Data Representation

1
Chapter 3

• Data Representation

2
Chapter goals

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

3
Data representation

• Humans have many symbolic forms to represent
  information
• Alphabet, numbers, pictograms ? ?
• Computer can only represent information with
  electrical signals
• Is a circuit on or off?

4
Computers, numbers, and binary data

• Computers only use on/off signals to represent
  information
• These signals can only represent numeric data
• Even character based data is represented as a
  number

5
Why binary data?

• Electricity has two states, on and off
• On 1
• Off 0
• Binary numbers only have 0s and 1s
• Data is stored as collections of binary numbers

6
Binary numbers are computer friendly

• Binary numbers are signals that can easily be
  transported
• Binary numbers can be easily processed
  (transformed) by two-state electrical devices
  that are easy to design and fabricate
• These devices (and/or gates, adders) are strung
  together like an assembly line to carry out a
  function

7
Logic gates
8
Boolean algebra

• System developed by George Boole (19th century
  mathematician) that can determine if two values
  are
• Equal, not equal, less than, greater than, etc.
• Boolean algebra allows the CPU to carry out
  binary arithmetic (see White p.36-37)

9
Binary numbers

• Can be combined into a positional numbering
  system
• Base for decimal numbers is 10, base for binary
  numbers is 2
• Each position to the left is an increasing factor
  of 2

10
Terminology for number systems

• Base is also referred to as the radix
• Binary numbers have a radix of 2
• Decimal numbers have a radix of 10
• Radix point separates whole values from
  fractional values
• Decimal point is a kind of radix point

11
Base 2 positional example
12
Numbering systems

• Higher base (radix) means fewer positions are
  needed to represent a number
• Base 2 needs many more positions than base 10
• Base 16 (hexidecimal) is often used to represent
  binary numbers

13
Computers binary numbers

• Each digit of a binary number is called a bit
• Bit string group of digits that describes a
  single value

14
Bit strings

• Left most bit (most significant bit) called high
  order bit
• Right most bit (least significant bit) called low
  order bit
• 8 bits make a byte
• Programming languages/spreadsheets/etc.
  automatically translate from base 10 to base 2
  and back again

15
Hexadecimal notation

• Base or radix is 16
• More compact than binary
• Symbols used are 0-9, A-F
• One hexadecimal position corresponds to 4 bits
• Used to designate memory locations, colors (html
  VB)

16
Goals of computer data representation

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

17
Balancing objectives

• Compactness and range are inversely related the
  more compact, the smaller the range
• Accuracy increases with of bits used,
  especially with real numbers example, 1/3, or
  0.33333333 (non-terminating fraction)

18
Other objectives

• Does information format make it easier for
  processor to perform operations?
• Is data in a standard format, allowing simple
  transfer between computers?

19
CPU standard data types

• Integer
• Real number
• Character
• Boolean
• Memory address

20
Integer data types

• Unsigned assumed to be positive
• Signed uses one bit (usually high order bit) to
  indicate sign
• 0 is positive, 1 is negative

21
Representing negative integers

• Excess notation and twos complement
• Allow subtraction to be carried out as addition
• Number is converted to its complement
• 1 is added to the result
• When added to another binary number, carry bit is
  ignored

22
Range and overflow

• Most CPUs use a fixed width of 32 or 64 bits to
  represent an integer
• For small numbers format is padded with leading
  zeros
• Machine processes fixed width information more
  easily than variable width

23
Integer overflow

• If number is too big for fixed width integer
  format CPU throws an overflow error
• Integer format width is tradeoff between overflow
  and wasted space (padded zeros)
• CPU often use double precision data types for
  arithmetic operations

24
Representing real numbers

• More complicated problem than storing integers
• Real numbers contain whole fractional
  components
• How to represent both parts together in one
  format?

25
Fixed format for real numbers
26
Floating point notation

• Any real number can be re-written using floating
  point (scientific notation)
• 12.555 becomes 1.2555 X 10¹
• Format stores 12555 (mantissa), 1 (exponent), and
  sign ()
• -143.99 becomes 1.4399 X 102
• Format stores 14399 (mantissa), 2 (exponent), and
  sign (-)

27
IEEE floating point format for real numbers
28
Floating point range

• Number of bits in floating point format limit
  range of exponent, mantissa
• Overflow (too large a number) always occurs in
  the exponent
• Underflow (too small a number, i.e. negative
  exponent does not fit)

29
Range for mantissa

• Number of bits for mantissa limit the number of
  significant digits stored for a real number
• 23 bits allows for approx. 7 decimal places of
  precision
• Mantissa is stored using truncation (information
  that does not fit is discarded)
• Does not throw an overflow condition

30
Processing complexity

• General rule is floating point operations (, -,
  , etc.) take CPU twice as long as integers
  (binary)
• Floating Point Operations Per Second (FLOPS) is a
  measure of processor speed

31
Character data

• Alphabetic letters (upper lower case),
  numerals, punctuation marks, special symbols are
  called characters
• Variable of type character contain only one
  symbol
• Sequence of symbols forming words, sentences,
  etc. called a string

32
How computers store characters

• Character data cannot be directly processed by a
  computer
• Must be translated into a number
• Characters are converted into numbers using a
  table of correspondences between a character and
  a bit string

33
Design issues for character coding schemes

• Table must be publicly available and all users
  must use the same table
• Coding scheme is a tradeoff among compactness,
  ease of manipulation, accuracy, range,
  standardization

34
Examples of character coding schemes

• BCD and EBCIDIC older IBM mainframe computers
• ASCII PCs
• Unicode larger format allows for expanded and
  international alphabets (Java and internet
  applications)

35
ASCII coding scheme

• 7 bit format allows for parity bit (used to check
  for errors over transmission lines)
• Has unique codes for all uppercase lowercase
  letters, numbers, other printable characters
• Also includes codes for device control

36
Device control

• In many applications that handle text, formatting
  commands to a device are included in the same
  stream of data as the text
• Examples word processors (reveal codes), HTML
  tags
• Examples CR (carriage return), tab, form feed

37
Limitations to ASCII

• Not robust enough to represent multiple languages
  and symbols
• 7 bit format allows for 128 unique codes, some
  languages have thousands of symbols
• Unicode (16 bit) has 65,536 entries

38
Boolean data

• Data types has two values, true and false
• Can be stored with one bit
• The results of many CPU operations (comparisons)
  generate a Boolean value stored in a register

39
Memory addresses

• Primary storage is a series of contiguous bytes
• CPU must be able to access sections of memory
  directly
• Sections of memory are accessed by their address
  (location)

40
Formats for memory addresses

• Flat memory model memory starts at address 0,
  goes to maximum capacity 1
• Simple integers used to store address
• Segmented memory model
• Memory is divided into equal sized segments
  called pages
• Address has two parts00FA0034 number for page,
  and location within page

41
Data structures

• These five primitive types are quite limited for
  representing real world data
• Words, sentences
• Dates
• Data base tables
• More complex data structures constructed from
  these five primitive types

42
Chapter summary

• 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.

43
Summary (cont.)

• 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.