# 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.

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### We know existence of Transcendental raised to Algebraic Irrational equals rational, but what about opposite?

Introduction: If we take $a=2^\sqrt[3]{2}$ which is transcendental by Gelfond-Schneider Theorem, and $b=\sqrt[3]{4}$ which is algebraic irrational because it is root of monic-irreducible polynomial ...
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### Proof that a base 2 logarithm of a rational number is irrational

How can I prove that if $a = \log_2 b, b \in \Bbb Q, b \neq 2^c$ and $c \in \Bbb Z$ then $a \notin \Bbb Q$ ? And could the proof be easily adapted to differently-based logarithms? I am familiar with ...
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### I'm not quite sure I understand this one. Show that the specified real number is rational: $7^{2/3}$

This is the first problem in this discrete math assignment, and I'm a little bit confused because I thought that the square root, cube root, nth root of a non-square, non-cube, etc. were not rational ...
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### Is there an elementary proof that $2^{\sqrt{2}}$ is irrational?

Is there an elementary proof that $2^{\sqrt{2}}$ is irrational? The Gelfond-Schneider theorem states that if $a$ and $b$ are complex algebraic numbers such that $a \not\in \{0, 1\}$ and $b$ is ...
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### Are there rational solutions $r,s \in \mathbb{Q}$ to the equation $\tan^2(\pi r) + \tan^2(\pi s) = 1$

I am seeking to understand the structure of solutions to the diophantine equation $$\tan^2(\pi r) + \tan^2(\pi s) = 1.$$ I am conjecturing that there are no rational solutions $r, s \in \mathbb{Q}$ to ...
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### If $a=b+c$ , $a^2$ is an integer, can $b^2$ or $c^2$ or both be irrational? [closed]

Say, $a=b+c$, $a$ may be rational or irrational. However, the constraint on $a$ is that $a^2$ is an integer. b>0 , c>0 which means a>0. Wanted to confirm that either $b^2$ or $c^2$ or both ...
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### Is $\frac{2 \cdot 4\cdot 8 \cdot 16 \cdots}{1\cdot 3\cdot 7\cdot15 \cdots}$ irrational?

The number is the infinite product:$$\frac{2 \cdot 4\cdot 8 \cdot 16 \cdots}{1\cdot 3\cdot 7\cdot15 \cdots},$$ or about $3.4628.$ I have only proven this number is finite. Can you either prove it is ...
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### Proof that there is no rational number $r$ satisfying $2^r=3$ [duplicate]

Hello guys it would be a massive help for me if you could take a look at my proof and point out any errors. I want to know whether its coherent and whether I need more or less detail. Suppose $2^r = 3$...
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### Prove or disprove a claim regarding irrational numbers

I am trying to prove the following claim: Let $0\leq n \in \Bbb Z$ and suppose that there exists a $k \in \Bbb Z$ such that $n=4k+3$. Prove or disprove: $\sqrt n \notin \Bbb Q$ . The problem I am ...
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### Question in details of the proof of irrationality of $\zeta (3)$

I am reading through Wadim Zudilin's lecture notes and at the beginning of page 76 (i.e. in last part of Lemma 5.9.) things are getting really confusing : How $g(y) = g(0) − 2^{3/2}y^2 + O(y^3)$ is ...
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### Is it possible for the rationality of a computable number to be unknowable?

Main question: Suppose we have some real number $x$ that is proven to be not only be computable, but has had such an algorithm explicitly found. Could it be the case that it may be impossible to prove ...
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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 ...
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### Can an infinite product of rationals be rational if the partial products become increasingly irreducible?

I've been curious about methods of proving the irrationality of some infinite products and had this idea. Suppose $$\prod_{n=1}^ka_n=\frac{b_k}{c_k}$$ where $a_n\in\mathbb{Q}$ and $\gcd(b_k,c_k)=1$. ...
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### If $\alpha$ is an irrational number, then the minimal polynomial of $\alpha$ over $\Bbb{Q}$ does not have a rational root?

I'm reading Ross' "Elementary Analysis". He shows the rational zeros theorem and a proof of it. And then there are some examples such as the following ones: The basic idea is that testing ...
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### Does my proof of the irrationality of $\ln(2)$ hold up?

At first, I thought that proving the irrationality of $\ln(2)$ was so easy that it was trivial. However, someone in the comments of the $115$ vote answer at Can an irrational number raised to an ...
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1 vote
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### How to prove that $\frac{\ln 5}{\ln 2}$ is irrational

I have to show that $\frac{\ln 5}{\ln 2}$ is irrational I have tried the following: Assume it's rational so $\frac{\ln 5}{\ln 2} = \frac{p}{q}$ which becomes $\log _2\left(5\right)=\frac{p}{q}$ ...
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### If b is rational and c is real, can $x^2 + bx + c$ have one rational root and one irrational root? Use only definition of rational numbers to prove.

If $b$ is rational and $c$ is real, can $x^2 + bx + c$ have one rational root and one irrational root? Use only the definition of rational numbers to prove. I think that it can't because since $b$ is ...
1k views

### Why was Apery's constant believed to be rational?

Apéry's theorem states that Apéry's constant $\zeta(3) := \sum_{n=0}^\infty n^{-3}$ is irrational. The Wikipedia article claims that this result was "wholly unexpected" to the point that the ...
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### Is $\sqrt[3]{7+5 \sqrt{2}}+\sqrt[3]{20-14 \sqrt{2}}$ a rational number?

Is there a way to show that $$\alpha=\sqrt[3]{7+5 \sqrt{2}}+\sqrt[3]{20-14 \sqrt{2}}$$ is a rational number? I found $\alpha=3$ from doing simplifications. But, I would like to known a different ...
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### How do I show that $\log_2(k)$ is an irrational number if $k$ is an odd integer greater than $1$?
Show that if $k$ is an odd integer grater than $1$, then $\log_2(k)$ is an irrational number. So my approach for this problem was to use proof by contradiction so I wrote "Assume by ...