Introductory course · Middle school

Rational Numbers

Learning objectives

Define a rational number and recognise one in different forms.
Locate rational numbers on the number line.
Explain why rational decimals either stop or repeat.
Show that between any two rationals there is another.

Definition

A number is rational if it can be written as \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b\neq 0\). The set of rational numbers is written \(\mathbb{Q}\).

Example

Each of these is rational:

\[7=\frac{7}{1},\qquad -\frac{5}{3},\qquad 0.75=\frac{3}{4}\]

Equivalent fractions

Different fractions can name the same number. Multiplying numerator and denominator by the same nonzero integer does not change the value:

\frac{a}{b}=\frac{ka}{kb}\qquad(k\neq 0)

Example

Divide top and bottom by their greatest common divisor, 6: \(\frac{18}{24}=\frac{3}{4}\). A fraction is in lowest terms when its numerator and denominator share no common factor other than 1.

Remark

Keep the denominator positive: write \(\frac{3}{-4}\) as \(-\frac{3}{4}\).

See it on the number line

Comparing rational numbers

Use cross-multiplication. For positive denominators \(b,d\):

\frac{a}{b}<\frac{c}{d}\iff ad<bc

Worked example

Which is larger, \(\frac35\) or \(\frac47\)? Compare \(3\cdot 7=21\) with \(5\cdot 4=20\). Since 21 > 20, \(\frac35>\frac47\).

Adding rational numbers

\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}

Worked example

\[\frac23+\frac34=\frac{2\cdot 4+3\cdot 3}{3\cdot 4}=\frac{17}{12}\]

Theorem · Density

If \(r<s\) are rational, then \(\dfrac{r+s}{2}\) is a rational number strictly between them.

Show proof

From \(r<s\) we get \(2r<r+s<2s\), so dividing by 2 gives \(r<\frac{r+s}{2}<s\). If \(r=\frac{a}{b}\) and \(s=\frac{c}{d}\), the midpoint equals \(\frac{ad+bc}{2bd}\), a quotient of integers with nonzero denominator. ∎

Worked example

Find a rational number between \(\frac13\) and \(\frac12\).

\[\frac{1}{2}\left(\frac13+\frac12\right)=\frac{1}{2}\cdot\frac{5}{6}=\frac{5}{12}\]

Example · Decimals

\(\frac18=0.125\) stops, while \(\frac17=0.\overline{142857}\) repeats a block of six digits. Set the sliders above to 1 and 7 to watch it appear.

Remark

The decimal form of a rational number always stops (like \(0.75\)) or repeats forever (like \(0.\overline{3}\)), because long division can only produce finitely many remainders.

Exercises

Write \(0.\overline{36}\) as a fraction in lowest terms.

Show solution

Let \(x=0.3636\ldots\). Then \(100x=36.3636\ldots\). Subtracting gives \(99x=36\), so \(x=\frac{36}{99}=\frac{4}{11}\).

Which is larger, \(\frac58\) or \(\frac{7}{11}\)?

Show solution

\(5\cdot 11=55\) and \(8\cdot 7=56\). Since 55 < 56, \(\frac58<\frac7{11}\).

Compute \(\frac12+\left(-\frac23\right)\).

Show solution

\(\frac{1\cdot 3+2\cdot(-2)}{6}=\frac{-1}{6}=-\frac16\)

Find a rational number between \(\frac25\) and \(\frac37\).

Show solution

Take the midpoint: \(\frac12\left(\frac{14}{35}+\frac{15}{35}\right)=\frac{29}{70}\).

Write \(0.\overline{7}\) as a fraction.

Show solution

Let \(x=0.777\ldots\). Then \(10x-x=7\), so \(x=\frac79\).

Is 0 a rational number? Is \(\frac10\)?

Show solution

Yes: \(0=\frac01\). No: division by zero is undefined, so \(\frac10\) is not a number at all.

Summary

Rational numbers are fractions of integers with nonzero denominator.
Their decimals stop or repeat.
Between any two rationals lies another, in fact infinitely many.