Rational numbers

Table of Contents

Understanding Rational Numbers

In this chapter we focus on one particular type of real number: rational numbers. Numbers and number systems in general are handled in the parent chapter; here we look specifically at what makes rational numbers special and how to recognize and work with them.

Definition of Rational Numbers

A rational number is any number that can be written as a fraction of two integers, with a nonzero denominator.

Formally, a number $q$ is rational if it can be written as
$$
q = \frac{a}{b}
$$
where

$a$ and $b$ are integers (positive, negative, or zero),
$b \neq 0$.

Examples:

$5$ is rational because $5 = \dfrac{5}{1}$.
$0$ is rational because $0 = \dfrac{0}{3}$ (or any nonzero denominator).
$-\dfrac{7}{4}$ is rational (both numerator and denominator are integers, denominator nonzero).
$0.25$ is rational because $0.25 = \dfrac{1}{4}$.

A number that cannot be written in this form is called irrational (these are handled in the “Irrational numbers” chapter, not here).

Fractions as Rational Numbers

Every rational number may be written as a fraction $\dfrac{a}{b}$ with integer $a$ and $b \neq 0$. In this chapter we treat “fraction” in this broad sense; the later “Fractions” chapter in Arithmetic will go into practical operations.

Important points specific to rational numbers:

The same rational number can be written in many ways:
$$
\frac{1}{2} = \frac{2}{4} = \frac{-3}{-6} = \frac{50}{100}.
$$
A simplest form (also called lowest terms) of a rational number is a fraction $\dfrac{a}{b}$ where:

$a$ and $b$ are integers,
$b > 0$ (by convention, we usually keep the denominator positive),
$a$ and $b$ have no common factor other than $1$.

For example:

$\dfrac{6}{8}$ is not in simplest form because both $6$ and $8$ are divisible by $2$.
Dividing top and bottom by $2$ gives $\dfrac{3}{4}$, which is in simplest form.

The process of getting to simplest form uses ideas from divisibility and prime factorization (covered later in Number Theory), but for now we only need the idea that we can “reduce” a fraction by dividing numerator and denominator by the same nonzero integer.

Decimal Forms of Rational Numbers

Rational numbers have characteristic decimal patterns. When a rational number is written in base-10 decimal form, its decimal expansion is always:

Terminating (it ends), or
Repeating (a finite block of digits repeats forever).

This is a key feature that distinguishes rational numbers from irrational numbers.

Terminating decimals

A terminating decimal has only finitely many nonzero digits after the decimal point.

Examples:

$\dfrac{1}{4} = 0.25$ (terminates after two decimal places),
$\dfrac{3}{2} = 1.5$ (terminates after one decimal place),
$\dfrac{7}{8} = 0.875$.

Every terminating decimal can be written as a rational number by reading it as a fraction with a power of $10$ in the denominator, for example:
$$
0.25 = \frac{25}{100}, \quad 1.5 = \frac{15}{10}, \quad 0.007 = \frac{7}{1000}.
$$
These can then be simplified to lowest terms.

Repeating decimals

A repeating decimal (also called recurring decimal) has a pattern of digits that repeats forever.

We often use a bar to show the repeating part:

$0.\overline{3}$ means $0.3333\ldots$ (3 repeats forever),
$0.\overline{142857}$ means $0.142857142857\ldots$,
$1.2\overline{7}$ means $1.27777\ldots$ (only the 7 repeats).

Every repeating decimal represents a rational number. For example:

$0.\overline{3} = \dfrac{1}{3}$,
$0.\overline{6} = \dfrac{2}{3}$,
$0.\overline{09} = \dfrac{1}{11}$.

The general method for turning a repeating decimal into a fraction is usually shown in an algebra course, but here it is for a simple case:

Let $x = 0.\overline{3}$.

Multiply both sides by $10$:
$$
10x = 3.\overline{3}.
$$
Subtract the original equation:
$$
10x - x = 3.\overline{3} - 0.\overline{3} \Rightarrow 9x = 3.
$$
So $x = \dfrac{3}{9} = \dfrac{1}{3}$.

