Prove whether root 17 is rational or irrational.
The square root of any positive non-square integer is irrational.
Proof:
Suppose that $\displaystyle n$ is a positive non-square integer.
If $\displaystyle \sqrt n$ were rational then the set $\displaystyle T = \left\{ {k \in \mathbb{Z}^ + :k\sqrt n \in \mathbb{Z}^ + } \right\}$ is not empty.
Let $\displaystyle j$ be the first in $\displaystyle T$. We see $\displaystyle j \ne 1$ because $\displaystyle n$ is a on-square.
That means that $\displaystyle j > 1$. Use the floor function: $\displaystyle 0 < \sqrt n - \left\lfloor {\sqrt n } \right\rfloor < 1\, \Rightarrow \,0 < j\sqrt n - \left\lfloor {\sqrt n } \right\rfloor j < j$.
That is a contradiction. Do you see it?
Is $\displaystyle j\sqrt n - \left\lfloor {\sqrt n } \right\rfloor j$ a positve integer?
What is $\displaystyle \left( {j\sqrt n - \left\lfloor {\sqrt n } \right\rfloor j} \right)\left( {\sqrt n } \right)$?
I have a similar question and not starting a new thread since they may be related:
Suppose that a and b are positive non-square integers.
$\displaystyle \sqrt a + \sqrt b = c$
where c is a rational number. No such a, b, c exists.
How can we prove this?
-O