Chapter 3 Data Representation

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Chapter 3Data Representation
2
Chapter Goals
Chapter 3
Systems Architecture

• Describe numbering systems and their use in data
  representation
• Compare and contrast various data representation
  methods
• Describe how nonnumeric data is represented
• Describe common data structures and their uses

3
Data Representation and Processing
Chapter 3
Systems Architecture

• Data can be manipulated in a variety of forms.
• Arabic numerals
• Roman numerals
• Lines or tick marks
• Pictorial characters
• Alphabetic characters
• Sound Waves

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Data Representation and Processing
Chapter 3
Systems Architecture

• To be processed by the brain, the data must be
  converted to an appropriate internal format.
• Sight
• Smell
• Taste
• Sound
• Skin Sensation
• Electrical Impulses

5
Data Representation and Processing
Chapter 3
Systems Architecture

• Data and information processors must be able
  to
• Recognize external data and convert it to an
  appropriate internal format
• Store and retrieve data internally
• Transport data among internal storage and
  processing components

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Automated Data Processing
Chapter 3
Systems Architecture

• Processing is implemented with electrical
  switches.
• Switches are combined to form processing
  circuits.
• For Example A B C

7
Automated Data Processing
Chapter 3
Systems Architecture
8
Binary Representation of Data
Chapter 3
Systems Architecture

• Computers represent data using binary numbers.
• Binary numbers correspond directly with values in
  boolean logic.
• Computers combine multiple digits to form a
  single data value to represent large numbers.

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Binary Data Representation
Chapter 3
Systems Architecture

• The symbol used to represent a digit and digit
  position within a string determine its value.
• For Example
• (5 x 1000) (6 x 100) (8 x 10) 9
• 5,000 600 80 9
  5,689

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Binary Data Representation
Chapter 3
Systems Architecture

• The multiplier that describes the difference
  between one position and the next is the base
  (radix).
• The base(radix) of a decimal number system is 10.
• The base(radix) of a binary number system is 2.

11
Binary Data Representation
Chapter 3
Systems Architecture

• The number of characters in the number system is
  equal to the base of the number system.
• There are 10 characters in the decimal number
  system. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9)
• There are 2 characters in the binary number
  system. (0, 1)

12
Binary Data Representation
Chapter 3
Systems Architecture

• The fractional part of a numeric value is
  separated from the whole number by a period
  (radix point).
• For Example 5,689.368
• (3 x .1) (6 x .01) (8 x .001)
• 0.3 0.06 0.008 0.368

13
Binary Data Representation
Chapter 3
Systems Architecture

• In the binary numbering system
• The first position to the right of the radix
  point represents halves ( 2-1)
• The second position to the right of the radix
  point represents quarters (2-2)
• The third position to the right of the radix
  point represents eights (2-3)

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Binary Data Representation
Chapter 3
Systems Architecture
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Binary Data Representation
Chapter 3
Systems Architecture

• You can convert a binary number to its decimal
  equivalent by multiplying the value of each
  position by the decimal weight of that position,
  then summing the results.

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Binary Data Representation
Chapter 3
Systems Architecture
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Binary Data Representation
Chapter 3
Systems Architecture
18
Hexadecimal Notation
Chapter 3
Systems Architecture

• The base(radix) of a hexadecimal number system is
  16.
• There are 16 characters in the hexadecimal number
  system.
• There are only 10 characters in the Arabic number
  system that can be used to represent some of the
  16 characters in the hexadecimal number system.
• The letters A, B, C, D, E, F are used to
  represent the last 6 characters in the
  hexadecimal number system.

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Hexadecimal Notation
Chapter 3
Systems Architecture
20
Hexadecimal Notation
Chapter 3
Systems Architecture

• The primary advantage of hexadecimal notation, it
  its compactness.
• Hexadecimal numbers are often used to represent
  memory addresses.
• Bit strings are usually expressed in binary,
  memory address are expressed in hexadecimal.

