and Data Representation

• 1. Number System
• 2. Data Representation
• 3. Integer Representation

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1. Number Systems

• Fundamental to understand how computers work is
  understanding the number system that computer use
  to store data and communicate with each other.
• Number system been used to understand computer
• Base 10 (decimal)
• E.g. 394510 / 3945d
• Base 2 (binary)
• E.g. 101010112 / 10101011b
• Base 16 (hexadecimal)
• E.g. 0A3E16 / 0A3Eh

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Number Systems (cont.)

• The Decimal System
• In everyday life we use a system based on decimal
  digits.
• Consider the number 4728 means four thousands,
  seven hundreds, two tens, plus eight
• 4728 (4 x 1000) (7 x 100) (2x10) 8
• The decimal system is said to have a base or
  radix of 10. Each digit in the number is
  multiplied by 10 raised to a power corresponding
  to that digits position
• 4728 (4 x 103) (7 x 102) (2 x 101) (8 x
  100)

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Number Systems (cont.)
The Binary System

• In the binary system, we have only two digits, 1
  and 0. Thus, number in the binary system are
  represented to the base 2.
• Each digits in a binary number also have a value
  depending on its position
• 1002 (1 x 22 ) (0 x 21 ) (0 x20 ) 410
• 101011b (1 x 25) (0 x 24) (1 x 23) (0 x
  22) (1 x 21)
• (1 x 20) 43d

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Number Systems (cont.)
The Hexadecimal System

• A computers world is a binary world and
  communication of instruction and data by the
  devices that process them is always in binary.
• Binary system is very difficult for human being.
  Human being are comfortable to decimal number
  system.
• However calculations to convert binary to decimal
  are relatively complex.
• A notation known as hexadecimal has been adopted.
  Binary digits are grouped into sets of four. Each
  possible combination of four binary digits is
  given a symbol (hexadecimal digits) as follows

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Number Systems (cont.)
The Hexadecimal System (cont.)
0000 0 0001 1 0010 2 0011 3 0100 4 0101
5 0110 6 0111 7
1000 8 1001 9 1010 A 1011 B 1100 C 1101
D 1110 E 1111 F
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Number Systems (cont.)
The Hexadecimal System (cont.)

• In the hexadecimal system, we have 16 hexadecimal
  digits. Thus, number in the hexadecimal system
  are represented to the base 16.
• Each digits in a hexadecimal number also have a
  value depending on its position
• 2C16 (2 x 161) (C x 160)
• (2 x 161) (12 x 160) 4410
• The reason for using hexadecimal notation are
  because it is more compact than binary notation
  and it is extremely easy to convert between
  binary and hexadecimal.

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Number Systems (cont.)
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Conversion Between Number Systems
Number Systems (cont.)

• Converting Binary to Decimal
• Converting Hex to Decimal
• Converting Decimal to Binary
• Converting Decimal to Hex
• Converting Between Hex and Binary

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Converting Binary to Decimal
Number Systems (cont.)

• 101001b to decimal
• 101001b (1 x 25) (0 x 24) (1 x 23) (0 x
  22) (0 x 21) (1 x 20)
• 32 0 8 0 0 1
• 41d

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Number Systems (cont.)
Converting Hex to Decimal

• A3F16 to decimal
• A3F16 (A x 162) (3 x 161) (F x 160 )
• (10 x 256) (3 x 16) (15 x 1)
• 262310

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Converting Decimal to Binary
Number Systems (cont.)

• Decimal can be converted in to a binary systems
  with the Remainder Method
• Example Convert 26d to base 2
• gt 26d 11010b

26/2 13 0 13/2 6 1 6/2 3
0 3/2 1 1 1/2 0 1
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Number Systems (cont.)
Converting Decimal to Binary (Floating point
number)

• How about floating point number?
• E.g. Convert 0.875d into base 2 number.
• gt 0.875d 0.1110b

0.875 x 2 1.75 1 0.75 x 2 1.5 1 0.5
x 2 1.0 1 0 x 2 0 0
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Converting Decimal to Hex
Number Systems (cont.)

• Decimal can be converted into a hex with the
  Remainder Method
• Example Convert 425d to base 16
• 425 / 16 26 9 -gt 9
• 26 / 16 1 10 -gt A
• 1 / 16 0 1 -gt 1
• gt 425d 1A9h

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Converting Between Hex and Binary
Number Systems (cont.)

• To convert a hex number to binary, we need only
  express each hex digit in binary
• E.g. Convert DE116 to binary
• D E 1
• 1101 1110 0001
• 110111100001b
• To go from binary to hex, just reverse this
  process
• 100100012 1001 0001 9116

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1. Number System 2. Data Representation 3.
Integer Representation
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2. Data Representation

• A common form of data are letters of the alphabet
  (A to Z), numerals (0 to 9), some symbols (_at_, ,
  ) and certain control character (Ctrl,Shift).
• This types of data is convenient for human being
  but all of the data in digital computer
  represented in binary form.
• Some coding systems are used to represent these
  data into the binary form. (Characters are
  represented by a sequence of bits.)

