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.
379 questions
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Is there a converse of the Copeland-Erdős theorem on normal numbers?
Copeland-Erdős theorem states that if $a_{1}, a_{2}, a_{3}, \ldots$ is an increasing sequence of integers such that for every $\theta < 1$ the number of $a$'s up to $N$ exceeds $N^{\theta}$ ...
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$n^n$ being rational for irrational $n$ within $x_{n+1} = x_n^{-x_n}$
Previously, in this question, I tried to prove the rationality of $n^n$ when $n$ is irrational, and I proved it using a recurrence relation rather than the answer I accepted, which was proving that $n$...
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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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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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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 an infinite series converges to a rational number, then does the series of terms divided by factorial always converge to an irrational number?
I was playing around with infinite series, specifically those that converged to rational numbers, and noticed (at least for the ones that I tried) that when I divided them by a factorial, they would ...
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If $0<x<1$ and $x$ is rational, does $2^x$ have to be irrational? [closed]
If $0<x<1$ and $x$ is rational, does $2^x$ have to be irrational? Why? Also, if $2^x$ is rational, and $0<x<1$ and $x$ is rational, does $x$ have to be irrational? (i.e. contrapositive of ...
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$ֿ\sqrt{2}$ is irrational. Proof by contradiction or Proof of Negation?
I am just learning about proofs in an introductory course. I came across an example of "proof by contradiction" (see attachment) about $ֿ\sqrt{2}$ being irrational. Some online sources have ...
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Prove that number $\sqrt{2}$ is an irrational number using this theorem "if $a^2$ is even, $a$ must be even"
I have to prove the following:
Prove $\sqrt{2}$ is an irrational number using this theorem "if $a^2$ is even, $a$ must be even"
I made a proof by contradiction for the statement above, but ...
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Does the maximum entropy Nash equilibrium with integer payoffs have rational probabilities?
I have a symmetric two-player zero-sum game, represented as an $n \times n$ skew-symmetric payoff matrix $M$. The components of $M$ are all integers. Are the probabilities in the maximum entropy Nash ...
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What are some techniques to prove that a set of numbers is rationally independent?
I am solving a math problem for fun and it amounts to proving that a specified (finite) set of numbers, each defined by an infinite series involving polynomials and the factorials, is a rationally ...
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If $\log_3(A)$ is rational, can $\log_3(A+1)$ be rational? (Where $A$ is a positive real number.)
Question:
If $\log_3(A)$ is rational, can $\log_3(A+1)$ be rational? (Where $A$ is a positive real number.)
My Working:
Starting by assuming
$$\log_3(A) = x \ {\rm{and}} \ \log_3(A+1) = y$$
it gives $...
user983206
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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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5
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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 ...
1
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2
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Is it true or false for all instances of $m>0$, $\sqrt{m^2+1}$ is irrational?
Is it true or false for all instances of $m>0$, $\sqrt{m^2+1}$ is irrational?
My textbook gives an answer
This is true for integers $m$:
$$m>0\implies m^2<m^2+1<m^2+2m+1=(m+1)^2$$
Since $...
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5
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Let $x^3+\frac{1}{x^3}$ and $x^4+\frac{1}{x^4}$ are rational numbers. Show that $x+\frac{1}{x}$ is rational.
$x^3+\dfrac{1}{x^3}$ and $x^4+\dfrac{1}{x^4}$ are rational numbers. Prove that $x+\dfrac{1}{x}$ is rational number.
My solution:
$x^3+\dfrac{1}{x^3}=\left(x+\dfrac{1}{x}\right)\left(x^2+\dfrac{1}{x^...
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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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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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Proving that $\sqrt[3] 7 -\sqrt 2$ is irrational.
I understand proving that $\sqrt{7}-\sqrt {2}$ is irrational, but how does the answer change if its cube root of $7$ instead of square root?
the way I solve $\sqrt{7}-\sqrt {2}$ is by assuming its ...
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If $x + y \in \mathbb Q$ and $x − 3 y \in \mathbb Q$, prove $x$ and $y$ are rational
Consider $𝑥, 𝑦 ∈ ℝ$, such that $𝑥 + 𝑦 ∈ ℚ$ and $𝑥 − 3𝑦 ∈ ℚ$. Then $𝑥, 𝑦 ∈ ℚ$.
Hey all. I'm recently working through a course-book that's involved with the math course I'm taking next semester ...
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1
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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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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 ...
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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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Two specific irrational numbers summing to rational number that is not zero?
There already exist pages that show how two irrational numbers can sum to a rational number. However, is there an actual example of this? What are some of irrational numbers $x$ and $y$ such that $x + ...
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How to make sure that solutions of an equation are rational multiple of $\pi$ or not?
I have this two-variable equation
$$\frac{\left(3 y^2+1\right)^2 \left(2 \cosh \frac{26 \pi y}{15} \cosh x y-\cosh (2 \pi -x) y\right)}{9 \left(y^2-1\right)^2 \cosh (x+2 \pi ) y+8 \left(3 y^2-1\...
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