Representation

A rational number can be represented as follows:

type Rational
    = Rational Int Int

So, for example, one-half can be represented by Rational 1 2.

Zero in the Denominator

By definition, a rational number is not supposed to have a zero in its denominator. But, the above representation doesn't prevent this possibility since, for example, Rational 3 0 is a valid value of type Rational. However, we can control the way values of type Rational get created by not exporting the type's constructor and instead exporting a custom function, in this case new, that constructs values which obey the definition of a rational number.

We call the Rational type an opaque type since its constructors are private to the module. It's opaque because users of the module can't see and thus access the internal details of the type.

We call new a smart constructor because it uses a bit of logic to decide how and when to construct a value of type Rational.

Equivalent Rational Numbers

One-half can also be represented by Rational 2 4, Rational 3 6, Rational 4 8, and so on. I decided to use the representation that had all common factors removed from both the numerator and denominator.

Negative Rational Numbers

Negative one-half can be represented by Rational -1 2 or Rational 1 -2. I decided to represent negative rational numbers such that the numerator always contains the negative value.