Computing a numeric hash

We do need to define the __hash__() method properly. See section 4.4.4 of the Python Standard Library for information on computing hash values for numeric types. That section defines a hash_fraction() function, which is the recommended solution for what we're doing here. Our method looks like the following:

    def __hash__( self ):
        P = sys.hash_info.modulus
        m, n = self.value, self.scale
        # Remove common factors of P.  (Unnecessary if m and n already coprime.)
        while m % P == n % P == 0:
            m, n = m // P, n // P
        if n % P == 0:
            hash_ = sys.hash_info.inf
        else:
            # Fermat's Little Theorem: pow(n, P-1, P) is 1, so
            # pow(n, P-2, P) gives the inverse of n modulo P.
            hash_ = (abs(m) % P) * pow(n, P - 2, P) % P
        if m < 0:
            hash_ = -hash_
        if hash_ == -1:
            hash_ = -2
        return hash_

This reduces a two-part rational fraction value to...