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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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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 ...
Math Admiral's user avatar
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83 views

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 ...
fedsavi's user avatar
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-1 votes
1 answer
81 views

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 ...
Hayden's user avatar
  • 97
9 votes
0 answers
297 views

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 ...
marty cohen's user avatar
6 votes
1 answer
166 views

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 ...
Mary_Smith's user avatar
0 votes
2 answers
166 views

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 $\...
Mračna Seka's user avatar
4 votes
0 answers
83 views

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 ...
Apuji's user avatar
  • 41
3 votes
2 answers
321 views

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 , ...
James Arathoon's user avatar
5 votes
3 answers
189 views

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 ...
James Jin's user avatar
0 votes
0 answers
44 views

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?
Inspector gadget's user avatar
6 votes
2 answers
307 views

$ֿ\sqrt{2}$ 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 ...
FeedMePi's user avatar
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1 vote
1 answer
93 views

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 ,...
Math Admiral's user avatar
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1 vote
1 answer
101 views

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 ...
rogemuggle's user avatar
2 votes
0 answers
77 views

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 ...
noname_lonestar's user avatar
1 vote
0 answers
70 views

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 ...
Shien's user avatar
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3 votes
1 answer
55 views

Regarding the rationality of $T_k = x^k + \frac{1}{x^k}$, given $T_m$ and $T_n$ are rational for some $m,n \in \mathbb{N}$.

We define $T_k = x^k + \frac{1}{x^k}$,for some non zero real number $x$. Then one can look at the algebra generated by $T_k$. Its simple to see that they satisfy $$T_k T_l = T_{k+l} + T_{|k-l|} $$ ...
Sagars's user avatar
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5 votes
3 answers
289 views

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 ...
Faraz Masroor's user avatar
1 vote
0 answers
233 views

Prove that when the denominator of a rational number is of the form $2^n * 5^m$ it is a terminating decimal

What is the proof for when the denominator of a rational number is of the form $2^n * 5^m$ it is a terminating decimal? For example: $7/8$, where $8$ is of the form $2^3 * 5^0$ Therefore, $7/8$ is a ...
Gaurav Chandrashekar's user avatar
1 vote
1 answer
59 views

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 $...
user avatar
0 votes
1 answer
51 views

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 ...
Shankar1719's user avatar
0 votes
1 answer
184 views

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 ...
AmoebaProteus's user avatar
3 votes
1 answer
263 views

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 $2^r = 3$...
Elimination_matrices's user avatar
2 votes
5 answers
335 views

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 ...
Curious Scientist's user avatar
2 votes
1 answer
252 views

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 ...
user avatar
1 vote
2 answers
86 views

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 $...
Cyan Turnip's user avatar
1 vote
1 answer
93 views

Rationalizing an expression with different nth-roots in the denominator

I was trying to help a friend of mine rationalize the following expression: $$\frac{b}{\sqrt[3]{a^4}-\sqrt[3]{c^2}} $$ Simply multiplying by the conjugate won't work, even with multiple iterations. Is ...
Someboby's user avatar
14 votes
5 answers
622 views

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^...
tompi2394's user avatar
  • 727
2 votes
1 answer
75 views

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 ...
Graviton's user avatar
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12 votes
0 answers
1k views

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 ...
James Propp's user avatar
0 votes
1 answer
38 views

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 $\gcd(b_k,c_k)=1$. ...
Graviton's user avatar
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2 votes
0 answers
197 views

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 ...
Red Banana's user avatar
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2 votes
2 answers
132 views

Is $\frac{\text{area of circle with diameter $d$}}{\text{area of square with side $d$}}$ irrational?

