# Questions tagged [rationality-testing]

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.

293 questions
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### 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 ...
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### 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 ...
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### 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 $(x+1)^n=s$ ...
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### 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 ...
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### Graph theoretic proof of irrational number.

Let there be six irrational numbers. Prove that there exists three irrational numbers among them such that the sum of any two of those irrational numbers is also irrational. I have tried to prove ...
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### 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 ...
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### Is the sum and difference of two irrationals always irrational?

If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational?
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### 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 ...
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### 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?
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### 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 ...
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### $\sin 1^\circ$ 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 ...
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### 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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### 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 ...
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### Prove that $5^{1/3}+7^{1/2}$ is irrational

Goal: Prove that $5^{1/3}+7^{1/2}$ is irrational. Idea: We can prove this is irrational by supposing it is rational and finding a contradiction. So, $5^{1/3}+7^{1/2} = p/q$ where $p$ and $q$ are ...
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### Why is $\sum_{i=1}^n a$ always irrational if $n>0$ and $a$ is irrational?

I'm asking this question because I was unable to find an answer elsewhere as most questions are about the summation of different irrational numbers, which is not what this question is about. Here, I'm ...
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### Why aren't all square roots irrational? [duplicate]

The more known proof of square root of $2$ is by contradiction when we assume it can be expressed as an irreducible fraction and later finding that it isn't irreducible, but... if we assume the same ...
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### Prove that $\sqrt 5$ is irrational

I have to prove that $\sqrt 5$ is irrational. Proceeding as in the proof of $\sqrt 2$, let us assume that $\sqrt 5$ is rational. This means for some distinct integers $p$ and $q$ having no common ...
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### Proving the irrationality of $e^n$

Let $n$ be a positive integer. I know the traditional proof that $e$ is irrational. How do we show that $e^n$ is irrational in some sort of similar line? I am of course assuming it is but I would be ...
795 views

### $0.101001000100001000001$… is irrational. [duplicate]

How do I show that $0.101001000100001000001...$ is irrational? How do I generalize this to decimal expansions of a similar type, i.e. what is a criterion for decimal expansions with $0$'s and $1$'s ...
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### Proof that $\sqrt6 - \sqrt2 - \sqrt3$ is irrational. [duplicate]

I want to prove that: $$\sqrt6 - \sqrt2 - \sqrt3$$ is irrational. I have tried using squares, the $p/q$ definition of rationality and the facts that 1)rational$\times$ irrational=irrational (unless ...
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### Prove or disprove that $\sqrt{2+\frac{1}{n}}$ is irrational for $n \in \mathbb{Z}^+$

I have good reason to suspect that $\sqrt{2+\frac{1}{n}}$ is irrational for all $n \in \mathbb{Z}^+$ but a proof of this eludes me. I've tried proof by contradiction have had no success. I've also ...
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### Show that $({\sqrt{2}\!+\!1})^{1/n} \!+ ({\sqrt{2}\!-\!1})^{1/n}\!\not\in\mathbb Q$

How could we prove that for every positive integer $n$, the number $$({\sqrt{2}+1})^{1/n} + ({\sqrt{2}-1})^{1/n}$$ is irrational? I think it could be done inductively from a more general expression,...
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### Proof that $\sin10^\circ$ is irrational

Today I was thinking about proving this statement, but I really could not come up with an idea at all. I want to prove that $\sin10^\circ$ is irrational. Any ideas?
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### Proving that $\sqrt{2}+\sqrt{3}$ is irrational [duplicate]

This is from Spivak. Prove that $\sqrt{2}+\sqrt{3}$ is irrational. So far, I have this: If $\sqrt{2}+\sqrt{3}$ is rational, then it can be written as $\frac{p}{q}$ with integral $p, q$ and in ...
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### Prove that 2.101001000100001… is an irrational number.

My try: This number is non-terminating and non-repeating, so this is an irrational number. But how do I prove it more formally in a more mathematically rigorous way?
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### Irrationality of $\pi^2$ and $\pi^3$

I wonder if there is any book and/or article you can recommend on the topic "Irrationality of $\pi^2$ and $\pi^3$" for me to study on. In case you are curious about why I ask these particular ...
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### Proof that at most one of $e\pi$ and $e+\pi$ can be rational

$e$ and $\pi$ are rather peculiar numbers. It turns out that, in addition to being irrational numbers, they are also transcendental numbers. Basically, a number is transcendental if there are no ...
283 views

### Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational?

Is $\sum_{n \ge 1}{\frac{p_n}{n!}}$ irrational, where $p_n$ is the $n^{\text{th}}$ prime number? This question is spurred by the comment thread on this question where I presented a rough idea of a ...
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### How does one prove that $\zeta(3)$ is irrational?

How does one prove that $\zeta(3)$ is irrational ? I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?
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### What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves? [duplicate]

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a known ...
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### Irrationality or Rationality of p+q given that pq=1

Let $p,q$ are irrational numbers such that $pq=1$. Then what can we say about character of $p+q$? i.e, $p+q$ is rational or irrational? Probably, I believe it is irrational but could not prove.
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### Is $\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+…$ irrational?

$$\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+\cdots$$ Is this infinite sum irrational? Is there a known way to prove it?
### Is it true that $\sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+$ such that $a^2-b=k^2$?
This is a curiosity question: Question Given two positive integers $a$ and $b$ do we have the following equivalence: \sum_{n=0}^{\infty}\frac{1}{n^2+2an+b}\in \Bbb Q \iff \exists k\in \Bbb N^+\...