Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such a rational number?

[I posted this only so that the useful answers at https://math.stackexchange.com/questions/414036/rationals-and-irrationals-on-the-real-number-line/414048#414048 could be merged here before that question was deleted.]

**Answer**

Let $x,y\in\mathbb{R}$, $x\neq y$. Without loss of generality, suppose $x<y$. Then there exists a positive $z$ such that $y-x=z$.

By Archimedes’ axiom, there exists a natural number $n$ such that

$$n > \dfrac{1}{z}$$

$$nz > 1$$

$$ny – nx > 1$$

So there exists an integer $m$ such that

$$nx < m < ny$$

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

i.e. $m/n$ is a rational number between $x$ and $y$.

Since $x$ and $y$ can be any real numbers, in particular they can be irrationals.

**Attribution***Source : Link , Question Author : MJD , Answer Author : joeA*