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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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1answer
49 views

Prove that $\sqrt[3]p+\sqrt[3]{p^5}$ is irrational if $p$ is prime [closed]

Prove that $\sqrt[3]p+\sqrt[3]{p^5}$ is irrational when $p$ is a prime. First I suppose $x=\sqrt[3]p+\sqrt[3]{p^5}$. Cubing gives $$x^3=p+p^5+p^2x$$ And then what properties of prime, and how to test ...
6
votes
4answers
658 views

Rational with finite decimals values for sine, cosine, and tangent

What are the possible combinations of sine, cosine, and tangent values such that all three are simultaneously rational with finite decimals? I am aware of the below two cases. $\sin(x) = 0, \cos(x) =...
2
votes
1answer
83 views

Is it rational or irrational?

I am a mathematical putz - please be kind. From what I know, a rational number is a non-imaginary number that can be written as $\frac{p}{q}$. A repeating number can also be a rational number (i.e. $...
1
vote
1answer
30 views

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 $...
0
votes
1answer
68 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
votes
1answer
147 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
votes
1answer
55 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
votes
1answer
147 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
votes
2answers
39 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
votes
7answers
181 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
votes
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
votes
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 ...
0
votes
3answers
56 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 ...
6
votes
3answers
369 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
votes
0answers
225 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
votes
3answers
400 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:...
-1
votes
4answers
57 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 ...
0
votes
6answers
145 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!
1
vote
0answers
51 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
vote
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 $...
4
votes
1answer
126 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
votes
2answers
191 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 ...
9
votes
1answer
355 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 ...
1
vote
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
votes
1answer
59 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 ...
11
votes
1answer
712 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
votes
2answers
130 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 ...
7
votes
6answers
182 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
votes
6answers
179 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
votes
1answer
139 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}}...
0
votes
1answer
65 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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votes
1answer
93 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 $x=30$.
4
votes
2answers
160 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 ...
1
vote
0answers
119 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
votes
2answers
648 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. ...
5
votes
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
votes
1answer
250 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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votes
3answers
101 views

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

Prove or disprove that $\sqrt{4+\pi} + \sqrt{4-\pi} \in \mathbb{Q}$.
0
votes
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.
10
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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
votes
2answers
703 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 ...
0
votes
1answer
499 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.
3
votes
3answers
204 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
votes
1answer
74 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
votes
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
vote
1answer
292 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 $$ 49 < 61 < 64$$ ...
0
votes
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. ...
1
vote
2answers
220 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 = ...
-2
votes
1answer
250 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
votes
1answer
189 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 ...