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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2
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1answer
40 views

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 ...
-2
votes
3answers
65 views

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 ...
0
votes
1answer
31 views

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 ...
2
votes
4answers
76 views

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 ...
1
vote
1answer
25 views

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 ...
0
votes
2answers
52 views

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 ...
0
votes
0answers
39 views

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 ...
1
vote
3answers
153 views

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 ...
1
vote
1answer
113 views

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 $\phi(t)=\frac{u}{v}$...
2
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2answers
112 views

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 ...
8
votes
1answer
278 views

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 ...
1
vote
4answers
9k views

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?
1
vote
1answer
75 views

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....
1
vote
1answer
91 views

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 ...
3
votes
0answers
37 views

Nonlinear regular bijection from $\mathbb Q$ to itself

Is there a bijection $\phi: \mathbb Q \to \mathbb Q$ such that $\phi$ is nonlinear (i.e. different from $ax+b$), $\phi$ is regular: the extension $\hat{\phi}$ of $\phi$ over $\mathbb R$ is $\mathcal ...
0
votes
2answers
55 views

Onto function with domain of rational numbers and co-domain of natural numbers

I'm trying to find an onto function $f: \mathbb{Q} \to\mathbb{N}$ I'm somewhere along the lines of $f(q) = |(1 - q)| + q$ for non integers, but I'm not sure where to go from there.
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2answers
58 views

Sets which are order-isomorphic to the (extended) rationals

Let $(S,<)$ be a totaly ordered set under the strict order relation $<$. Suppose that, for any $a,b\in S$, if $a<b$, then there exists $c\in S$ such that $a<c<b$. We also assume that $...
1
vote
1answer
46 views

Does the inner pentagon inside a Robbins pentagon $also$ have a rational area?

The Heron triangle has integer sides and area. The Robbins pentagon is just the generalization: it also has integer sides and area. The example below has sides $78, 126, 66, 50, 32$ and area $A_R = \...
0
votes
2answers
71 views

Given a rational number and an irrational number, both greater than 0, prove that the product between them is irrational.

Does this proof I made make sense? Proof// $\mathbf a$ is the rational number, $\mathbf b$ is the irrational number. Assume that $\mathbf {a * b}$ is rational due to proof by contradiction. ...
2
votes
1answer
56 views

Proof of part of properties of exponentiation Tao proposition $4.3.12$

If you let $x,y$ be non-zero rational numbers, and let $n, m$ be integers, I need to prove that if $x \geq y>0$, then $x^{n} \geq y^{n}>0$ if n is positive, and $0< x^{n} \leq y^{n}$ if $n$ ...
1
vote
1answer
65 views

The set of rational numbers has continuum many subsets that are order-wise inequivalent

How to show that there are continuum many subsets of $\mathbb Q$, no two of which are similar? Two sets are called similar if there is an order-preserving bijection between them.
1
vote
1answer
51 views

Is this proof correct? Show $\mathbb{Q}$ is dense in $\mathbb{R}$

I like proof by contradictions in showing that $\mathbb{Q}$ is dense in $\mathbb{R}$. But I can't understand this one> https://math.dartmouth.edu/archive/m54x12/public_html/m54densitynote.pdf ...
1
vote
1answer
28 views

Property of a sequence being an enumeration of the rationals.

Let $(r_n)$ be an enumeration of the rationals and $x\in\mathbb{R}$. Is it possible to find out whether the set $\left\{n\in\mathbb{N}:\left|x-r_n\right|<\frac{1}{2^n}\right\}$ is finite or ...
1
vote
1answer
48 views

When $\frac{\pi ^{x}}{\zeta (x)}$ is rational?

When $n$ is a positive integer, we know $$\zeta (2n)=\frac{(-1)^{n+1}B_{2n}(2\pi )^{2n}}{2(2n)!}$$ Now let's say $x>1$ is a real number. Can we say if $\frac{\pi ^{x}}{\zeta (x)}$ is a rational ...
3
votes
1answer
78 views

How to find out the number of repeating digits of a rational number in decimal form?

Upon dividing two integers, I would like to programmatically predict the number of decimal places that repeat after the decimal point. For example in $\frac{1}{3}=0.\overline{3}$, I want to know that ...
4
votes
1answer
100 views

Proving that the Calkin-Wilf tree enumerates the rationals.

The Calkin-Wilf tree is an infinite undirected graph (tree) which is constructed as follows: starting from the root at $\frac{1}{1}$, each node $\frac{a}{b}$ has two children: a left child $\frac{a}{...
5
votes
2answers
92 views

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+...
0
votes
1answer
31 views

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 ...
28
votes
10answers
951 views

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 ...
4
votes
1answer
67 views

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, ...
6
votes
2answers
100 views

Is there a first order formula $\varphi[x]$ in $(\mathbb Q, +, \cdot, 0)$ such that $x≥0$ iff $\varphi[x]$?

In the first-order language $\mathscr L$ having $(+, \cdot, 0)$ as signature, it is easy to define a formula $\phi[x]$, namely $\exists y \; x = y^2$, satisfying : $$\text{for all } x \in \Bbb R, \...
0
votes
1answer
27 views

Why is it that for any rational numbers $a < b$, the interval $[a, b]$ in $\mathbb{Q}$ is not compact with respect to this metric?

Suppose $q$ is any nonzero rational number and $p$ is a fixed prime. If $q = p^k\frac{n}{m}$ for integers $n$ and $m$, neither of which has $p$ as a factor, then we define $|q|_p := p^{−k}$. We can ...
2
votes
1answer
49 views

$\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$

I'd appreciate if you could please check to see if my proof is valid. Find $\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$. Solution: $q^2 > p^2/5 \iff q > ...
64
votes
2answers
2k views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that $$r=\frac{a^\color{red}{3}+b^\color{red}{3}}{...
2
votes
1answer
61 views

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 ...
1
vote
0answers
39 views

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 ...
0
votes
1answer
113 views

Relation on rational numbers that defines a total order

Define the relation on $\mathbb{Q}$ by $$[m,n]<[j,k]$$ if and only if $jn-mk$ belongs to $\mathbb{N}$, $j$ and $m$ belong to $\mathbb{Z}$, $n$ and $k$ belong to $\mathbb{N}$. (a) Show that $&...
3
votes
2answers
50 views

For a.e. $x \in [0, 1]$, there are finitely many $p/q$ such that $\left| x - p/q \right| < 1 / \left( q \log q \right)^2$

I am stuck on a qualifying exam problem and was hoping to get some help. Show for a.e. $x \in [0, 1]$ that there are finitely many $p/q \in \mathbf{Q}$ in reduced form such that $q \geq 2$ and $\...
0
votes
2answers
46 views

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 ...
1
vote
3answers
72 views

Proving that there is no continuous function $f:\Bbb R\to\Bbb R$ satisfying $f(\Bbb Q)\subset\Bbb R-\Bbb Q$ and $f(\Bbb R-\Bbb Q) \subset\Bbb Q$. [duplicate]

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 \...
2
votes
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?
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votes
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 ...
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votes
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^{...
4
votes
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 $...
0
votes
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 ...
7
votes
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 $$p\cdot \pi = q\cdot e$$
1
vote
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!
12
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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 ...
0
votes
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{...
4
votes
1answer
110 views

A rational orbit that's provably dense in the reals?

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 ...