I understand that the area of a circle (based on diameter) with respect to the area of a square (same side length as circle diameter) is approximately 0.7854 the area of the square. My question: Is $$...
Mark Lee's user avatar
4 votes
3 answers
833 views

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 ...
user904299's user avatar
-3 votes
3 answers
1k views

Prove that $ 4 + 3 \sqrt 2 $ is irrational

I am trying to prove by contradiction that $ 4 + 3 \sqrt 2 $ is irrational. (I may use the fact that $ \sqrt 2 $ is irrational, and must use the definition of rational numbers.) Proofs by ...
Slowly_Learning's user avatar
1 vote
4 answers
98 views

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 ...
MathsNoob's user avatar
  • 117
0 votes
0 answers
53 views

Can I use Euler's formula to prove a series converges to a rational value?

Given the following infinite series with rational $x < 1$ && integer $n \geqslant 2$: $$1-\frac{x}{n}+\frac{(1+n)x^2}{2n^2}-\frac{(1+n)(1+2n)x^3}{6n^3}+\frac{(1+n)(1+2n)(1+3n)x^4}{24n^4}-\...
kayle's user avatar
  • 109
1 vote
1 answer
286 views

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 ...
Some Guy's user avatar
  • 2,685
1 vote
1 answer
151 views

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 $\log _2\left(5\right)=\frac{p}{q}$ ...
Cheeze's user avatar
  • 69
0 votes
2 answers
75 views

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 ...
Intradiction's user avatar
6 votes
1 answer
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 ...
tparker's user avatar
  • 6,280
1 vote
2 answers
200 views

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 ...
Kevin Duran's user avatar
1 vote
2 answers
474 views

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 ...
iceiscold's user avatar
-1 votes
2 answers
51 views

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 + ...
John Liu's user avatar
  • 419
0 votes
0 answers
122 views

About union of two algebraically independent sets

My question might be very simple for those that have a deep understanding for algebraically independent sets. Definition 1.1. Let $F$ be an extension field of $K$ and $S$ a subset of $F$. $S$ is ...
00GB's user avatar
  • 2,423
0 votes
1 answer
39 views

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\...
charmin's user avatar
  • 180
2 votes
2 answers
165 views

If $(a-b^2)b>0$, then $\sqrt[3]{a+\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}+\sqrt[3]{a-\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}$ is rational

From Hardy´s "A course of pure mathematics" 10th edition, problem 31 miscellaneous problems of chapter I. If $(a-b^2)b>0$, then $$ \sqrt[3]{a+\frac{9b^3+a}{3b}\sqrt{\frac{a-b^3}{3b}}}+\...
SAUL ROJAS's user avatar
0 votes
3 answers
467 views

When is $\log_{b}(a)$ irrational, for natural numbers $a,b$?

I've seen this result usually given as "if $a$ and $b$ are coprime, then $\log_{b}(a)$ is irrational." Wolfram|Alpha even says it is "unknown" whether or not $\log_{12}(6)$ is ...
Jonathan Hebert's user avatar
1 vote
0 answers
334 views

Miscellaneous Problem Chapter I ex.22 on G.H.Hardy's book "A course of pure mathematics"

If all the values of x and y given by $$ ax^{2}+2hxy+by^{2}=1, a'x^{2}+2h'xy+b'y^2=1 $$ (where $a, h, b, a', h', b'$ are rational) are rational, then $$ (h-h')^2 - (a-a')(b-b'), (ab'-a'b)^2+4(ah'-a'h)(...
SAUL ROJAS's user avatar
5 votes
2 answers
808 views

Spivak's Calculus: chapter 2, problem 18(c)

In Spivak's calculus book, I cannot understand the solution proposed for question (c) of problem 18 in chapter 2: Prove that $\sqrt{2}+\sqrt[3]{2}$ is irrational. Hint: Start by working out the first ...
Alexandros's user avatar
2 votes
0 answers
264 views

Why is the irrationality of $\gamma$ so ellusive? (Euler–Mascheroni constant)

Perhaps I'm in way over my head asking this; regardless, considering the abundance of known formulas and properties of $\gamma$, seemingly comparable in quantity to the likes of $\pi$ or $e$, why is ...
Graviton's user avatar
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