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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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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
60 views

Is there a way to prove numbers irrational in general?

I'm familiar with the typical proof that $\sqrt2\not\in\mathbb{Q}$, where we assume it is equivalent to $\frac ab$ for some integers $a,b$, then prove that both $a$ and $b$ are divisible by $2$, ...
5
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1answer
136 views

A proof that $\sqrt{2}$ is not a rational number.

Is this proof correct? Suppose that $\sqrt{2}=\frac{a}{b}$, where $a,b \in \mathbb{N}$ and $a$ is as small as possible. Then $\sqrt{2}b=a$ which means $2b=\sqrt{2} a$. So we rewrite $\sqrt{2}=\frac{a}...
2
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1answer
48 views

Is it possible to cut the unit square in triangles of the same area such that one of the vertices has irrational coordinates? [duplicate]

This may be an easy question, although it stumped some of us over lunch. Consider the unit square and cut it up in several triangles, such that all the triangles have the same area. Is it possible ...
5
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1answer
145 views

Show that $\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x=0$ [duplicate]

The Dirichlet function is defined as the indicator function of rational numbers. I have also seen this function described by: $$f(x)=\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x$$ How does this ...
0
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2answers
36 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 ...
4
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7answers
162 views

Proof that $\sqrt[3]{17}$ is irrational

Consider $\sqrt[3]{17}$. Like the famous proof that $\sqrt2$ is irrational, I also wish to prove that this number is irrational. Suppose it is rational, then we can write: $$ 17 = \frac{p^3}{q^3}.$$ ...
3
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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 ...
17
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2answers
3k views

Why is $\sum_{i=1}^n a$ always irrational if $n>0$ and $a$ is irrational?

I'm asking this question because I was unable to find an answer elsewhere as most questions are about the summation of different irrational numbers, which is not what this question is about. Here, I'm ...
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3answers
55 views

Is there anything wrong with my interpretation of these exercices about Dedekind left set?

There seems to be serious misunderstanding between me and my textbook ("Classic Set Theory. A guided independent study" by Derek Golderi). I must provide you with definition of Dedekind left set ...
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3answers
365 views

Is this proof correct (Rationality of a number)?

Is $\sqrt[3] {3}+\sqrt[3]{9} $ a rational number? My answer is no, and there is my proof. I would like to know if this is correct: Suppose this is rational. So there are positive integers $m,n$ such ...
0
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0answers
201 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, ...
5
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3answers
375 views

Spivak Calculus 4-th Ed., Chapter 2, Exercise 13a, Understanding the proof of $\sqrt3$ being irrational.

Problem Statement: Prove that $\sqrt3$ is irrational. Hint: To treat $\sqrt3$, for example, use the fact that every integer is of the form $3n$, $3n+1$ or $3n+2$. Solution from the "Answers" Chapter:...
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4answers
54 views

For the expression $\sqrt{\frac{x}{y}}$ to be rational, is it necessary for both to be squares?

QUESTION: Let $x,y$ be positive integers such that the expression $\sqrt{\frac{x}{y}}$ is rational. Is it necessary that $x$ and $y$ have to be perfect squares? So I've tried some numbers and the ...
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6answers
114 views

Can we find smallest positive $x$ such that $\pi^x$ is rational? [closed]

Can we find smallest positive $x$ such that $\pi^x$ is rational? Is this possible if $x$ is rational? My attempt: I don't know how to start!
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0answers
42 views

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 $\Gamma(a)$, ...
1
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0answers
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 $...
3
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1answer
117 views

$\sqrt{\frac{\pi e}{2}}=\frac{1}{1+\mathrm{K}_{i=1}^{\infty}{\frac{i}{1}}}+\sum_{n=0}^{\infty}{\frac{1}{(2n+1)!!}}$ implies $\sqrt{\pi e/2}\notin Q$?

On the OEIS Wiki immediately after the formula $$\sqrt{\frac{\pi e}{2}}=\frac{1}{1+\mathrm{K}_{i=1}^{\infty}{\frac{i}{1}}}+\sum_{n=0}^{\infty}{\frac{1}{(2n+1)!!}}$$ (where I am using $\mathrm{K}$ as ...
5
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2answers
185 views

Is there an essential difference between Cartwright's and Niven's proofs of the irrationality of $\pi$?