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Octal Notation
Chapter 3
Systems Architecture

• Some operating systems and machine language
  programs use octal notation.
• The base(radix) of an Octal number system is 8.
• There are 8 characters in the octal number
  system. (0, 1, 2, 3, 4, 5, 6, 7)

22
Goals of Computer Data Representation
Chapter 3
Systems Architecture

• Any representation format for numeric data
  represents a balance among several factors,
  including
• Compactness
• Accuracy
• Range
• Ease of manipulation
• Standardization

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Goals of Computer Data Representation
Chapter 3
Systems Architecture

• Compactness
• Compact data representation requires less
  storage space.
• Compact data representation requires less
  expensive processing and storage devices.

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Goals of Computer Data Representation
Chapter 3
Systems Architecture

• Accuracy
• The accuracy of representation increases with the
  number of data bits used.
• Routine calculations can generate quantities that
  are either too large or too small to be stored
  within finite circuitry.

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Goals of Computer Data Representation
Chapter 3
Systems Architecture

• Ease of manipulation
• The efficiency of a processor depends on its
  complexity.
• Efficient processor circuits perform their
  function quickly.

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Goals of Computer Data Representation
Chapter 3
Systems Architecture

• Standardization
• Data formats must be suitable for a wide variety
  of devices and computer systems.
• Various organizations have created standard data
  encoding methods for communication among computer
  systems and their components.

27
CPU Data Types
Chapter 3
Systems Architecture

• Five Primitive Data Types
• Integer
• Excess Notation
• Twos Complement Notation
• Real number
• Floating Point Notation
• Character
• Boolean
• Memory address

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CPU Data Types
Chapter 3
Systems Architecture

• Integers
• An integer is a whole number (For example 3, 5,
  6)
• Integers can be signed or unsigned
• A signed integer uses one bit to represent the
  sign
• The sign bit is the high order bit

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CPU Data Types
Chapter 3
Systems Architecture

• Excess Notation
• Excess notation is used to represent signed
  integers
• A fixed number of bits is used to represent the
  number
• The leftmost digit represents the sign
• The leftmost digit is 1 for positive values
• The leftmost digit is 0 for negative values

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CPU Data Types
Chapter 3
Systems Architecture
31
CPU Data Types
Chapter 3
Systems Architecture

• Twos Complement Notation
• Nonnegative integer values are represented as
  ordinary binary numbers
• Negative integer values are represented using
  (Complement of positive value 1)
• The complement of a number is formed by changing
  all 1 bits to 0 and all 0 bits to 1

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CPU Data Types
Chapter 3
Systems Architecture

• Twos Complement Notation
• For Example
• Decimal number -710
• Binary representation 01112
• Twos Complement 10002
• Add one 00012
• 10012 (-710)

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CPU Data Types
Chapter 3
Systems Architecture

• Twos Complement Notation
• The leftmost bit represents the sign
• A fixed number of bit positions are used
• Only two logic circuits are required to perform
  addition on single-bit values
• Subtraction can be performed as addition of a
  negative value

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CPU Data Types
Chapter 3
Systems Architecture

• Range and Overflow
• Most modern CPUs use either 32 or 64 bits to
  represent a twos complement value
• The numeric range of a twos complement value is
  (2 n-1) t o(2 n-1-1)
• For example the range for a 32 bit number is 2
  32-1 to 2 32-1-1 -2 31 to 2 31 - 1

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CPU Data Types
Chapter 3
Systems Architecture

• Range and Overflow
• If data is too large to store in the 32 or 64
  bits, then overflow occurs
• Overflow is treated as an error by the CPU
• To avoid overflow some computers and programming
  languages define additional data types as double
  precision (long integer)

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CPU Data Types
Chapter 3
Systems Architecture

• Real Numbers
• A real number can contain both whole and
  fractional components
• The whole portion appears to the left of the
  radix point
• The fractional portion appears to the right of
  the radix point

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CPU Data Types
Chapter 3
Systems Architecture

• Real Numbers
• For Example
• 18.56 (18 whole portion)
• (.56 fractional portion)

