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 one, and is also true when rational numbers are considered to be p-adic numbers rather than real numbers. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational. A real number that is not a rational number is called an irrational number.
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Two fractions are added as follows
The rule for multiplication is
Additive and multiplicative inverses exist in the rational numbers
It follows that the quotient of two fractions is given by
Any positive rational number can be expressed as a sum of distinct reciprocals of positive integers, such as
For any positive rational number, there are infinitely many different such representations, called Egyptian fractions, as they were used by the ancient Egyptians. The Egyptians also had a different notation for dyadic fractions.
and if c ≠ 0, division by
This equivalence relation is a congruence relation: it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set of ~, i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.)
We can also define a total order on Q by writing
The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.
The rationals are a densely ordered set: between any two rationals, there sits another one, in fact infinitely many other ones. Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.
The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.
Let p be a prime number and for any non-zero integer a let | a | p = p − n, where pn is the highest power of p dividing a;