For questions on determining whether a number is rational, and related problems. If applicable, use this tag instead of (rational-numbers) and (irrational-numbers). Consider adding a tag (radicals) or (logarithms), depending on what the question is about.

# Questions tagged [rationality-testing]

358 questions

**164**

votes

**14**answers

### How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?

It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing 'even' with 'divisible by $3$'), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly $\sqrt{n^2} = n$ for any positive integer $n$. …

anonymous

**126**

votes

**1**answer

### Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ?
We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can not be rational:
Denote…

lsr314

- 15,458
- 5
- 34
- 87

**111**

votes

**2**answers

### Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?"
Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but…

users31526

- 1,851
- 2
- 13
- 13

**111**

votes

**19**answers

### What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational?
Here is my favorite:
Theorem: $\sqrt{2}$ is irrational.
Proof:
$3^2-2\cdot 2^2 = 1$.
(That's it)
That is a corollary of
this result:
Theorem:
If $n$ is a positive…

marty cohen

- 101,285
- 9
- 66
- 160

**98**

votes

**4**answers

### A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational.
What I have tried:
Denote $x^n=r$ and $(x+1)^n=s$ with $r$, $s$ rationals. For each $k=0,1,\ldots,…

Dan Ismailescu

- 1,163
- 8
- 9

**63**

votes

**17**answers

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

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?

John Gietzen

- 3,281
- 4
- 24
- 25

**63**

votes

**7**answers

### Is the sum and difference of two irrationals always irrational?

If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational?

Miguel Mora Luna

- 2,959
- 5
- 25
- 30

**50**

votes

**7**answers

### $\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is .....

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and using the strong induction principle. (Problem on…

user21436

**50**

votes

**2**answers

### Irrationality of $\sqrt{2\sqrt{3\sqrt{4\cdots}}}$

In this question it is stated that Somos' quadratic recurrence constant
$$\alpha=\sqrt{2\sqrt{3\sqrt{4\sqrt{\cdots}}}}$$
is an irrational number. [update: the author of that question is no longer claiming to have a proof of this]
This fact seems by…

Mizar

- 5,731
- 18
- 46

**50**

votes

**2**answers

### Prove that $\sum_{n=1}^\infty \frac{n!}{n^n}$is irrational.

Let $$S= \sum_{n=1}^\infty \frac{n!}{n^n}$$
(Does anybody know of a closed form expression for $S$?)
It is easy to show that the series converges.
Prove that $S$ is irrational.
I tried the sort of technique that works to prove $e$ is irrational, but…

Mark Fischler

- 40,850
- 2
- 36
- 70

**45**

votes

**1**answer

### irrationality of $\sqrt{2}^{\sqrt{2}}$.

The fact that there exists irrational number $a,b$ such that $a^b$ is rational is proved by the law of excluded middle, but I read somewhere that irrationality of $\sqrt{2}^{\sqrt{2}}$ is proved constructively. Do you know the proof?

user53216

**43**

votes

**9**answers

### Prove that $\sqrt 2 + \sqrt 3$ is irrational

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. My first question is, is this reasoning…

ankush981

- 1,943
- 3
- 22
- 33

**41**

votes

**5**answers

### Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}, \ \ a \in \mathbb{Q}$, which I don't think is…

Spine Feast

- 4,420
- 3
- 35
- 62

**37**

votes

**9**answers

### Prove $2^{1/3}$ is irrational.

Please correct any mistakes in this proof and, if you're feeling inclined, please provide a better one where "better" is defined by whatever criteria you prefer.
Assume $2^{1/2}$ is irrational.
$2^{1/3} * 2^{x} = 2^{1/2} \Rightarrow x = 1/6$.…

user4536

**37**

votes

**6**answers

### Test of being a rational number for $(1-\frac13+\frac15-\frac17+\cdots)/(1+\frac14+\frac19+\frac1{16}+\cdots)$

Is the following expression a rational number?
$$\frac{1-\dfrac13+\dfrac15-\dfrac17+\cdots}{1+\dfrac14+\dfrac19+\dfrac1{16}+\cdots}$$
My thoughts:
The sum and product of two rational numbers is a rational number.
So is the difference.
As well as…

user148849

- 525
- 4
- 7