This reasoning uses algebraic manipulation, which is studied more systematically later in the course.

Characterization via decimals

Putting this together:

If a number’s decimal expansion terminates or repeats, then it is rational.
If a number is rational, then its decimal expansion must terminate or repeat.

Numbers whose decimal expansions neither terminate nor have a repeating pattern (such as $\pi = 3.14159265\ldots$) are irrational and not rational.

Positive, Negative, and Zero Rational Numbers

Rational numbers include:

Positive numbers (e.g. $\dfrac{3}{4}, 5, 2.1$),
Negative numbers (e.g. $-\dfrac{3}{4}, -5, -2.1$),
Zero ($0$).

Using the fraction form, a negative rational number is one where the fraction has an overall negative sign:
$$
-\frac{3}{4}, \quad \frac{-3}{4}, \quad \frac{3}{-4}
$$
all represent the same rational number. By convention, we usually write the minus sign in front of the whole fraction:
$$
-\frac{3}{4}.
$$

Zero is special:

$0$ is rational because $0 = \dfrac{0}{b}$ for any integer $b \neq 0$.
But you can never have $0$ in the denominator: $\dfrac{a}{0}$ is not defined and is not a rational number.

Comparing and Ordering Rational Numbers

Rational numbers can be placed on a number line and compared. There are several equivalent ways to compare two rational numbers.

Let $r = \dfrac{a}{b}$ and $s = \dfrac{c}{d}$, with $b > 0$ and $d > 0$.

Common denominator method

Rewrite both fractions with a common positive denominator.
Compare the numerators.

Example:
$$
\frac{3}{4} \quad \text{and} \quad \frac{5}{6}.
$$
A common denominator is $12$:
$$
\frac{3}{4} = \frac{9}{12}, \quad \frac{5}{6} = \frac{10}{12}.
$$
Since $9 < 10$, we have
$$
\frac{3}{4} < \frac{5}{6}.
$$

This method fits naturally with the idea of equivalent fractions.

Cross-multiplication method

When both denominators are positive, another way is:

$r < s$ if and only if
$$
ad < bc.
$$

Example with $\dfrac{3}{4}$ and $\dfrac{5}{6}$:
$$
\frac{3}{4} < \frac{5}{6} \quad \text{?}
$$
Compute:
$$
3 \cdot 6 = 18, \quad 4 \cdot 5 = 20.
$$
Since $18 < 20$, we conclude $\dfrac{3}{4} < \dfrac{5}{6}$.

This approach avoids explicitly finding a common denominator.

Number line interpretation

Rational numbers can be placed on a number line as exact points. Between any two different rational numbers, there is always another rational number. For example, between $\dfrac{1}{3}$ and $\dfrac{1}{2}$, the number
$$
\frac{\frac{1}{3} + \frac{1}{2}}{2} = \frac{\frac{5}{6}}{2} = \frac{5}{12}
$$
is also rational and lies strictly between them on the number line.

This property—that between any two rational numbers lies another rational number—shows that rational numbers are densely packed on the number line, even though, as will be seen later, they still do not fill the line completely (there are also irrational numbers).

Closure Properties of Rational Numbers

A set of numbers is said to be closed under an operation if performing that operation on numbers in the set never takes you outside the set.

Rational numbers are closed under the basic arithmetic operations (with a restriction for division):

Let $r$ and $s$ be rational numbers.

Addition: $r + s$ is rational.
Subtraction: $r - s$ is rational.
Multiplication: $r \cdot s$ is rational.
Division: If $s \neq 0$, then $\dfrac{r}{s}$ is rational.

These statements will be revisited later with more formal proofs, but intuitively:

Adding, subtracting, or multiplying fractions with integer numerators and denominators gives another fraction with integer numerator and denominator.
Dividing by a nonzero fraction is the same as multiplying by its reciprocal, which is again a fraction with integer numerator and denominator.

This closure is one reason rational numbers form a very useful and stable number system for arithmetic.