# Tagged Questions

Questions about numbers expressible as the quotient of two integers. For questions on determining whether a number is rational, use the (rationality-testing) tag instead.

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### Countable dense subset claim in Arzela Ascoli proof

In every Arzela Ascoli proof you see the following: Let $S = \mathbb{Q} \cap [a,b]$, where $[a,b]$ is an interval in $\mathbb{R}$, then $S$ is a countable dense subset and there exists a ...
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### If $x$ is rational and $xy$ is irrational, then $y$ is irrational. [closed]

This is a statement that I need to prove. Let $x$ and $y$ be real numbers. If $x$ is rational and $x\times y$ is irrational, then $y$ is irrational. I believe you have to prove this using ...
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### Proving that rational equivalence is an equivalence relation on any set.

I seek to prove that the rational equivalence relation is an equivalence relation, in that it is reflexive, symmetric, and transitive. The rational equivalence relation is as follows "Two numbers in ...
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### Prove that $\frac{p}{q}$ is a rational number with a finite decimal expression if $p$ is an integer and $q=(2^n)(5^m)$

Let $p,q$ be two integers and $q=(2^n)(5^m)$. Then $\frac pq$ is a rational number with a finite decimal expression. Any ideas how to do this? I've been thinking about it all day but I have no idea ...
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### Basel problem over $\mathbb{Q}_{\geq 1}$

Let $\mathbb{Q}_{\geq 1}=\{r\in\mathbb{Q}\,|\,r\geq 1\}$. Once $\mathbb{Q}$ is enumerable, $\mathbb{Q}_{\geq 1}$ is also enumerable. Let $\{r_1,r_2,\ldots\}$ be such an enumeration. What can we say ...
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### What is a good way to show that $[0,1]$ is not complete in $\mathbb{Q}$

To show a set is not complete, the best way is always produce a Cauchy sequence that does not converge in the set. I wish to show $[0,1]$ is not complete in $\mathbb{Q}$ I am a little stucking ...
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### How many rational values of x are not integers and satisfy the following equation?

How many rational values of x are not integers and satisfy the following equation: $$x^7 - 6x^6 + 5x^5 - 4x^4 + 3x^3 - 2x^2 + 1 = 0 ?$$ Well, I got this question from one of the Mathcounts ...
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### Why is the set of Rational numbers countably infinite? [duplicate]

Why is the set of Rational numbers ,$\mathbb Q$, a countably finite set? I think that - if we assign $n$ to a rational number, and $n+1$ to another rational number, Then I can surely find a rational ...
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### Image of ring homomorphism $\phi : \mathbb{Z}[t] \to \mathbb{Q}$?

Here is a problem I face practicing the theory of rings: Define $\phi : \mathbb{Z}[t] \to \mathbb{Q}$, a ring homomorphism (it does map $1$ to $1$). I'm trying to show that if $\phi(t)=\frac{u}{v}$...
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### T/F: any number that can be written as a fraction is rational.

Any number that can be written as a fraction is rational. I am being asked this question, and I believe it is true but for some reason,I feel that there is a trick. However, the definition of ...
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### For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?

It is well known that there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational. By the way, I've been interested in the following two propositions. Proposition 1 : For each ...
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### What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
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### Set of Rational numbers a countable set?

How can we say that rational numbers is a countable set? I can divide a rational number by infinite different number of natural numbers so shouldn't there be infinite rational numbers. http://www....
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### Which set is more dense: set of irrational numbers or set of rational numbers? [duplicate]

Is the infinity of irrational numbers equal to the infinity of rational numbers? Or is one is greater than other? And what is the proof? I could not find out a rigorous proof about this. P.S. I am ...
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### Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$?

Is it true that $\mathbb{Q}(\sqrt{2}) \cap \mathbb{Q}(i) = \mathbb{Q}$? I know that \begin{align*} \mathbb{Q}(\sqrt{2}) &= \{a+b\sqrt{2} \mid a,b \in \mathbb{Q}\}, \\ \mathbb{Q}(i) &= \{a+...
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### Rational solutions for $\sin(n)$ in radians

This is completely for my own curiosity. Does $y = \sin(n)$ have rational solutions for $n$, an integer number of radians. I know that this is strange because usually integers are only used in ...
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### What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
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### if $r,s$ are rational numbers, then $r+s\sqrt2$ is irrational unless $s=0$?

if $r,s$ are rational numbers, Prove $r+s\sqrt2$ is irrational unless $s=0$? I need to prove this simple question, but not sure if my method is acceptable I'm trying to prove it by contradiction, ...
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### How can all of them be irrational ??

