Case Study - Fractions

1
Case Study - Fractions

• Timothy Budd
• Oregon State University

2
Objectives

• To easily create new instances of the rational
  number abstraction.
• To manipulate rational numbers using arithmetic
  operations, yielding new rational number results.
  
• It should be possible to mix rationals and other
  arithmeticquantities in the same expression.
• It should be possible to assign a rational number
  value to arational number variable.
• The modification forms of assignment should be
  supported.
• To compare one rational number to another.
• It should be possible to perform input and output
  operations with rational numbers.

3
Example of Rational Numbers

• // probability one is 1 in 8rational p1 (1,8)
• // probability two is 2 in 3rational p2 (2,
  3)
• // probably of both together is their
  productrational p3 p1 p2
• // probably of either independent is their
  sumrational p4 p1 p2
• // what is this probability?cout ltlt "combined
  probability is " ltlt p3 ltlt endlcout ltlt
  "independent probability is " ltlt p4 ltlt endl

4
Definition of Operations on Rational Numbers
5

• // class rational// rational number data
  abstraction
• class rational public // constructors rationa
  l () top(0), bottom(1) rational
  (int t) top(t), bottom(1) rational
  (int t, int b) top(t), bottom(b) normalize()
  rational (const rational r)
  top(r.top), bottom(r.bottom) // accesser
  functions int numerator () const return
  top int denominator () const return
  bottom // assignments void operator (const
  rational r) top r.top bottom
  r.bottom void operator (const rational
  ) // other operations operator double ()
  const return top / (double) bottom const
  rational operator() top bottom return
  this const rational operator(int)private
  int top // data areas int
  bottom void normalize () //
  operation used internally

6
Interface and Implementation

• The code associated with a component is separated
  into two files.
• The interface file describes how to use the
  component.
• The implementation file contains the actual code
  to perform the actions.
• The class description is divided into two parts.
• Public behaviors and data fields that users of
  the data abstraction can access.
• Private behavior and fields that are accessible
  only within the component and are off-limits to
  other users.

7
Constructors

• Every class should define a default constructor.
• Use initializes whenever possible.
• The declaration that uses an empty parentheses
  list is not syntactically incorrect simply not
  doing what the programmer expects.
• Every class should define a copy constructor.

8
Member Function

• An accessor function provides access to the
  internal state of an object.
• left.numerator() right.denominator()
  right.numerator() left.denominator()
• A function defined as part of the behavior of a
  class is called a member function.
• The const keyword in C is not the same as the
  final keyword in Java.

9
Operators

• In C, operators can potentially have overloaded
  meanings.
• Requirement the definition must not require
  arguments that match any existing definition.
• Can be achieved if one or both arguments are a
  new data type, since no existing definition can
  be using these types.

10
Operators

• const rational operator
• (const rational left, const rational
  right)
• // return sum of two rational
  numbers rational result ( left.numerator()
  right.denominator() right.numerator()
  left.denominator(), left.denominator()
  right.denominator()) return result
• The address operator in the argument list
  indicates that the left and right values will be
  passed by reference.

11
Operators

• A list of prototypes for the arithmetic
  operations.
• // prototypes for arithmetic operations,
  including unary negationconst rational operator
  (const rational , const rational )const
  rational operator - (const rational , const
  rational )const rational operator (const
  rational , const rational )const rational
  operator / (const rational , const rational
  )const rational operator - (const rational
  )bool operator lt (const rational , const
  rational )bool operator (const rational ,
  const rational )

12
Unary Negation Operator

• const rational operator - (const rational val)
• // return negation of argument value return
  rational (- val.numerator(), val.denominator())

13
Comparison Operators

• bool operator lt (const rational left, const
  rational right)
• // less than comparison of two rational
  numbers return left.numerator()
  right.denominator() lt right.numerator()
  left.denominator()
• bool operator (const rational left, const
  rational right)
• return left.numerator() right.denominator()
  right.numerator() left.denominator()

14
Increment and Decrement

• The variable this is a value in Java but is a
  pointer in C.
• class rational ... const rational operator
  () top bottom return this ...
• const rational rationaloperator (int) //
  increment fraction, but return original value,
  make clone rational clone(this) top
  bottom // make change return clone // return
  clone

15
Functions

• Functions cannot access the internal (private)
  structure of any class, unless declared as a
  friend.
• const rational abs (const rational num) //
  return the absolute value of a rational
  number int newtop int newbottom
  num.denominator() // get non-negative
  numerator part if (num.numerator() lt 0) newtop
  - num.numerator() else newtop
  num.numerator() // create and return
  result return rational(newtop, newbottom)

16
Member Function Operators

• void rationaloperator (const rational
  right)
• // modify by adding right hand side top
  top right.denominator() bottom
  right.numerator()
• bottom right.denominator() // normalize
  the result, ensuring lowest denominator
  form normalize()

17

• void rationalnormalize() // normalize
  rational by (a) moving sign to numerator // b)
  making sure numerator and denominator have no
  common divisorsint sign 1 // sign is 1 if
  non-negative, -1 if negativeif (top lt 0) sign
  -1 top - topif (bottom lt 0) sign -
  sign bottom - bottom// make sure we are
  not dividing by zeroif (bottom 0)throw
  range_error("fraction with zero numerator")//
  find greatest common divisor int d gcd(top,
  bottom)// move sign to top and divide by
  gcdtop sign (top / d)bottom bottom / d

18
Member Function Operators

• try ... // computation involving rationals
  catch (range_error e) printf("got exception
  s", e.what())

19
Conversion Operations

• Constuctors are also used implicitly.
• rational x, y ...x y 3
• Temporary values can also be created directly by
  the programmer, by invoking the constructor as if
  it were an ordinary function.
• x y rational(3, 4)

20
Conversion Operations

• Conversions the other direction, from an object
  type to another type, are defined using a
  conversion operator.
• A conversion operator is an operator whose name
  is a type.
• class rational ... operator double () const
  return top / (double) bottom

21
Conversion Operations

• Avoid cast if possible, but if unavoidable use a
  static or dynamic cast.
• rational x(3, 4)cout ltlt "3/4 of pi is " ltlt
  (3.14 double(x)) ltlt '\n'

22
Input and Output Streams

• Output is easily accommodated by redefining the
  left shift operator ltlt, and providing a new
  meaning.
• ostream operator ltlt
• (ostream out, const rational value)
• // print representation of rational number on
  // an output stream out ltlt value.numerator() ltlt
  '/' ltlt value.denominator() return out

23
Input and Output Streams

• Predefined precedence is low enough to allow
  arithmetic expressions to appear in output
  without using parenthesis, as in
• cout ltlt "a b c is " ltlt a b c ltlt '\n'
• The left shift operator with its conventional
  meaning can be used in anoutput statement, by
  surrounding it with parenthesis
• cout ltlt " a left shift by 3 is " ltlt (a ltlt 3) ltlt
  '\n'

24
Stream Input

• A loop that would read values repeatedlyfrom the
  input until end of file could be written as
• while (cin gtgt intval) // process
  intval ...// reach this point on end of
  input...
• An easy way to remember, the stream I/O
  operations is to visualize them as arrows.
• The input operator, gtgt x, points data into x,
  whilethe output operator, ltlt x, copies data out
  of x.

25

• istream operator gtgt (istream in, rational
  r) // read a rational number from an input
  streamint t, b// read the topin gtgt t// if
  there is a slash, read the next numberchar cin
  gtgt cif (c '/') in gtgt b // read bottom
  partelse in.putback(c) b 1// do the
  assignmentrational newValue(t, b)r
  newValue// return the streamreturn in