Lab exercise #3: Rational numbers

The objective

Our goal for today is to design, implement, and test a Fraction class -- a way to represent exact rational numbers as objects in Java.

Part 1: The BigInteger class

The java.math.BigInteger class allows programmers to compute with arbitrarily large integer values, as in Scheme. The limitation is that all of the operations are formulated as method calls rather than expressions with infix or prefix operators. (For instance, to compute and print out the sum of three octillion and five octillion, one might write

BigInteger augend = new BigInteger("3000000000000000000000000000");
BigInteger addend = new BigInteger("5000000000000000000000000000");
System.out.println(augend.add(addend))

a. Look over the documentation for the BigInteger class and acquaint yourself with the methods for addition, subtraction, negation, multiplication, and division.

b. Write, compile, and run a test program that prints out the quotient and the remainder when the product of seven trillion and ninety-six trillion is divided by seventeen million three hundred and nineteen. (You'll need to import the BigInteger class from the java.math package.)

Part 2: Designing the Fraction class

To represent a rational number, we can use an object that has two fields, each holding a BigInteger value. One of these fields will be the numerator of a fraction expressing the rational number and the other will be the denominator.

a. In a file called Fraction.java, write the definition for a class that has two BigInteger fields called numerator and denominator. Should these fields be static? Justify your answer.

b. Write a two-argument constructor for the Fraction class. It should take two arguments, each a BigInteger, and store the arguments in its fields. (Hint: You can invoke a this-constructor to do the second part of this job.)

c. Write a second two-argument constructor for the Fraction class. It should take two arguments, each an int. From each argument, the constructor should then build a BigInteger with the same value and store that BigInteger into the corresponding field. Note that the BigInteger constructor needs a String as its argument.

d. Revise your constructors so that it throws an ArithmeticException (which is an unchecked exception type) when an attempt is made to construct a Fraction in which the denominator is zero. (If you followed the hint in part b, you'll only have to change one of the constructors.)

Part 3: Establishing invariants

Let's establish some invariants for our rational numbers. These are conditions that, though perhaps not logically necessary, are conveniently enforced as conventions of our representation and will simplify some of the coding later on.

For instance, in ordinary arithmetic, fractions are usually represented as having only positive numbers as denominators; when, in our computations, we find a negative number in the denominator, we reverse its sign and the sign of the numerator in presenting our result. For instance, we write "-3/5" rather than "3/-5", even though theoretically these express the same rational number. Similarly, we'd always write "5/7" rather than "-5/-7".

Similarly, fractions are usually expressed "in lowest terms" -- any divisors common to the numerator and denominator are cancelled in the final form. For instance, we'd write "2/5" rather than "24/60". That is to say, we divide both the numerator and the denominator by their greatest common divisor, in this case 12, to express the fraction in lowest terms.

a. Revise your constructors so that, if the proposed denominator is negative, both the proposed numerator and the proposed denominator are negated before they are stored the corresponding fields.

b. Revise your constructors so that the fraction is stored in lowest terms, with the numerator and the denominator having no common divisors greater than 1. (Hint: The BigInteger class supports a method that computes the greatest common divisor of two BigIntegers.)

Part 4: Rational arithmetic

The product of two rational numbers a/b and c/d is (ac)/(bd).

a. Add a multiply method to your Fraction class that takes a Fraction as argument and multiplies the fraction to which the message is sent by the fraction supplied as argument, returning the product (as a Fraction, of course). Will the product that you return always be expressed in lowest terms?

The sum of of two rational numbers a/b and c/d is (ad + bc)/(bd).

b. Add an add method to your Fraction class that takes a Fraction as argument and adds it to the fraction to which the message is sent, returning the sum as a Fraction.

c. Add a similar subtract method to your Fraction class.

d. Add a negate method that takes no arguments and returns the negative of the fraction to which the message is sent.

e. Add a reciprocal method that takes no arguments and returns the reciprocal of the fraction to which the message is sent (i.e., a new fraction with the numerator and denominator swapped). Ensure that your method throws an ArithmeticException if it is sent to a fraction in which the numerator is zero.

f. Add a similar quotient method to your Fraction class, throwing an ArithmeticException if the numerator of the divisor is zero. Note that the quotient of any two non-zero rational numbers can always be expressed exactly as a rational number, so there is no concept of a remainder in this case.

Part 5: Converting fractions to strings

To test these methods, it would be really convenient to have string representations of Fraction objects that show what their numerators and denominators are.

a. Add a toString method to your Fraction class that takes no arguments and returns a string formed by concatenating the string representation of the numerator, a slash character, and the string representation of the denominator.

b. Test your work by writing a main method that constructs some fractions, prints out their string representations, performs some arithmetic on them, and prints out the results.