Assume that $\{x,y,x^2,y^2,xy\}$ are all irrational. Can it be true that all of $\{x-y,x+y,x^2-y^2,x^2+y^2\}$ are irrational? Details: $|x|\ne|y|$ and $x,y\in\mathbb R$. In the question ...
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### Is a value that tends to infinity considered rational?

This really confused me. I know a rational number is any number that can be written in the form of ${a\over b} \space \space \forall a,b\in Z$. We also know all Integers are clearly Rational. But ...
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### Rational mean of irrational numbers?

My teacher tells me that in the vicinity of any rational number, an irrational exists. To elucidate, I presume, he further went on to say, if a function, if defined to give 1 for every rational number ...
How can I prove that there is no continuous function $f:\mathbb{R}\to \mathbb{R}$ satisfying $f(\mathbb{Q}) \subset \mathbb{R}\backslash \mathbb{Q}$ and $f(\mathbb{R}\backslash \mathbb{Q} ) \subset \... 1answer 38 views ### Rational points on a line This question is quite unique. Does there exist some point in the coordinate system such that any line passing through it has at most 2 rational points lying on it? 3answers 149 views ### Is there a rational surjection$\Bbb N\to\Bbb Q$? The question is in the title. Is there a one-dimensional rational function$f\in\Bbb R(X)$which restricts to$\Bbb N\to\Bbb Q$, which is a surjection onto$\Bbb Q$? My guess is no. Expanding the ... 1answer 145 views ### Can I belive that :$e^{e^{e^{e^{\cdots}}}}$is$\infty$? [closed] Definetly this number :$e^{e^{e^{e^{\cdots}}}}$is not an integer this implies that is not prime number or perfect number , now i would like to know really what is the nature of this number :$$e^{e^{... 1answer 90 views ### Is there a choice homomorphism? Let \pi : \mathbb{R} \to \mathbb{R}/ \mathbb{Q} be the canonical projection. With the axiom of choice we "know" that there are choice functions \alpha : \mathbb{R}/ \mathbb{Q} \to \mathbb{R} with ... 1answer 164 views ### Cluster points of the sequence a_n(x):=nx-\lfloor nx \rfloor I have a_n(x):=nx-\lfloor nx \rfloor where x is real. I want to show that if x is rational, then a_n(x) has finitely many cluster points, if x is irrational, then every real a with 0\leq ... 1answer 183 views ### Are \frac{\pi}{e} or \frac{e}{\pi} irrational? Is it clear whether \displaystyle \frac{\pi}{e} or \displaystyle \frac{e}{\pi} are irrational or not? If not, then there would exist q,p\in \mathbb{Z} such that$$p\cdot \pi = q\cdot e$$1answer 54 views ### Is there an irrational number arbitrarily close to another irrational number? I know that there is a rational number arbitrarily close to an irrational, due to the density of real number. But what about an irrational number? Thanks! 0answers 128 views ### Do all rational numbers repeat in Fibonacci coding? In a regular, positional number system (like our decimal numbers), every rational number ends in a repeating sequence (even if it's just 0). For example, 1/3 in base 10 is 0.33333..., in base 5 it's ... 2answers 79 views ### Let a,b be rationals and x irrational. Show that if$\frac{x+a}{x+b}$is rational, then$a=b$. I'm trying to solve the following problems: Let$a$,$b$be rationals and$x$irrational. Show that if$\frac{x+a}{x+b}$is rational, then$a=b$Let$x$,$y$be rationals such that$\frac{x^2+x+\sqrt{...
Iterating the map $\ \ x\ \mapsto\ x-\frac{1}{x},\ \$ the orbit of initial point $2$ is "probably" dense in $\mathbb{R}$. Is there an explicit rational mapping together with an initial rational ...