Rational numbers

Lowest terms and canonical form

A rational number is in lowest terms if and only if the GCD of its numerator and denominator is \(1\).

A rational number is in canonical form if and only if

• it is in lowest terms;
• the denominator is positive; and
• if the rational number is negative, then the numerator is negative (as in \(\tfrac{-2}{3}\), or the negative sign is written before the entire rational number (as in \(-\tfrac{2}{3}\).

Comparison

For two (or more) rational numbers to be compared, they must have the same denominator. This is often called a common denominator. The common denominator of two (or more) rational numbers is the LCM of their individual denominators.

When two rational numbers have a common denominator, they are compared based on their numerators.

Multiplication

When two rational numbers are multiplied, we multiply their numerators to get the numerator of the result, and their denominators to get the denominator of the result.

\[\dfrac{\color{blue}{a}}{\color{red}{b}}\times\dfrac{\color{blue}{c}}{\color{red}{d}}=\dfrac{\color{blue}{a}\times\color{blue}{c}}{\color{red}{b}\times\color{red}{d}}\]

For example, instead of evaluating \(\frac{15}{18}\times\frac{14}{35}\) as \(\frac{210}{630}\) and then putting the result in lowest terms, we should reduce before the multiplication, as follows.

\[\require{cancel}\dfrac{\cancelto{3}{15}}{18}\times\dfrac{14}{\cancelto{7}{35}}=\dfrac{\cancelto{1}{3}}{\cancelto{6}{18}}\times\dfrac{\cancelto{2}{14}}{\cancelto{1}{7}}=\dfrac{1}{\cancelto{3}{6}}\times\dfrac{\cancelto{1}{2}}{1}=\dfrac{1}{3}\]

Reciprocal

A number is the reciprocal of another if and only if their product is equal to one. The reciprocal of any number \(n\) is the rational number \(\frac{1}{n}\).

\[n\times\frac{1}{n}=1\]

The reciprocal of a rational number \(\frac{a}{b}\) is the rational number \(\frac{b}{a}\), obtained by switching the numerator and the denominator.

\[\dfrac{a}{b}\times\dfrac{b}{a}=1\]

Division

Dividing by a rational number is equivalent to multiplying by the reciprocal of this number.

\[\dfrac{\color{blue}{a}}{\color{red}{b}}\div\dfrac{\color{blue}{c}}{\color{red}{d}}=\dfrac{\color{blue}{a}}{\color{red}{b}}\color{green}{{}\times{}}\dfrac{\color{red}{d}}{\color{blue}{c}}\]

Addition and subtraction

Two rational numbers can be added or subtracted if and only if they have a common denominator. We then add or subtract their numerators.

\begin{align} \dfrac{\color{blue}{a}}{\color{red}{c}}+\dfrac{\color{green}{b}}{\color{red}{c}}&=\dfrac{\color{blue}{a}+\color{green}{b}}{\color{red}{c}} \\ \dfrac{\color{blue}{a}}{\color{red}{c}}-\dfrac{\color{green}{b}}{\color{red}{c}}&=\dfrac{\color{blue}{a}-\color{green}{b}}{\color{red}{c}} \end{align}

Raising to a power

When a rational number is raised to a power, both its numerator and denominator are raised to that power.

\[\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\]