Fractions, aka Rational Numbers
Read Problem 9, chap. 6 (p. 380) for the full specification of this assignment
It may come as a bit of a surprise when the C++ floating-point types
(float, double), fail to accurately
capture a particular value. Certainly double, which is usually stored
as a 64-bit value, is far better than the old float, which is
only 32 bits, but problems do arise. For example:
float n = 2.0;
float d = 3.0;
cout.precision(17);
cout << n / d << endl;
Produces 0.66666668653488159, which is accurate to 8 decimal
places. Not bad, but a bit dirty for a discipline that prides itself
on precision.
A solution that is often used when precision is of greatest importance
and all of the numbers involved are going to be rational (that is,
expressible as a fraction) is to use a custom type that implements
fractions / rational numbers. This is what you will do in this assignment.
The spec
Write a C++ program that performs the rational number operations addition,
subtraction, multiplication, division, less than, and equals operations on two
fractions.
You will have to design a rational number class whose value will be a fraction
(e.g. 1/128, or 22/7), with appropriate constructors, accessor and mutator functions.
A fraction will be specified as a numerator and a denominator - e.g. the pair (8, 109)
represents the fraction 8/109.
Your program must implement the rational number operations listed above,
and test both the class and the operations thoroughly.
You need to use constructors for:
1) correctly initializing member variables, and
2) creating objects with given initial values.
Note that a fraction may have both a numerator and a denominator, for example:
(8/3), or may simply be a number, for example: 6, in which case you will
actually use (6/1) to represent it. Thus, the constructors with two
and one arguments should be included in the class definition.
Question: what would be a useful default constructor
- i.e. what would you consider to be the null Rational Number?
Your program will have a function for each one of the six operations
add, sub, mul, div, less, equals.
For example: Suppose we have defined objects a and b as two fractions,
then to compute a + b, we will use:
a.add(b);
When you perform an operation, you do not have to simplify the result,
i.e., 4/5 * 5/10 = 20/50.
Here are the rules you will need:
(a/b) + (c/d) = (a*d + b*c) / (b*d)
(a/b) - (c/d) = (a*d - b*c) / (b*d)
(a/b) * (c/d) = (a*c) / (b*d)
(a/b) / (c/d) = (a*d) / (c*b)
(a/b) < (c/d) => (a*d) < (c*b)
(a/b) == (c/d) => (a*d) == (c*b)
Extra credit (1 point)
Extend your class and operator functions to properly handle either positive
or negative rational numbers, including the case of both numerator and
denominator being negative.