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### How can you prove that the square root of two is irrational? [closed]

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
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### The sum of square roots of non-perfect squares is never integer [duplicate]

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
99k views

### Prove that $\sqrt 2 + \sqrt 3$ is irrational [duplicate]

I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. ...
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### Novel (?) proof of the irrationality of $\sqrt3$ [duplicate]

A student of mine offered the following proof of the irrationality of $\sqrt{3}$: Suppose $(a/b)^2 = 3$ with $a,b$ having no common factor. Since $a^2=3b^2$, an easy parity argument (using the fact ...
210 views

### Find number of integral solutions of $\sqrt{n}+\sqrt{n+7259}$

How many integers $n$ are there such that $\sqrt{n}+\sqrt{n+7259}$ is an integer? No idea on this one.
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### Prove $\log_7 n$ is either an integer or irrational

I have been trying to prove a certain claim and have hit a wall. Here is the claim... Claim: If $n$ is a positive integer then $\log_{7}n$ is an integer or it is irrational Proof (so ...
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331 views

### irrationality proof of $\sqrt{n}+\sqrt{n+1}$ for any $n>0$

irrationality proof of $\sqrt{n}+\sqrt{n+1}$ for any $n>0$ My attempt: $\sqrt{n}+\sqrt{n+1}=\frac{p}{q}$ $2n+1+2\sqrt{n(n+1)}=\frac{p^2}{q^2}$ Now we have to show $2n+1+2\sqrt{n(n+1)}$ cannot ...
288 views

### Show that there are no rationals $r$ such that $r^3 = 6$

Show that there are no rationals $r$ such that $r^3 = 6$ We were asked this for a real analysis assignment. I just assumed that it would be the same as if it were $r^2$, but now I'm second guessing ...
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### Proving the irrationality of $\sqrt{5}$: if $5$ divides $x^2$, then $5$ divides $x$

I am working on proving that $\sqrt{5}$ is irrational. I think I have the proof down, there is just one part I am stuck on. How do I prove that $x^2$ is divisible by 5 then $x$ is also divisible by ...
### Show that $(\sqrt{2}-1)^n$ is irrational
Show that for all $n\in \mathbb{N}$ the number $(\sqrt{2}-1)^n$ is irrational. I do not get the idea of the proof at all, any help appreaciated. edit: I am also thinking whether it will be possible ...