rational number

The set of rational numbers, usually denoted by \(\mathbb{Q}\), is the subset of real numbers that can be expressed as a ratio of integers, that is, as ratios of positive or negative whole numbers, like \(\frac{2}{3}\) or \(\frac{-13}{5}\). Real numbers that cannot be so expressed are called irrational numbers (e.g., \(\pi\), \(\sqrt{2}\), etc.)

The arithmetic of rational numbers takes some getting used to. For any rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\) the operations of addition, multiplication, and division work as follows:

\[\begin{eqnarray*} \frac{a}{b} + \frac{c}d{} & = & \frac{ad+bc}{bd} \\ & & \\ \frac{a}{b}\cdot\frac{c}{d} & = & \frac{ac}{bd} \\ & & \\ \frac {a}{b}\div \frac{c}{d} & = & \frac{ad}{bc} \\ \end{eqnarray*}\]

These rules are most commonly remembered by saying that to add fractions we need a common denominator, to multiply them we just multiply straight across the numerators and straight across the denominators, and to divide them we simply turn the divisor over and then multiply. So for example we have:

\[\begin{eqnarray*} \frac{1}{2} + \frac{3}{5} & = & \frac{1\times 5+3\times 2}{2\times 5} = \frac{11}{10} \\ & & \\ \frac{1}{2}\cdot\frac{3}{5} & = & \frac{1\times 3}{2\times 5} = \frac{3}{10} \\ & & \\ \frac {1}{2}\div \frac{3}{5} & = & \frac{1\times 5}{2\times 3} = \frac{5}{6} \end{eqnarray*}\]

Formally, the rational numbers are defined as a set of equivalence classes of ordered pairs of integers, where the first component of the ordered pair is the numerator and the second is the denominator. The equivalence classes arise from the fact that a rational number may be represented in any number of ways by introducing common factors to the numerator and denominator. For instance, \(\displaystyle \frac{2}{3}\) and \(\displaystyle \frac{6}{9}\) are the same number:

\[\begin{eqnarray*} \frac{6}{9} & = & \frac{2\times 3}{3\times 3} \\ & & \\ & = & \frac{2}{3}\times \frac{3}{3} \\ & & \\ & = & \frac{2}{3} \times 1 \\ & & \\ & = & \frac{2}{3} \end{eqnarray*}\]

The equivalence class \(\sim\) between ordered pairs of integers that determines when two fractions represent the same rational number is given by:

\[\frac{a}{b}\sim\frac{c}{d} \Leftrightarrow ad = bc\]

Thus for instance, we see that \( \displaystyle \frac{2}{3}=\frac{6}{9}\) because \(2\times 9=3\times 6\).

Notice that this similarity definition breaks down if either of the denominators is zero, so this must be outlawed.

So, finally, the complete set-theoretic definition of the rational numbers is as follows:

\[ \mathbb{Q}= \left. \left\{ \left. \frac{p}{q} \, \right| \,\, p,q \in \mathbb{Z}, q\neq 0 \right\} \right/ \sim \]

It is common for students to find fractions to be “hard,” and there is a good reason for this. It is because they are hard. When working with a ratio of integers one is working not with a number per se, but with a representative of an equivalence class of ordered pairs. Of course it’s hard. But once you get the knack, a facility with fractions is a great boon because from then on you can frequently avoid frittering away your time with decimals and calculators. (And you can impress the socks off your friends, too.)