# Tagged Questions

0answers
120 views

### Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
3answers
232 views

### Prove that $\sqrt[3]{5-\sqrt{2}}$ is not a rational number

My attempt: Consider the polynomial $(x^3-5)^2 - 2 = x^6 -10x^3 + 23 = 0$. By the rational zeros theorem, we can conclude that $\pm 1$ and $\pm 23$ are the only possible rational solutions*. ...
5answers
226 views

### Different ways to prove $\sqrt p$ irrational for $p$ prime.

I know this fact can be proved by contradiction(reductio ad absurdum) but please give proofs by different methods.
0answers
143 views

### Direct proof that $\sqrt{2}$ is irrational? [duplicate]

Possible Duplicate: Irrationality proofs not by contradiction I've been puzzled for some days now, and I can't come up with an answer. I'm trying to come with a direct proof that $\sqrt{2}$ ...
3answers
565 views

### Is this proof that $\sqrt 2$ is irrational correct?

Suppose $\sqrt 2$ were rational. Then we would have integers $a$ and $b$ with $\sqrt 2 = \frac ab$ and $a$ and $b$ relatively prime. Since $\gcd(a,b)=1$, we have $\gcd(a^2, b^2)=1$, and the fraction ...
1answer
3k views

### Sum of irrational numbers

Well, in this question it is said that $\sqrt[100]{\sqrt3 + \sqrt2} + \sqrt[100]{\sqrt3 - \sqrt2}$, and the owner asks for "alternative proofs" which do not use rational root theorem. I wrote an ...
3answers
289 views

### How to prove $e$ isn't a $\frac {a}{b}$. Not irrationality with other ways or about transcendental, only about fractions

I would like a proof that $e$ isn't a fraction $\frac{a}{b}$, for $a,b \in Z$ and $mdc(a,b)=1$. Just a observation =) I'd like a proof with fractions, not about $e$ irrationality or if $e$ is ...