Each rational number can be written in infinitely many forms,
for example 3 / 6 = 2 / 4 = 1 / 2. The simplest form is when a
and b have no common divisors, and every non-zero rational
number has exactly one simplest form of this type with positive
denominator. The decimal expansion of a rational number is
eventually periodic (in the case of a finite expansion the
zeroes which implicitly follow it form the periodic part). The
same is true for any other integral base above 1. Conversely, if
the expansion of a number for one base is periodic, it is
periodic for all bases and the number is rational. Two rational
numbers and are equal if and only if ad = bc Additive and
multiplicative inverses exist in the rational numbers. Any
positive rational number can be expressed as a sum of distinct
reciprocals of positive integers. For any positive rational
number, there are infinitely many different such
representations. These representations are called Egyptian
fractions, because the ancient Egyptians used them. The Egyp