In an appendix to the third edition of Scientific Inference, Harold Jeffreys wrote: The following was set as an example in the Mathematics Preliminary Examination at Cambridge in 1945 by Dame Mary ...
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1answer
354 views

What is Euler doing?

In Euler's paper "De Fractionibus Continuis Dissertatio" (English Translation) he proves that the constant $e\approx 2.71828$ is irrational.1 One step in the proof threw me for a loop, though. In the ...
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4answers
27 views

Find an equation where all 'y' is always irrational for all integer values of x

Intuitively it appears to me that if $x$ is an integer, $y$ has to be an irrational number for the following equation. $10y^2-10x-1 = 0$ Can someone prove me right or wrong?
3
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1answer
58 views

Irrationality of $\log_2(2015)$

Claim: $\log_2(2015)$ is irrational Proof: Let's assume it is rational. Then: $$\log_2(2015)=a/b, \quad a,b\in \mathbb Z \setminus \{0\}, b\neq 0$$ Question 1: is it okay how I chose a,b? then we ...
12
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1answer
679 views

Irrationality of $\pi^2$ and $\pi^3$

I wonder if there is any book and/or article you can recommend on the topic "Irrationality of $\pi^2$ and $\pi^3$" for me to study on. In case you are curious about why I ask these particular ...
2
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2answers
116 views

Show that $\sqrt{2} \notin \mathbb{Q}(i)$ using infinite descent

Standard exercise is to show $\sqrt{2} \notin \mathbb{Q}$ (e.g. Wikipedia). There are examples on Math.SE such as [1, 2, 3, 4]. If we adjoin an element to $\mathbb{Q}$ does the same proof by ...
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6answers
179 views

Is $n^\frac{1}{n}$ ever rational?

Sorry if this is a duplicate, as usual I'm struggling with how to search for this. I was wondering to myself how to prove that you can't get a square number that is twice another square number, I.e. $...
4
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6answers
177 views

Show that there are no rationals $r$ such that $r^3 = 6$

Show that there are no rationals $r$ such that $r^3 = 6$ We were asked this for a real analysis assignment. I just assumed that it would be the same as if it were $r^2$, but now I'm second guessing ...
2
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1answer
125 views

Prove that $1+3^{\frac{1}{5}}+3^{\frac{2}{5}}+3^{\frac{3}{5}}+3^{\frac{4}{5}}$ is irrational

I tried to follow this example to resolve this exercise, this example to resolve this exercise, however if: $1+3^{\frac{1}{5}}+3^{\frac{2}{5}}+3^{\frac{3}{5}}+3^{\frac{4}{5}}=x$,then $3^{\frac{1}{5}}...
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1answer
53 views

Prove that an infinite decimal whose digits are natural numbers strung end to end is irrational [duplicate]

The question is from the book The Foundation of Mathematics by Ian Stewart. Let $y = 0·1234567891011121314151617181920 . . . $, whose digits are the natural numbers in decimal form, strung end to ...
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1answer
74 views

Rational value of sine [duplicate]

Given $x$, $\sin(x) \in \mathbb{Q}$, where $x$ is in degrees, we want to find all $x$ in the range $(0,90)$. One trivial solution is $x=30$.
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2answers
157 views

Is $\sum_{n\ge0}\frac1{2^{2^n}}$ rational? [duplicate]

How might I determine whether $\sum_{n\ge0}\frac1{2^{2^n}}$ is rational? I have seen in the answers to this question that the sum converges and it was also mentioned that this is called Kempner's ...
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0answers
106 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 ...
5
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2answers
588 views

Prove the series $\sum_{n=1}^{\infty}\frac{1}{{5^n}^!}$ converges to an irrational number

This is the sum $$\sum_{n=1}^{\infty}\frac{1}{{5^n}^!}$$ My first attempt was to assume that the series does converge to a rational number $a/b$. But the $n!$ bothered me and I failed in my proof. ...
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0answers
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.
2
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1answer
225 views

iEquation in Rudin's 1.1.3 (Principles of Mathematical Analysis)

