Chapter 6, Problem 1NT

To determine

To explain:

Why there are infinitely many rational numbers between any two rational numbers.

Answer to Problem 1NT

Solution:

Yes, there are infinitely many rational numbers between any two rational numbers.

Explanation of Solution

Given:

There are infinitely many rational numbers between any two rational numbers.

Approach:

Use denseness property of rational numbers.

Calculation:

Suppose there are two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$,

where $\frac{a}{b}<\frac{c}{d}$.

By denseness property of rational numbers, we can find a rational number $x_1$ between the two fractions. Since $\frac{a}{b}<x_1$, there is a rational number $x_2$ between $\frac{a}{b}$ and $x_1$. We can next find a rational number $x_3$ between $\frac{a}{b}$ and $x_2$ and so on. Hence there can be infinitely many rational numbers between $\frac{a}{b}$ and $\frac{c}{d}$.