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CPU Data Types
Chapter 3
Systems Architecture
39
CPU Data Types
Chapter 3
Systems Architecture

• Floating Point Notation
• Floating point notation is used to represent very
  small numbers and very large numbers
• Values can either be very large or very small,
  but not both at the same time

40
CPU Data Types
Chapter 3
Systems Architecture

• Floating Point Notation
• The exponent attached to the base can be
  interpreted as the number and the direction of
  positional moves of the radix point
• Negative exponents indicate a movement to the
  left, and positive exponents indicate movement to
  the right

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CPU Data Types
Chapter 3
Systems Architecture
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CPU Data Types
Chapter 3
Systems Architecture

• Range, Overflow and Underflow
• The number of bits in the string represent the
  range of values
• Number of bits in the mantissa
• Number of bits in the exponent

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CPU Data Types
Chapter 3
Systems Architecture
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CPU Data Types
Chapter 3
Systems Architecture

• Range, Overflow and Underflow
• Overflow occurs when the exponent is larger than
  the allocated space.
• Underflow occurs when a negative exponent is too
  large in absolute value to fit within the bits
  allocated to store it.

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CPU Data Types
Chapter 3
Systems Architecture

• Precision accuracy is reduced as the number of
  digits available to store the mantissa is
  reduced.
• Truncation if the number of digits in the
  mantissa is larger than the allocated space,the
  number of digits in the mantissa is truncated.

46
CPU Data Types
Chapter 3
Systems Architecture

• Character Data
• An individual symbol is a character.
• Characters grouped together form a string.
• Character data can only be represented in the
  computer system using a coding scheme.

47
CPU Data Types
Chapter 3
Systems Architecture

• Character Data
• Coding Scheme Characteristics
• All users must use the same table of
  correspondence.
• The values used to represent the data must be
  capable of being encoded, transmitted and decoded.

48
CPU Data Types
Chapter 3
Systems Architecture

• Character Data
• Coding Scheme Characteristics
• The specific coding method represents a tradeoff
  among compactness, ease of manipulation,
  accuracy, range and standardization.

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CPU Data Types
Chapter 3
Systems Architecture

• BCD
• Binary Coded Decimal (BCD)
• Character coding method used by early IBM
  mainframe computers.
• Characters are encoded as strings of six bits.
• 64 - (26) symbols are represented.

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CPU Data Types
Chapter 3
Systems Architecture

• EBCDIC
• Extended Binary Coded Decimal Interchange
  Code
• 8 bit coding method used by IBM mainframe
  computers.
• Characters are encoded as strings of eight bits.
• 256 - (28) symbols are represented.

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CPU Data Types
Chapter 3
Systems Architecture

• ASCII
• American Standard Code of Information
  Interchange
• Coding method used in data communication that has
  been adopted by the United States.
• 7-bit and 8-bit formats
• Includes device control codes

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CPU Data Types
Chapter 3
Systems Architecture
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CPU Data Types
Chapter 3
Systems Architecture
54
CPU Data Types
Chapter 3
Systems Architecture

• Unicode
• Multilingual character encoding standard
  encompassing all of the worlds written
  languages.
• Characters are coded using 16 bit strings.
• 65,535 (216) characters are represented.

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CPU Data Types
Chapter 3
Systems Architecture

• Boolean Data
• Two data values true and false.
• Data is represented using a single bit.
• Binary 1 can represent true and binary 0 can
  represent false.

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CPU Data Types
Chapter 3
Systems Architecture

• Memory Addresses
• Flat memory model
• Memory addresses can be represented using a
  single integer.
• Segmented memory model
• Memory addresses require multiple integers.

57
Data Structures
Chapter 3
Systems Architecture

• A data structure is a related group of
  primitive data elements that is organized for
  some type of processing.
• Data structures are defined and manipulated
  within software.

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Data Structures
Chapter 3
Systems Architecture
59
Data Structures
Chapter 3
Systems Architecture

• Virtually all data structures make extensive
  use of pointers and addresses.
• Pointer a data element that contains the
  address of another data element.
• Address the location of some data element
  within a storage device.

