A rational number is a ratio of integers, and rational numbers constitute an important sub-class of real numbers. A rational number such as 1/3 or 17/29 is typically written as:

<numerator>/<denominator>

where both the <numerator> and <denominator> are placeholders for integer values. Both parts are needed to exactly characterize the value of the rational number. Actually dividing integers produces a float approximation, losing the exact precision of integers.

>>> 1/3
0.3333333333333333
>>> 1/3 == 0.333333333333333300000  # Dividing integers yields an approximation
True

However, we can create an exact representation for rational numbers by combining together the numerator and denominator.

We know from using functional abstractions that we can start programming productively before we have an implementation of some parts of our program. Let us begin by assuming that we already have a way of constructing a rational number from a numerator and a denominator. We also assume that, given a rational number, we have a way of selecting its numerator and its denominator component. Let us further assume that the constructor and selectors are available as the following three functions:

• rational(n, d) returns the rational number with numerator n and denominator d.
• numer(x) returns the numerator of the rational number x.
• denom(x) returns the denominator of the rational number x.

We are using here a powerful strategy for designing programs: wishful thinking. We haven't yet said how a rational number is represented, or how the functions numer, denom, and rational should be implemented. Even so, if we did define these three functions, we could then add, multiply, print, and test equality of rational numbers:

>>> def add_rationals(x, y):
        nx, dx = numer(x), denom(x)
        ny, dy = numer(y), denom(y)
        return rational(nx * dy + ny * dx, dx * dy)

>>> def mul_rationals(x, y):
        return rational(numer(x) * numer(y), denom(x) * denom(y))

>>> def print_rational(x):
        print(numer(x), '/', denom(x))

>>> def rationals_are_equal(x, y):
        return numer(x) * denom(y) == numer(y) * denom(x)

Now we have the operations on rational numbers defined in terms of the selector functions numer and denom, and the constructor function rational, but we haven't yet defined these functions. What we need is some way to glue together a numerator and a denominator into a compound value.