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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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Proving that things are or aren't rational numbers

My question comes in three parts: Suppose $x,y\in \Bbb Q$. Prove that $2x-5y\in \Bbb Q$ Prove that $3^{1/2}\not\in \Bbb Q$ Suppose $x\in \Bbb Q$. Prove that $x^2+3^{1/2}\not\in \Bbb Q$ In the ...
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39 views

Rational or irrational sum and the integral

I wanted to ask you is it possible to define that the number n is rational or irrational from analysis of integral form of function of series, for e. x. we have ...
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143 views

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^+\...
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189 views

Alternative Proof to irrationality of $\sqrt{2}$ using linear algebra

I am taking my first Proof course, and have been researching alternative proofs to the irrationality of $\sqrt{2}$. One that particularly interested me could be found on this site as number $10$, by ...
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1answer
67 views

Decide if radical expression equals a given rational number

Given a radical expression an a rational number. Is there an algorithm to decide if the expression equals the number? Example: $(\sqrt{\sqrt[5]{74} - \sqrt[14]{78}})^{356}+\sqrt[6]{63} \overset{?}= ...
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1answer
57 views

If the slopes of bisectors are rational numbers, the slopes of sides are also rational numbers

Let $m_1, m_2, m_3$ be the slopes of the cartesian equations of the sides of a triangle of which no side is parallel to y-axis, and let $s_1,s_2,s_3$ be the slopes of the cartesian equations of the ...
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1answer
30 views

Shifting roots in infinite sums of polynomials

Define $$P(n)=(n-r_1)(n-r_2)...(n-r_k)$$ where $r_i\in\mathbb{Q}$ and $r_i\neq r_j$ for $i\neq j$. Now, define $$Q(n)=(n-(r_1+m))(n-r_2)...(n-r_k)$$ where $m\in\mathbb{Z}$ and $r_1+m\neq r_j$ for $...
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1answer
68 views

Rational solutions for $\sin(n)$ in radians

This is completely for my own curiosity. Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians. I know that this is strange because usually integers are only used in ...
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1answer
82 views

Goodness-of-Fit tests for Multinomial and Binomial Data

A box has 4000 red, 5000 blue and 1000 orange balls. A selection of 70 balls is made, with 25 reds, 35 blues, and 10 oranges being observed. Can one essentially prove that the selection was NOT a ...
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1answer
54 views

Prove that $\sqrt m$ is irrational by showing that the set $\{n\in\mathbb N: n\sqrt m\in\mathbb N\}$ is empty

Let $m\in\mathbb N$ be such that $m\neq k^2$ for all $k\in\mathbb N$. Prove that $\sqrt m$ is irrational by showing that the set $\{n\in\mathbb N: n\sqrt m\in\mathbb N\}$ must be empty.
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Irrationality of $\min$ and $\arg\,\min$ of $\Gamma|_{[1, 2]}$

The Gamma function achieves a local minimum at $x^* \approx 1.46163$, where $\Gamma(x^*) \approx 0.88560$. Can $x^*$ and $\Gamma(x^*)$ easily be proven irrational? Are they transcendental?
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439 views

Alternative Proof of $\sqrt{2}$ is irrational

Can anyone check if this proof is correct. Thank you. Proof that $\sqrt{2}$ is irrational. Let $x = \sqrt{2}$ then $x^2=2$ and $x^2-2=0$ By the Rational Root Theorem, we have: the number $1$ ...
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87 views

Are there any proofs for (ir)rationality of the numbers $\sin(e)$, $\cos(e)$?

Are there any proofs for the (ir)rationality of the numbers $\sin(e)$, $\cos(e)$, $\tan(e)$, and $\cot(e)$? Thanking in advance for any references.
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When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ block}}\overbrace{...
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How to find if $n$th root of $m$ is rational?

I know that if $m$ is an integer, then the $n$th root of $m$ must be an integer, else it is irrational. But what if $n$ and $m$ both are decimals - is there a way of easily telling if the $n$th root ...
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125 views

Irrationality of $\zeta(\frac{3}{2})$

Is $$ \zeta\left(\frac{3}{2}\right) = \sum_{n=1}^\infty \frac{1}{n^{3/2}} $$ an irrational number?
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328 views

Very simple proof that $\sqrt{2}$ is irrational.

I came across a nice-looking proof that $\sqrt{2}$ is irrational here. It somehow seems to good to be true. What are the assumptions being made in the proof and if this proof is indeed correct, why is ...
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59 views

Proof Verification: for $u,v\in\mathbb Q$, $u^{\frac{1}n} + v^{\frac{1}n}$ is rational iff $u^{\frac{1}n}$ and $v^{\frac{1}n}$ are both rational

The chosen answer here claims that for rational $u$ and $v$, $u^{\frac{1}n} + v^{\frac{1}n}$ is rational iff $u^{\frac{1}n}$ and $v^{\frac{1}n}$ are both rational. However, the link to a proof seems ...
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Rationality of the Gamma function

I am wondering whether it is correct to say, or whether there is a way to prove that, when $a$ is a positive rational number but not an integer, the complete gamma function of $a$, namely $\Gamma(a)$, ...
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28 views

Rationality of circumference of an ellipsis with rational semi-axes

We all know that the ratio of circumference of a circle to the radius is a transcendental number, but how about ellipses? It is well known that the circumference of an ellipse with semi-axes lengths $...
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119 views

Monty Hall Problem - Extended

So, take the usual Monty Hall setting - 3 doors and one car, 2 goats. I can reveal a door once you've chosen yours. Then extend it to a situation, where,if you SWITCH, the door you switched from is ...
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Is it possible that some combination of partial sums of $\sum \sqrt{k}$ be a rational number? $\text{where } k \text { not a perfect square}$

Is it possible that some combination of partial sums of $\sum \sqrt{k}$ be a rational number? $k=2,3,5,\cdots \text{where } k \text { not a perfect square}$ More ever : can a linear combination of $\...
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61 views

On $p^{\log_q n}$, where $p$ and $q$ are distinct primes

Let $p,q$ be distinct primes, $n>1$ an integer with $\log_q n $ irrational. It was, and probably still is, a conjecture that $p^{\log_q n}$ is non-integer. What progress has been made towards it?
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850 views

The cube of at least one irrational number is rational

The cube of at least one irrational number is rational I am supposed to prove the statement above. Here is what I have so far: Suppose that the cube of at least one irrational number $n$, is ...
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226 views

Showing $\pi$ is irrational using taylor's theorem

To prove the irrationality of $$e = \sum ^\infty _ {n=0} \frac{1}{n!}$$ we can show that $e \lt 3$ by using a suitable geometric series. By Taylor's theorem (applied to $a=0$ and $b=1$) we know that, ...
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71 views

Can a log be a root?

I want to prove that if we let $a,b\in\Bbb N$ such that $log_{a}b\in \Bbb R/\Bbb Q$ (i.e., irrational number) and $$(log_{a}b)^c=D,$$ where $c\in \Bbb Z/\{0\}$, then $D$ must be irrational.
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Proving that if $m,n,p,q\in\mathbb{Z^+}, \sqrt[p]{m}\in\mathbb{R}\setminus\mathbb{Q}$ then $\sqrt[p]{m}+\sqrt[q]{n}\in\mathbb{R}\setminus\mathbb{Q}$

If $\sqrt{m}+\sqrt[q]{n}=r$ rational, the rationality of $\sqrt{m}$ is derived expanding $(r-\sqrt{m})^q$ using the binomial theorem: after rearrangement, isolating the terms containing odd powers of $...
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133 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...