So I've been working on Rudin's 1.1.3 for a while and I think I'm close to a figuring it out. In the set $A$ there are positive rationals $p$ such that $p^2<2$ and we can find a rational $q$ in $A$...
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3answers
100 views

Is $\sqrt{4+\pi} + \sqrt{4-\pi} \in \mathbb{Q}$? [closed]

Prove or disprove that $\sqrt{4+\pi} + \sqrt{4-\pi} \in \mathbb{Q}$.
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0answers
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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5answers
2k views

Irrationality or Rationality of p+q given that pq=1

Let $p,q$ are irrational numbers such that $pq=1$. Then what can we say about character of $p+q$? i.e, $p+q$ is rational or irrational? Probably, I believe it is irrational but could not prove.
5
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2answers
599 views

Prove that the square root of pq when p and q are distinct primes is irrational

I am trying to prove the following: If p and q are distinct primes, then $\sqrt{pq}$ $\notin$ $\Bbb{Q}$. Here is my proof thus far: Suppose towards a contradiction that if p and q are distinct ...
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1answer
406 views

Is this proof by contradiction that 9+13√3 is irrational correct?

If there is a fault in the logic, please explain it in simple terms. Thank you.
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3answers
202 views

Is the number $\sum\limits_{n=1}^\infty2^{-n^2}$ rational?

Is the number $\sum\limits_{n=1}^\infty2^{-n^2}$ rational? I could prove that the series is convergent (as it is bounded above by the geometric series with common ratio $\frac{1}{2}$. But how do I ...
2
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1answer
73 views

Rational values for $\sin\left(\frac{2\pi }{n}\right)$

I want to find for what $n\in \mathbb{N}$ a $n$-sided polygon has rational area, assuming the polygons' "long" radius is $1$. This reduces to whether or not $\sin\left(\frac{2\pi }{n}\right)$ is ...
4
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2answers
2k views

If $a$ is rational and $b$ is irrational, is $ab$ necessarily irrational?

There are some similar questions, but not exactly this one. First: As, Zero is a rational number. So, by using the counterexample $a=0$ and $b=x$ ($x$ being an irrational number) we get $ab = ...
1
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1answer
226 views

Square root of a prime is irrational [duplicate]

How can one prove that the square root of a prime number (e.g. $ \sqrt 61$) is irrational. First we need to prove that $61$ is prime. This can be done by simply showing that $$ 49 < 61 < 64$$ ...
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1answer
1k views

Properties of Irrational and Rational Numbers [closed]

What is the properties that helps us when we are proving a number whether rational or irrational? It is better if you can give several properties. If you can give links, that will helps too. ...
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2answers
202 views

Error in proving $e + \pi$ is irrational.

From a proofs class: Note that $2^{1/n}$ is irrational for all $n>1$, since 2 is prime. Therefore, if $m$ is a positive, rational number, then $m*2^{1/n}$ is irrational for all $n>1$. Let $m = ...
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1answer
222 views

Prove $\sqrt[m]{D}$ irrationality simply, without unique prime factorization

I apologize if this has been asked already. I am aware of proofs that $\sqrt[m]{D}$ is either integer or irrational for $m,D\in\mathbb{N}$, all of which that I recall and understand make use of the ...
5
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1answer
185 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 ...
6
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3answers
208 views

Please critique my proof that $\sqrt{12}$ is irrational

I would like critiques on correctness, conciseness, and clarity. Thanks! Proposition: There is no rational number whose square is 12 Proof: Suppose there were such a number, $a = \in \mathbb{Q}$ s.t....
3
votes
2answers
411 views

Proof that $2^{1/3} - 1/( 2 ^ {1/3} $) is irrational

I know that in most cases you assign the expression to a number alpha and then do some algebra in order to reach a polynomial with rational coeficients that alpha is a root of, proceding to prove that ...
0
votes
0answers
68 views

Is $\sum_{p} \frac{1}{2^p}$ irrational? [duplicate]

$$ \sum_{p \, \in \text{ primes}} \frac{1}{2^p} $$ My friend came across a book which claimed this to be irrational and claimed it could be solved using the pigeonhole principle.