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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9 votes
2 answers
472 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 ...
4 votes
7 answers
775 views

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

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: $$ 17 = \frac{p^3}{q^3}.$$ ...
1 vote
0 answers
56 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 ...
1 vote
3 answers
125 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[...
0 votes
1 answer
35 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 $a$...
2 votes
1 answer
48 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 $...
3 votes
0 answers
190 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 ...
3 votes
1 answer
186 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. ...
-4 votes
1 answer
635 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 $8$
21 votes
1 answer
36k views

The square root of a prime number is irrational [duplicate]

If $p$ is a prime number, then $\sqrt{p}$ is irrational. I know that this question has been asked but I just want to make sure that my method is clear. My method is as follows: Let us assume that ...
8 votes
6 answers
24k 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} ...
43 votes
9 answers
123k views

Prove that $\sqrt 2 + \sqrt 3$ is irrational [duplicate]

I have proved in earlier exercises of this book that $\sqrt 2$ and $\sqrt 3$ are irrational. Then, the sum of two irrational numbers is an irrational number. Thus, $\sqrt 2 + \sqrt 3$ is irrational. ...
28 votes
4 answers
16k views

When is $\sin(x)$ rational?

Obviously, there are some points (e.g. $\pi$, $30^\circ$) but I am unsure if there are more. How can it be proved that there are no more points, or, if there are, what those points will be? EDIT: I ...
0 votes
0 answers
26 views

Can there exist two different prime numbers, such that their roots added together equal a rational number? [duplicate]

Can there exist two (or more) different prime numbers, such that their roots added together equal a rational number? This came up while trying to prove that two square roots, added together, can only ...
1 vote
1 answer
97 views

Can the b-ary power series of an irrational number evaluate to a rational number?

Assume we have a base $b$ and a power series $f(x)$ with the coefficient of each power $x^n$ being equal to the nth non-zero $b$-ary digit times $b^{-i}$ (where $i$ is the index of the b-ary digit) in ...
3 votes
3 answers
306 views

For distinct odd primes $p, q$ we have $\sqrt{p^2 - 4q}$ is always irrational?

I am trying to show that $\sqrt{p^2 - 4q}$ is always irrational with $p, q$ as described above. Any hints?
1 vote
2 answers
33 views

Is a definite integral of $f(x)$ rational if $f(x)$ is rational at the endpoints and f(0)=0?

It seems that in many situations, the fact that $f(x)$ is rational at $x=a,b$ and $f(0)=0$ and $f$ is integrable is enough to say that $$\int_a^bf(x)dx$$ is rational. Is this true? If not, can you ...
101 votes
4 answers
4k views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
3 votes
2 answers
212 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 ...
1 vote
0 answers
40 views

If $\sum_{i=0}^\infty p_1^{i}p_2^{-a_i}$ converges and is rational does that imply that $\sum_{i=0}^\infty p_2^{-a_i}$ is also rational

For primes $p_1$ and $p_2$ and a monotonically increasing sequence of natural numbers $a_i$, if $\sum_{i=0}^\infty p_1^{i}p_2^{-a_i}$ converges and is rational does that imply that $\sum_{i=0}^\infty ...
0 votes
1 answer
778 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 $2$, ...
19 votes
5 answers
4k views

Prove that $5^{1/3}+7^{1/2}$ is irrational

Goal: Prove that $5^{1/3}+7^{1/2}$ is irrational. Idea: We can prove this is irrational by supposing it is rational and finding a contradiction. So, $5^{1/3}+7^{1/2} = p/q$ where $p$ and $q$ are ...
6 votes
4 answers
668 views

Are there any natural proofs of irrationality using the decimal characterization?

Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates ...
5 votes
2 answers
16k views

Value of cos 1 (angle is in radians)?

How can we calculate the value of $\cos 1$ where the angle is in radians (and not degrees). If this isn't possible, can we somehow find whether this value would be rational or irrational? P.S: I know ...
1 vote
2 answers
142 views

What is the reasoning behind $y=\frac{x(x^2+3D)}{ 3x^{2}+D }$ in Dedekind's "Essays On The Theory Of Numbers"?

In the section IV "Creation Of Irrational Numbers"of Richard Dedekind's "Essays On The Theory Of Numbers" the author shows that domain of rational numbers is incomplete, but can be made so. To show ...
7 votes
4 answers
3k views

Can transcendental to the power transcendental be rational?

Can a transcendental number to the power of a transcendental number be a rational number?
2 votes
6 answers
4k 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, ...
-1 votes
1 answer
90 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 ...
7 votes
4 answers
927 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
1 answer
2k 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
4 answers
727 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}$. ...
0 votes
0 answers
155 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 ...
1 vote
1 answer
100 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
1 answer
270 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 ...
16 votes
1 answer
2k 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
2 answers
73 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 ...
3 votes
1 answer
82 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 ...
18 votes
2 answers
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
3 answers
393 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
3 answers
101 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
0 answers
359 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 ...
0 votes
0 answers
515 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
2 answers
312 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 = ...
55 votes
7 answers
13k 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
3 answers
197 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 ...
6 votes
3 answers
713 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
4 answers
91 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 ...
7 votes
3 answers
3k 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
4 answers
9k 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
6 answers
231 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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