Each rational number can be written in infinitely many forms, for example . The simplest form is when and 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
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 Egyptians also had a different notation for dyadic fractions. See also Egyptian numerals.
To conform to our expectation that , we define an equivalence relation upon these pairs with the following rule:
This equivalence relation 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 quotient field.)
We can also define a total order on Q by writing
The rationals are the smallest field with characteristic 0: every other field of characteristic 0 contains a copy of .
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.
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 expressions of continued fraction.
By virtue of their order, the rationals carry an order topology. The rational numbers also carry a subspace topology. The rational numbers form a metric space by using the metric , and this yields a third topology on . All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metric space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of .
In addition to the absolute value metric mentioned above, there are other metrics which turn into a topological field:
in addition write . For any rational number , we set .
Then defines a metric on .
The metric space is not complete, and its completion is the p-adic number field .bg:Рационално число bn:মূলদ সংখ্যা ca:Nombre racional cs:Racionální číslo da:Rationale tal de:Rationale Zahl et:Ratsionaalarvud es:Número racional eo:Racionala nombro eu:Zenbaki arrazional fr:Nombre rationnel ko:유리수 hr:Racionalni brojevi is:Ræðar tölur it:Numero razionale he:מספר רציונלי lt:Racionalieji skaičiai jbo:fi'urna'u nl:Rationaal getal ja:有理数 no:Rasjonalt tall pl:Liczby wymierne pt:Número racional ro:Număr raţional ru:Рациональное число scn:Nummuru razziunali simple:Rational number sl:Racionalno število sr:Рационалан број fi:Rationaaliluku sv:Rationella tal th:จำนวนตรรกยะ tr:Rasyonel sayılar uk:Раціональні числа zh:有理数