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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.

2
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6answers
893 views

Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational

Prove that if $n$ is a positive integer then $\sqrt{n}+ \sqrt{2}$ is irrational. The sum of a rational and irrational number is always irrational, that much I know - thus, if $n$ is a perfect square, ...
8
votes
13answers
6k views

Are there any irrational numbers that have a difference of a rational number?

Are there any irrational numbers that have a difference of a rational number? For example, if you take $\pi - e$, it looks like it will be irrational ($0.423310\ldots$) - however, are there any ...
-1
votes
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 ...
139
votes
12answers
32k views

How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?

It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing 'even' with 'divisible by $3$'), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly ...
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 $...
6
votes
4answers
657 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
80 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
4answers
230 views

Proof that $2\sqrt2 + \sqrt7$ is an irrational number

Prove that $2\sqrt2 + \sqrt7$ is an irrational number. I am trying to use contradiction to show that this is irrational. Also I am using the fact that $ 2\sqrt2 + \sqrt7 = \frac{1}{2\sqrt2 - \sqrt7}$. ...
29
votes
5answers
3k views

Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}...
0
votes
0answers
133 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
0
votes
1answer
67 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}...
49
votes
13answers
12k views

How can you prove that the square root of two is irrational?

I have read a few proofs that $\sqrt{2}$ is irrational. I have never, however, been able to really grasp what they were talking about. Is there a simplified proof that $\sqrt{2}$ is irrational?
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 ...
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 ...
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 ...
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
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 ...
11
votes
0answers
317 views

What is known about the sum $\sum\frac1{p^p}$ of reciprocals of primes raised to themselves? [duplicate]

Consider the following series: $$\sum_{p\in\mathcal{P}}\frac{1}{p^p}$$ where $\mathcal{P}$ is the set of all prime numbers: $\mathcal{P}=\{2,3,5,7,11,13,\ldots\}$. My question is: Is this a known ...
21
votes
10answers
8k views

Critiques on proof showing $\sqrt{12}$ is irrational.

My only exposure to proofs was in a math logic class I took in University. I was wondering if my attempt at proving that $\sqrt{12}$ is irrational is OK. $$\Big(\frac{m}{n}\Big)^2 = 12$$ $$\Big(\frac{...
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, ...
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 = ...
13
votes
7answers
1k views

Proof that $\sqrt6 - \sqrt2 - \sqrt3$ is irrational. [duplicate]

I want to prove that: $$\sqrt6 - \sqrt2 - \sqrt3$$ is irrational. I have tried using squares, the $p/q$ definition of rationality and the facts that 1)rational$\times$ irrational=irrational (unless ...
48
votes
7answers
8k views

$\sin 1^\circ$ is irrational but how do I prove it in a slick way? And $\tan(1^\circ)$ is …

In the book 101 problems in Trigonometry, Prof. Titu Andreescu and Prof. Feng asks for the proof the fact that $\cos 1^\circ$ is irrational and he proves it. The proof proceeds by contradiction and ...
3
votes
3answers
132 views

Prove that $\sqrt[n]{\frac{b^{n}}{a^{n}}+1}$ is irrational for every $b,a,n\in\mathbb{ N}$ and $n>2$.

Prove that $\sqrt[n]{\frac{b^{n}}{a^{n}}+1}$ is irrational for every $b,a,n\in\mathbb{ N}$ and $n>2$. I tried a long time using proof by contradiction and induction but it didn't work, it seem so ...
5
votes
3answers
399 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 ...
84
votes
18answers
6k views

What is the most unusual proof you know that $\sqrt{2}$ is irrational?

What is the most unusual proof you know that $\sqrt{2}$ is irrational? Here is my favorite: Theorem: $\sqrt{2}$ is irrational. Proof: $3^2-2\cdot 2^2 = 1$. (That's it) That is a ...
4
votes
4answers
1k views

Can transcendental to the power transcendental be rational?

Can a transcendental number to the power of a transcendental number be a rational number?
4
votes
3answers
797 views

The sum of square roots of non-perfect squares is never integer [duplicate]

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
-1
votes
4answers
6k views

Is the sum of two rationals or two irrationals irrational?

1. I know this statement is false (if I am correct) but how to prove it's false? "The sum of two rational numbers is irrational." 2. I know this statement is true (if I am correct) but how to ...
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
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 $...
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 $\Gamma(a)$, ...
11
votes
5answers
2k views

Prove that 2.101001000100001… is an irrational number.

My try: This number is non-terminating and non-repeating, so this is an irrational number. But how do I prove it more formally in a more mathematically rigorous way?
7
votes
5answers
460 views

Shorter proof of irrationality of $\sqrt{2}$?

Euclid's proof of the irrationality of $\sqrt{2}$ via contradiction involves arguments about evenness or odness of $a^2 = 2 b^2$ which is then lead to contradiction in using few more steps. I wonder ...
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.
15
votes
5answers
34k views

Prove that $\sqrt 5$ is irrational

I have to prove that $\sqrt 5$ is irrational. Proceeding as in the proof of $\sqrt 2$, let us assume that $\sqrt 5$ is rational. This means for some distinct integers $p$ and $q$ having no common ...
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 ...
8
votes
5answers
19k views

How to show that the product of two irrational numbers may be irrational?

Show that the product of two irrational numbers may be irrational. You may use any facts you know about the real numbers. All we know is that $\sqrt{2}$ is irrational and that $\sqrt{2}\cdot \sqrt{2} ...
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 ...
21
votes
4answers
3k views

Homework 8th grader: $\pi^2$ is irrational

I'm tutoring a girl in 8th grade (so she is 14 years old) and she recently had a mathematics chapter about numbers. In the last paragraph they introduced the difference between rational and irrational ...