Rational Number

number-theory algebra

Definition

Rational Number

A rational number is a number that can be written as the quotient of two integers with a non-zero denominator.

$$\mathbb{Q}=\left\{\frac{p}{q}|p,q\in \mathbb{Z},q\ne 0\right\}$$

The set of all rational numbers is denoted by $\mathbb{Q}$. Any rational number has a decimal expansion that either terminates or repeats periodically.

Countability

Countability via reduced forms (implicit composition)

Every rational number can be written uniquely in reduced form as

$$\frac{p}{q}\text{with }p\in \mathbb{Z},q\in {\mathbb{N}}_{>0},\gcd (p,q)=1.$$

Here, $\gcd$ denotes the greatest common divisor. Consider the set

$$S=\{(p,q)\in \mathbb{Z}\times {\mathbb{N}}_{>0}:\gcd (p,q)=1\}.$$

The map

$$f:\mathbb{Q}\to S,\frac{p}{q}\mapsto (p,q)$$

is a bijection, because each rational number has exactly one reduced representation. Since $S\subseteq \mathbb{Z}\times {\mathbb{N}}_{>0}$ and $\mathbb{Z}\times {\mathbb{N}}_{>0}$ is countable, it follows that $S$ is countable. Therefore, $\mathbb{Q}$ is countable.

Countability via grouping

First write each rational number in unique reduced form,

$$x=\frac{p}{q}\text{with }p\in \mathbb{Z},q\in {\mathbb{N}}_{>0},\gcd (p,q)=1.$$

For such a reduced pair $(p,q)$, define

$$k=\max (|p|,q).$$

Then $x$ belongs to the single group

$$G_k=\left\{\frac{p}{q}\in \mathbb{Q}|q\in {\mathbb{N}}_{>0},\gcd (p,q)=1,\max (|p|,q)=k\right\},k\ge 1.$$

This is unique because the reduced form of $x$ is unique, so the value of $\max (|p|,q)$ is unique as well. Each $G_k$ is finite. Indeed, if $\max (|p|,q)=k$, then $1\le q\le k$ and $|p|\le k$, so only finitely many pairs $(p,q)$ can occur. In fact, $|G_k|=4k-1$.

Since

$$\mathbb{Q}=\bigcup _{k=1}^{\infty }{G}_k,$$

and each $G_k$ is finite, $\mathbb{Q}$ is a countable union of finite sets. Therefore, $\mathbb{Q}$ is countable.