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# RATIONAL NUMBERS

## Introduction

A rational number is a number that can be represented as a fraction $m/n$, where $m$ and $n$ are integers and $n\ne 0$.

### Terminology

Rational numbers may be referred to as “rationals”.  The name comes from the fact that they represent ratios. It is not related to the meaning “able to reason” or “sane”.

##### Examples
• The numbers $3/4$ and $–11/5$ are rational.
• $6$ is rational because $6 = 6/1$. In fact, any integer is a rational number by the same reasoning.
• $0.33$ is rational because $0.33=33/100$.

Note that $0.33\neq1/3$. See Approximations.

## Representations of rational numbers

Rational numbers have two familiar representations, as fractions and as decimals.

Decimals are discussed in the section on real numbers.

The definition of rational number says that it must be a number that can be represented as a fraction of integers.

• Can be represented” does not mean “is represented”!

• The number $0.25$ is a rational number because it can be represented as $1/4$, but the expression “$0.25$” is not itself a fraction representation.
• An expression "$a/b$" does not automatically denote a rational number. You must check that $a$ and $b$ are integers.

• For example if you see the expression $\pi/e$, you cannot conclude that it denotes a rational number, because $\pi$ and $e$ are not integers.

### Properties of the fraction representation

#### Non-uniqueness

The representation of a rational number as a fraction is not unique. For example,$\frac{3}{4}=\frac{6}{8}=\frac{-9}{-12}$

Two representations $m/n$ and $r/s$ denote the same rational number if and only if $ms=nr$.

##### Example

$3/4=6/8$ because $3\cdot 8=6\cdot 4$.

#### Lowest terms

Let $m/n$ be the representation of a rational number with $m\ne 0$ and $n\gt0$. The representation is in lowest terms (or is reduced) if there is no integer $d\gt1$ for which $d$ divides $m$ and $d$ divides $n$.  (See parenthetic assertion.)

#### Usage

The symbols $3/4$ and $6/8$ are two different representations of the same number.  One of the representations, $3/4$, is in lowest terms and the other is not. So when someone says “$3/4$ is in lowest terms”, the symbol “$3/4$” refers to the representation, not the rational number. See Context-sensitive interpretation.

##### Examples
• $3/4$ is in lowest terms, but $6/8$ is not, because $6$ and $8$ have $2$ as a common divisor.
• $74/111$ is not in lowest terms because $74$ and $111$ have $37$ as a common divisor.  In fact, $74/111 = 2/3$.

Every nonzero rational number has a representation in lowest terms. (To get it, divide out the largest integer that divides both numerator and denominatos.)

## Closure properties of rationals

Rational numbers are closed under addition, subtraction, multiplication, as well as division by a nonzero rational.

These operations are carried out according to the familiar rules for operating with fractions. For any two rational numbers $a/b$ and $c/d$, the values of the operations are given by these rules: $\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd},\,\,\, \frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd} \text{ and } \frac{a}{b}\div \frac{c}{d}=\frac{ad}{bc}\,\,\,\,\,\,\text{(AMD)}$

The expressions $ac/bd$, $(ad+bc)/bd$, and $ad/bc$ denote rational numbers because integers are closed under addition and multiplication, so that $ac$, $bd$, $ad$,$bc$ and $ad+bc$ are all integers.

If  “$a/b$” and “$c/d$” are in lowest terms, “$ac/bd$” and “$(ad+bc)/bd$” may nevertheless not be in lowest terms. For example, the formulas give

$\frac{1}{4}+\frac{1}{4}=\frac{8}{16}$ and $\frac{2}{3}\times \frac{3}{4}=\frac{6}{12}$

In other words, “being in lowest terms” is not closed under addition and multiplication.

## Images and metaphors for the rationals

### Ratios

Suppose we have a line segment $L$ that is $5/8$ of a unit long.  Then we can take a line segment $M$ that is $1$ unit long and divide it into exactly $8$ segments of length $1/8$ unit each, and we can divide $L$ into exactly $5$ segments of length $1/8$ unit each.  That is because the ratio of $L$ to $M$ is $5:8$. This is the sense in which rational numbers represent ratios of integers.

Remarkably, we cannot take a line of length $1$ and a line of length $\sqrt{2}$ and divide them into whole numbers of segments of the same length, no matter how small we make the segments. This means that $\sqrt{2}$ is not a rational number. This is discussed in the Wikipedia article on irrational numbers.

### Density of the rational numbers

The integers can be thought of as beads or points in a row going to infinity in both directions.  The rational numbers go to infinity in both directions, too, but:

You must not think of the rational numbers as a row of points.

That is because: between any two distinct rational numbers there is another one. Specifically: If $r$ and $s$ are any distinct rational numbers, then $\frac{r+s}{2}$ is a rational number between them.  This number $\frac{r+s}{2}$ is the average of $r$ and $s$, so it makes sense that it is between them. (There is a formal proof of this in the Glossary.) For example, you can check that if $r=\frac{5}{12}$and $s=\frac{1}{2}$, then $\frac{r+s}{2}=\frac{11}{24}$. Note that $r=\frac{10}{24}$ and $s=\frac{12}{24}$.

#### Infinitely many between any two

Observe that $\frac{r+s}{2}$ is not the only rational number between $r$ and $s$.  In fact, between any two distinct rational numbers there are infinitely many other rational numbers, because the fact that there is a rational number between any two distinct rational numbers means there is a rational number $t$ between $r$ and $\frac{r+s}{2}$, a rational number $u$ between $r$ and $t$, a rational number $v$ between $r$ and $u$, and so on forever.

There are in fact uncountably many rational numbers between any two distinct rational numbers, although the argument I just gave does not prove that.

This means that if you are given a rational number $r$, then there is no “next largest” rational number (or next smallest, either). That is the sense in which the set of rational numbers is not a row of points.

The same argument shows that the set of all real numbers is also dense. 