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Arrays and Lists
Chapter 3
Systems Architecture

• An array is an ordered list in which each
  element can be referenced by an index to its
  position.

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Arrays and Lists
Chapter 3
Systems Architecture
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Arrays and Lists
Chapter 3
Systems Architecture

• Array elements can be stored in contiguous, or
  sequential memory locations.
• Each character is stored in a single byte of
  memory.
• Characters are ordered in sequential byte
  locations.
• Access to individual elements of the array can be
  achieved by using the starting address of the
  array and the index of the element.

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Arrays and Lists
Chapter 3
Systems Architecture
64
Arrays and Linked Lists
Chapter 3
Systems Architecture

• Linked List
• A linked list is a data structure that uses
  pointer so list elements can be scattered among
  nonsequential storage locations.

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Arrays and Linked Lists
Chapter 3
Systems Architecture

• Linked List
• In a singly linked list each element occupies
  two storage locations
• Data value of a list element
• Address of the next element value

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Arrays and Linked Lists
Chapter 3
Systems Architecture

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Arrays and Lists
Chapter 3
Systems Architecture
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Arrays and Lists
Chapter 3
Systems Architecture

• The procedure to add a new element
• Allocate storage for the new element.
• Copy the pointer from the element preceding the
  new element into the pointer field of the new
  element.
• Write the address of the new element into the
  pointer field of the preceding element.

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Arrays and Lists
Chapter 3
Systems Architecture
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Arrays and Lists
Chapter 3
Systems Architecture

• Insertion of an element into a list stored
  in contiguous memory
• Allocate a new storage location at the end of the
  list.
• For each element past the insertion point, copy
  the element value into the next storage location.
• Write the new element value in the storage
  location at the insertion point.

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Arrays and Lists
Chapter 3
Systems Architecture
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Arrays and Lists
Chapter 3
Systems Architecture

• A doubly linked list
• Each element has two pointers
• One pointer points to the next element in the
  list
• One pointer points to the previous element in the
  list

73
Arrays and Lists
Chapter 3
Systems Architecture

• Doubly linked list
• Advantage
• The list can be traversed in any order
• Disadvantage
• More pointers must be updated each time an
  element is inserted or deleted
• More storage locations are required to hold the
  extra set of pointers

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Arrays and Lists
Chapter 3
Systems Architecture
75
Records and Files
Chapter 3
Systems Architecture

• A record is a data structure composed of other
  data structures or primitive data elements.
• Records are used as a unit of input and output to
  files or databases.

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Records and Files
Chapter 3
Systems Architecture
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Records and Files
Chapter 3
Systems Architecture

• A sequence of records on secondary storage is
  called a file.
• A sequence of records stored within main memory
  is called a table.
• Sequential files suffer the same problems as
  contiguous arrays when inserting and deleting
  records.
• To eliminate this problem, linked lists and
  indexed files are used.

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Records and Files
Chapter 3
Systems Architecture
79
Classes and Objects
Chapter 3
Systems Architecture

• An alternative view of computer and software
  behavior has been developed.
• This view involves classes and objects.
• A class is a data structure that contains both
  data elements and programs that manipulate that
  data.
• The programs in a class are methods.
• An object is one instance or variable of the
  class.

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Classes and Objects
Chapter 3
Systems Architecture
81
Summary
Chapter 3
Systems Architecture

• To be processed by any device, data must be
  converted from its native format into a form
  suitable for the processing device.
• All data, including nonnumeric data, are
  represented within a modern computer system as
  strings of binary digits, or bits.
• Each bit string has a specific data format and
  coding method.

82
Summary
Chapter 3
Systems Architecture

• Numeric data is stored using integer, real
  number, and floating point formats.
• Characters are converted to numbers by means of a
  coding table.
• Boolean vales can have only two values, true and
  false.
• Programs often need to define and manipulate data
  in larger and more complex units than primitive
  CPU data types.