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.
373
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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 ...
- 35
-1
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1
answer
81
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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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0
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297
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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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1
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166
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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 ...
- 69
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2
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166
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Irrationality of $\sqrt 3$ proof verification
I'm new to proofs and I'd like to check if this proof is valid and if there is anything I'm overlooking.
We have to prove that $\sqrt 3$ is an irrational number.
We can suppose the opposite; suppose $\...
- 115
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0
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83
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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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2
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321
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Compact Proof that the constant $\sum_{k=1}^\infty \frac{(-1)^{k-1}}{k.k!}$ is irrational
In an answer some time ago I referenced a short proof that the constant $e$ is irrational by A. R. G. MacDivitt and Yukio Yanagisawa (The Mathematical Gazette , Volume 71 , Issue 457 , October 1987 , ...
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3
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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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Show that equation has no rational solutions
How do I prove that the equation $a^2-2b^2-3c^2+6d^2=0$ has no non trivial rational solutions?
What techniques are there to solve general problems like this?
6
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2
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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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1
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1
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Prove $\sqrt{2}+\sqrt{3}+\sqrt{5}+...+\sqrt{p_{n}}$ is irrational, where $p_{n}$ is the nth prime.
My motivation is making general proof , instead of trying to prove special cases.
To which branch of mathematics does my question belong?
I am highly interested in irrational numbers.
Is it good idea ,...
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1
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1
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101
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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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2
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0
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Directions for a proof of the irrationality of $\pi$ from Archimedes' approximation
I'm a complete amateur with respect to mathematics, but I looked up a few proofs of the irrationality of $\pi$ and was unsatisfied by the lack of proofs that would be elementary enough to be able to ...
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0
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70
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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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