Density of Numbers Math Example 2

Solution

1. 1
   We need $x$ irrational with $1 < x < 2$.
2. 2
   Consider $\sqrt{2}$: we know $1^2 = 1 < 2 < 4 = 2^2$, so $1 < \sqrt{2} < 2$.
3. 3
   $\sqrt{2}$ is irrational (proved by contradiction: if $\sqrt{2} = \frac{p}{q}$ in lowest terms, then $2q^2 = p^2$, so $p^2$ is even, $p$ is even, say $p=2k$; then $2q^2 = 4k^2$, so $q^2 = 2k^2$, meaning $q$ is even — contradicting lowest terms).
4. 4
   Therefore $\sqrt{2}$ is an irrational number strictly between $1$ and $2$.