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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Is the minimizer of this function always rational?

Let $M$ be a skew-symmetric $n \times n$ matrix of integers. Let $S$ be the (convex) set of all $p \in \mathbb R^n$ such that: $p_{i} \ge 0$ for all $i$ $\sum_{i} p_{i} = 1$ $(Mp)_{i} \ge 0$ for all $...
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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 ...
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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 $...
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1 answer
43 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 ...
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0 votes
1 answer
165 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 ...
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4 votes
1 answer
98 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 $2^r = 3$...
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3 votes
5 answers
279 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 ...
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2 votes
1 answer
232 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 ...
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1 vote
2 answers
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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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1 vote
1 answer
26 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 ...
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14 votes
5 answers
586 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^...
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2 votes
1 answer
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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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12 votes
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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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1 answer
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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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2 votes
0 answers
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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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2 votes
2 answers
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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 $$...
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4 votes
3 answers
337 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 ...
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-3 votes
3 answers
178 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 ...
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2 votes
4 answers
71 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 ...
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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}-\...
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1 vote
1 answer
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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
1 answer
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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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2 answers
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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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5 votes
1 answer
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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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1 vote
2 answers
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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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0 votes
2 answers
92 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 ...
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-1 votes
2 answers
47 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 + ...
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0 answers
84 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 ...
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0 votes
1 answer
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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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2 votes
2 answers
146 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}}}+\...
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3 answers
144 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 ...
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1 vote
0 answers
112 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)(...
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5 votes
2 answers
449 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 ...
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2 votes
0 answers
142 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 ...
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0 votes
2 answers
111 views

How To Show The Irrationality Of $0.0110001111...$

Show that $F=0.0110001111...$ is irrational, with $F$‘s decimal expansion being $1,3,5,7,9,...$ consecutive $0$s interspersed with $2,4,6,8,...$ consecutive $1$s. A friend sent me this recently. The ...
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1 vote
1 answer
151 views

Determining the parity of the coefficients of a quadratic given that there is a rational solution

Que. If the quadratic $ax^2 + bx + c$ has a rational root, and $a$, $b,$ and $c$ are integers, then A) at least one of $a, b, c$ is even B) all of $a,b,c$ are even C) at most one of $a,b,c$ is odd D)...
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0 votes
0 answers
34 views

Prove rationality of a number by proof by contradiction

So I was going through proof of irrationality of $\sqrt2$ and tried to construct a proof whether $\sqrt4$ is rational by proof of contradiction. Proof: (By Contradiction) For the purposes of ...
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0 answers
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Showing some decimal representations are not rational

I have solutions for the following two problems and I am hoping to obtain some feedback on my solutions. Does the number $0.123456789101112l314\ldots$ which is obtained by writing successively all ...
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0 votes
2 answers
98 views

Rational Zero Test of Polynomials

Say I'm trying to find the factors of $r^3 - 3r^2 + 3r - 4 = 0$ I thought I could do the rational zero test by finding the factors of p and q and then doing $\frac{p}{q}$ So, in this case, $p = \pm ...
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0 votes
0 answers
42 views

Not proving the irrationality of $e^e$

How would one go about proving that there does not exist such a product that satisfies $$ e^k=\prod_{n=1}^{k}\frac{a_n}{b_n} $$ for $k$, $a_n$, $b_n\in\mathbb{N}$ and $\lim_{n\to\infty}\frac{a_n}{b_n}=...
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10 votes
2 answers
442 views

Proof that $\log_23 +\log_52$ is irrational number

Problem is to prove that $$\log_23 +\log_52$$ is irrational number. My attempt: I try to write number like $$\log_23 +\frac{1}{\log_25}$$ but I didn't get anything(proof by contradiction). I also try ...
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1 vote
0 answers
49 views

Find all rational angles in degrees whose cosine is also rational [duplicate]

I know it is called Niven's Theorem, but I don't know how to prove it. I am looking for a clue or a direction for where to begin. Here is what I do know: $\cos{\theta^ {\circ}}$ can be expanded to ...
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2 votes
3 answers
102 views

Prove $\sqrt{2}$ is irrational using prime number properties

I found the following argument in a textbook that uses a number theory approach. However I do not understand the last two sentences that seems to use some number theory properties, can someone please ...
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1 vote
3 answers
117 views

Show $\frac{\sqrt{\sqrt[4]8-\sqrt{\sqrt2+1}\;}}{\sqrt{\sqrt[4]8+\sqrt{\sqrt2-1}\;} -\sqrt{\sqrt[4]8-\sqrt{\sqrt2-1}\;}}=\frac1{\sqrt2}$

Days ago, I tried to demonstrate this equality, reducing radicals, multiplying by the conjugate of the denominator, etc. But, I did not reach anything similar to the right side. $$ \frac{\sqrt{\sqrt[...
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2 votes
1 answer
143 views

Rationality and transcendence of zeros of Bessel functions

Is the first zero 2.40482555769577... of the Bessel function $J_0(x)$ transcendental? (Is anything known about its status?)
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0 votes
1 answer
27 views

Equivalent conditions for a logarithm to be rational

My question is the following: Question: Is it true that for any two integers $a,b \ge 2$, the following five conditions are equivalent? There exists an integer $c \ge 2$ that is a common power of $a$...
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3 votes
0 answers
145 views

Is $\zeta^2(3)$ irrational?

It is known that Apéry's constant defined as $$\zeta (3)=\sum _{n=1}^{\infty }\frac {1}{n^3}= \frac{5}{2}\sum _{{n=1}}^{\infty }{\frac {(-1)^{{n-1}}}{n^{3}{\binom{2n}{n}}}}=1.2020569\cdots$$ was ...
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  • 5,009
5 votes
0 answers
56 views

Area inside $|x|^G+|y|^G=r^G$ is an integer multiple of $r^2$

The problem is to find the numbers $G_i$ such that the area inside the curve $|x|^{G_i}+|y|^{G_i}=r^{G_i}$ is an integer multiple of $r^2$. Because the curve defined by this equation is always ...
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  • 199
2 votes
1 answer
111 views

Prove that the ratio of acute angles in a $3:4:5$ triangle is irrational

Inspired by a comment by @QC_QAOA on Question 3458920, which mentioned the ratio between the acute angles in a $3:4:5$ triangle, I would like to know if we can prove that this ratio is irrational. ...
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-4 votes
1 answer
453 views

Show that there is no rational number whose square is $2$ or $8$ [duplicate]

Show that there is no rational number whose square is $2$ or $8$
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