# 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.

293 questions
46 views

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

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

30 views

51 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 ...
146 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 ...
37 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 ...
175 views

### Proof that $\sqrt{17}$ is irrational

Consider $\sqrt{17}$. 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: $$17 = \frac{p^3}{q^3}.$$ ...
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 ...
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 ...
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 ...
366 views

### Is this proof correct (Rationality of a number)?

Is $\sqrt {3}+\sqrt{9}$ 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 ...
223 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, ...
390 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:...
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 ...
140 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!
49 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 $\Gamma(a)$, ...
28 views

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 ...
133 views

273 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 $$49 < 61 < 64$$ ...