Sorry is a little confuse, please help.

Question

Is the square root of $2$ divided by any integer (except zero), a rational or irrational number?

Answer 1

Here is a thorough explanation.

Take an arbitrary integer $n\not=0$.

Then $\frac{n}{\sqrt{2}}$ is the desired expression.

Assume for sake of contradiction that $\frac{n}{\sqrt{2}}=\frac{p}{q}$ for relatively prime integers $p$ and $q$.

Square both sides.

$$\frac{n^2}{2}=\frac{p^2}{q^2}$$

Multiply both sides by $2q^2$.

$$n^2q^2=2p^2$$

So $2|n$ or $2|q$.

Case 1: Let $n=2k$ for positive $k$.

Then, $\frac{n}{\sqrt{2}}=k \sqrt{2}$ which is obviously irrational.

Case 2: Let $q=2a$ for integer $a$.

Then, $$4n^2a^2=2p^2$$

$$2n^2a^2=p^2$$

Hence $2|p$ which is a contradiction as we have assumed $p$ and $q$to be relatively prime.

Hence it will always be irrational.

This is VERY long-winded, but hopefully it will completely clear it up for you.

Answer 2

It is a irrational, as a irrational diveded by any integers (but 0) is still a irrational.