When Fractions Are Not Enough: Why Real Numbers Matter

Can a fraction represent every point on the number line? In this illustrated precalculus lesson, three university students investigate that question using a counterexample: the length represented by \(\sqrt{2}\). Their discussion introduces the distinction between rational numbers and irrational numbers, and explains why both belong to the real number system.

Why \(\sqrt{2}\) cannot be a fraction

A rational number can be written as \(\frac{p}{q}\), where \(p,q\in\mathbb{Z}\) and \(q\ne 0\). Suppose, for a contradiction, that \(\sqrt{2}\) is rational. We can choose a representation in lowest terms:

\[\sqrt{2}=\frac{p}{q},\qquad \gcd(p,q)=1.\]

Squaring both sides and multiplying by \(q^2\) gives \(p^2=2q^2\). Therefore \(p^2\) is even, which implies that \(p\) is even. Writing \(p=2k\), with \(k\in\mathbb{Z}\), produces

\[4k^2=2q^2\quad\Longrightarrow\quad q^2=2k^2.\]

Consequently \(q\) is even as well. Both numerator and denominator would then be divisible by \(2\), contradicting \(\gcd(p,q)=1\). The original assumption is false:

\[\boxed{\sqrt{2}\notin\mathbb{Q}.}\]

This proof by contradiction shows that the rational numbers cannot account for every point on the number line.

Where rational and irrational numbers belong

The illustration places the familiar number sets inside one another. The natural numbers lie within the integers; the integers lie within the rational numbers; and the rational numbers lie within the real numbers:

\[\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}.\]

For example, \(-3\), \(\frac12\), and \(0.75=\frac34\) are rational. By contrast, \(\sqrt{2}\) and \(\pi\) are examples of irrational real numbers: they belong to \(\mathbb{R}\setminus\mathbb{Q}\). The accompanying number-line panel illustrates how rational and irrational values occupy positions on the same real number line.

The central idea

The counterexample answers the students’ opening question: not every real number is a ratio of integers. The set \(\mathbb{R}\) includes rational values and irrational values alike, providing the number system represented by the complete number line.