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Data Representation (cont.)

• The famous coding system been used for data
  representation are
• ASCII (American Standard Code for Information
  Interchange)
• EBCDIC (Extended Binary Coded Decimal Interchange
  Code )
• BCD (Binary Coded Decimal)

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ASCII
Data Representation (cont.)

• ASCII is used in almost all present-day personal
  computers.
• Each alphabetic, numeric, or special character is
  represented with a 7-bit binary number (a string
  of seven 0s or 1s).
• 128 possible characters can be represented.
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• The eight bit may be set to 0 or used as a parity
  bit for error checking on communication lines or
  other device-specific functions.
• Example char A65 in decimal,41 in hex,
  0100 0001 in binary.

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Data Representation (cont.)
ASCII Control Characters
The first thirty-two codes (numbers 0-31 decimal)
in ASCII are reserved for control characters
codes that may not themselves represent
information, but that are used to control devices
(such as printers) that make use of ASCII. For
example, character 10 represents the "line feed"
function (which causes a printer to advance its
paper), and character 27 represents the "escape"
key found on the top left of common keyboards.
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Data Representation (cont.)
ASCII Printable Characters
Code 32 is the "space" character, denoting the
space between words, which is produced by the
large space bar of a keyboard.
Codes 33 to 126 are called the printable
characters, which represent letters, digits,
punctuation marks, and a few miscellaneous
symbols.
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Data Representation (cont.)
ASCII Printable Characters
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Data Representation (cont.)
ASCII Printable Characters (cont.)
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Data Representation (cont.)
ASCII Printable Characters (cont.)
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EBCDIC
Data Representation (cont.)

• EBCDIC (pronounced either "ehb-suh-dik" or
  "ehb-kuh-dik") is a binary code for alphabetic
  and numeric characters that IBM developed for its
  larger operating systems
• Each alphabetic or numeric character is
  represented with an 8-bit binary number (a string
  of eight 0's or 1's).
• 256 possible characters (letters of the alphabet,
  numerals, and special characters) are defined.
• EBCDIC uses more or less the same characters as
  ASCII, but different code points. Example A C1
  in hex, 1100 0011 in binary.
• Today outside IBM everyone uses ASCII instead
  EBCDIC is considered a bit of a dinosaur.

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Data Representation (cont.)
BCD

• In BCD, 4 bit binary number were used to
  represent 1 decimal number ( e.g 3d0011b,
  9d1001b)
• Highest decimal number were coded to BCD is 9
  (1001). Thus, 1010, 1011, 1110 and 1111 were not
  used.
• To encode the number such as 43 use
• 43d 0100 0011b
• The BCD format usually used in the BIOS at
  Personal Computer (PC) to keeps the date and time
  for historical reason.

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1. Number System 2. Data Representation 3.
Integer Representation
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3. Integer Representation (cont.)

• For the purpose of computer storage and
  processing, only binary digits (0 and 1) may be
  used to represent numbers (negative or positive).
• For a 8-bit number, there are 28 256 possible
  bit patterns.
• For unsigned number, we can represent 0 to 255
  using 8-bit number.
• For signed number, the most significant
  (leftmost) bit usually used as a sign bit. ( 0
  for positive and 1 for negative number).

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Integer Representation (cont.)

• There are several alternative conventions used to
  represent negative integers. Some of them are

• Signed magnitude
• Ones complement
• Twos complement

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Signed Magnitude
Integer Representation (cont.)

• Also know as sign and magnitude, the leftmost
  bit is the sign and the rest are magnitude
• 0 positive
• 1 negative
• Sign Magnitude

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Integer Representation (cont.)
Signed Magnitude (cont.)

• Example (for 8-bit number)
• 25d 0 0011001b
• -25d 1 0011001b
• Largest number is 127 and smallest number is
  127
• ProblemsTwo representations for zero
• 0 00000000b
• -0 10000000b

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Ones Complement
Integer Representation (cont.)

• The leftmost bits is the sign ( 0 positive, 1
  negative)
• Negative number is obtained by complementing each
  bit from 0 to 1 or from 1 to 0
• Example (8-bit number)
• 25d 00011001b
• -25d 11100110b
• Two representation of zero 0d 00000000b and
  -0d 11111111
• Largest number is 127 and smallest number is -127

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Integer Representation (cont.)
Twos complement

• The leftmost bit is the sign bit (0positive, 1
  negative)
• Negative of the number is obtained by adding 1 to
  the ones complement negative,
• Example(8-bit number)
• 25d 00011001b
• -25d 11100111b
• One representation for zero 0000000b
• Largest number is 127 and smallest number is -128

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Range in Integer Representation
Integer Representation (cont.)

• For n bit number, highest integer value can be
  represent is 2n-1 -1.

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• Number System
• Data Representation
• Integer Representation