Density of Q in R

The density of the rational numbers in the real numbers—the fact that between any two distinct reals we can always find a rational—is a fundamental result in real analysis. In the HSC Mathematics Extension 2 course, this proof illustrates a classic technique: magnifying the gap by a large integer so that it exceeds 1, then trapping an integer inside the scaled interval. Mastering this existence argument will strengthen your ability to reason about the real numbers using the Archimedean property.

Problem Statement

Prove that for any real numbers $x$ and $y$ with $x < y$, there exists a rational number $q$ such that

$$x < q < y.$$

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Hints

Attempt the proof independently first. Focus on the key theorem, algebraic transformation, or contradiction setup that links the hypothesis to the target conclusion.

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Solutions

Existence Proof

The plan: find integers $n>0$ and $m$ such that $nx < m < ny$. Because then dividing by $n$ gives a rational $q=m/n$ between $x$ and $y$. For an integer $m$ to lie strictly between $nx$ and $ny$, the length of that interval must be greater than $1$.

Since $y-x>0$, we can choose a positive integer $n$ such that

$$n(y-x)>1.$$

Equivalently,

$$ny-nx>1.$$

Now the scaled interval $(nx,ny)$ has length exceeding $1$, so it must contain at least one integer. To pick one explicitly, let $m$ be the smallest integer greater than $nx$ (think of $m$ as the ‘ceiling’ of $nx$). Then

$$m-1 \le nx < m.$$

So in particular,

$$nx < m \le nx+1.$$

But $ny-nx>1$ gives

$$nx+1<ny.$$

Hence

$$nx<m\le nx+1<ny,$$

so

$$nx<m<ny.$$

Dividing through by $n>0$,

$$x<\frac{m}{n}<y.$$

Therefore, with $q=\frac{m}{n}$, we have found a rational number satisfying $x<q<y$.

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Takeaways

• Magnification Idea: Multiply the interval by a large integer so its length becomes greater than $1$
• Key Step: Once the gap exceeds $1$, an integer can be placed between the endpoints
• Density of $\mathbb{Q}$: Rational numbers occur between every two distinct real numbers
• Big Picture: This is an existence proof rather than a constructive formula for a specific rational
• For any real numbers $x$ and $y$ with $x < y$, there exists infinitely many rational numbers $q$ such that $x < q < y$.

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Further Readings

If you found this proof interesting, be sure to check out these relevant HSC booklets to sharpen your reasoning skills: HSC Proofs, HSC Polynomials, HSC Differential Equations