Answer
See explanation
Work Step by Step
A concise way to show \(\mathbb{Q}\) is dense is by taking the **average** of two rationals:
1. Let \(r_1\) and \(r_2\) be rational numbers with \(r_1 < r_2\).
2. Define
\[
x \;=\; \frac{r_1 + r_2}{2}.
\]
3. Since \(r_1\) and \(r_2\) are rational, their sum \(\,(r_1 + r_2)\) is rational, and so is \(\tfrac{r_1 + r_2}{2}\).
4. Clearly,
\[
r_1
\;=\; \frac{r_1 + r_1}{2}
\;<\; \frac{r_1 + r_2}{2}
\;<\; \frac{r_2 + r_2}{2}
\;=\; r_2.
\]
Hence \(r_1 < x < r_2\), and \(x\) is a rational number. Because we can always find such an \(x\) between any two rationals \(r_1 < r_2\), the set of rationals \(\mathbb{Q}\) is **dense** on the real number line.
