# Thread: Check whether root 17 is rational or irrational?

1. ## Check whether root 17 is rational or irrational?

Prove whether root 17 is rational or irrational.

2. The square root of any positive non-square integer is irrational.

3. ## Rational or irrational

Thanks a lot friend... but i need this to be proved by proof by contradiction.. Anybody there to help me?

4. The square root of any positive non-square integer is irrational.
Proof:
Suppose that $n$ is a positive non-square integer.
If $\sqrt n$ were rational then the set $T = \left\{ {k \in \mathbb{Z}^ + :k\sqrt n \in \mathbb{Z}^ + } \right\}$ is not empty.
Let $j$ be the first in $T$. We see $j \ne 1$ because $n$ is a on-square.
That means that $j > 1$. Use the floor function: $0 < \sqrt n - \left\lfloor {\sqrt n } \right\rfloor < 1\, \Rightarrow \,0 < j\sqrt n - \left\lfloor {\sqrt n } \right\rfloor j < j$.
That is a contradiction. Do you see it?
Is $j\sqrt n - \left\lfloor {\sqrt n } \right\rfloor j$ a positve integer?
What is $\left( {j\sqrt n - \left\lfloor {\sqrt n } \right\rfloor j} \right)\left( {\sqrt n } \right)$?

5. 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.

$\sqrt a + \sqrt b = c$
where c is a rational number. No such a, b, c exists.

How can we prove this?

-O

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### prove that root 17 is irrational

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