# 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 $\displaystyle n$ 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 $\displaystyle \left( {j\sqrt n - \left\lfloor {\sqrt n } \right\rfloor j} \right)\left( {\sqrt n } \right)$?

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.

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

How can we prove this?

-O

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# Show whether sqaure root 17 is rational or not

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