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Line 1: − A '''"rational" number''' is a quotient of the form <math>\frac{a}{b}</math> where ''a'', ''b'' are [[integer]]s and ''b'' ≠ 0. The set of rational numbers, usually denoted by <math>\mathbb{Q}</math> is an example of a [[totally disconnected set]] that is not [[locally compact]]. The rational numbers are [[countable]]. Rational numbers can be identified by their fractional form, or a terminating or repeating decimal.+ − + + + + + + + + + + + + + + + + + + + + + + +
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{{Math-m}}
[[Category: Mathematics]]
The '''rational numbers''' are the numbers representable as ratios of integers, that is, fractions. Mathematicians denote the set of rational numbers with an ornate capital letter: <math>\mathbb{Q}</math>. They are the 3<sup>rd</sup> item in this hierarchy of types of [[number]]s:
*The "[[natural number]]s", 1, 2, 3, ... (There is controversy about whether zero should be included. It doesn't matter.)
*The "[[integer]]s"—positive, negative, and [[zero]]
*The "rational numbers", or [[fraction]]s, like 355/113
*The "[[real number]]s", including irrational numbers
*The "[[complex number]]s, which give solutions to polynomial equations
__TOC__
Rational numbers are actually defined as ''[[equivalence class]]es'' of ratios of integers, so that 2/3 and 4/6 are the same number.
If a rational number were to be represented as a decimal to infinite precision, that decimal would either terminate at some point (e.g. 1.25) or would eventually get into an endless repeating pattern (e.q. 1.250909090909..., which is 1376/1100).
Zero in the denominator of a rational number is not allowed. All rational numbers are [[infinity|finite]].
==Countability and density==
An important theoretical property of the rationals is that they are countable. That is, the entire set of rational numbers can be put into a one-to-one correspondence with the integers. This is surprising at fist glance, since the integers are "sparse" while the rationals seem to fill out the real line. To see this correspondence, we need to list all rational numbers in some order. List the rationals (that is, the rationals in which the fraction has been fully reduced) that have the sum of their numerators and denominators equal to one. 0/1 is the only such. Then follow those with the rationals that have the sum of their numerators and denominators equal to two. 1/1 is the only such, since 0/2 is not reduced. Follow those with the rationals that have the sum of their numerators and denominators equal to three. They are 1/2 and 2/1. Continue without end.
A more subtle theoretical property is that the rationals comprise a countable [[dense set|dense subset]] of the reals. This is used in some advanced theorems of topology.
[[Category:Mathematics]]
[[Category:Calculus]]