# rational number

The rational numbers $\mathbb{Q}$ are the fraction field of the ring $\mathbb{Z}$ of integers. In more elementary terms, a rational number is a quotient $a/b$ of two integers $a$ and $b$, where $b$ is nonzero. Two fractions $a/b$ and $c/d$ are equivalent if the product of the cross terms is equal:

 $\frac{a}{b}=\frac{c}{d}\iff ad=bc$

Addition and multiplication of fractions are given by the formulae

 $\displaystyle\frac{a}{b}+\frac{c}{d}$ $\displaystyle=$ $\displaystyle\frac{ad+bc}{bd}$ $\displaystyle\frac{a}{b}\cdot\frac{c}{d}$ $\displaystyle=$ $\displaystyle\frac{ac}{bd}$

The field of rational numbers is an ordered field, under the ordering relation $\leq$ defined as follows: $a/b\leq c/d$ if

1. 1.

the inequality $a\cdot d\leq b\cdot c$ holds in the integers, and $b$ has the same sign as $d$, or

2. 2.

the inequality $a\cdot d\geq b\cdot c$ holds in the integers, and $b$ has the opposite sign as $d$.

Under this ordering relation, the rational numbers form a topological space under the order topology. The set of rational numbers is dense when considered as a subset of the real numbers.

Title rational number RationalNumber 2013-03-22 11:50:30 2013-03-22 11:50:30 djao (24) djao (24) 15 djao (24) Definition msc 13B30 msc 11A99 msc 03E99 $\mathbb{Q}$ Fraction ProofThatTheRationalsAreCountable rational