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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1answer
29 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
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4answers
75 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
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1answer
24 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 ...
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2answers
50 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 ...
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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 ...
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3answers
59 views

Conditions under which $\frac{ax+b}{cx+d}$ will be rational.

Suppose $x$ is an irrational number and $a,b,c,d$ are rational numbers. If we know that $$ \frac{(ax+b)}{(cx+d)} $$ is rational, then it follows that: a.) $a=c=0$, b.) $a=c$ and $b=d$, c.) $a+b = ...
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3answers
151 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 ...
2
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2answers
106 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 ...
1
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1answer
83 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 ...
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1answer
63 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. ...
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0answers
36 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 ...
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2answers
51 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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1answer
37 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 = ...
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2answers
57 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. ...
1
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2answers
53 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 ...
2
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1answer
50 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
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1answer
57 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.
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1answer
49 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
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1answer
47 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 ...
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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 ...
5
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2answers
90 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) &= ...
4
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1answer
80 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 ...
4
votes
1answer
64 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 ...
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 ...
0
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1answer
30 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 ...
2
votes
1answer
45 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 > ...
5
votes
2answers
89 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, ...
2
votes
1answer
49 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 ...
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0answers
37 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 ...
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2answers
42 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 ...
3
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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 ...
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3answers
69 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
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1answer
37 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?
2
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0answers
47 views

How to generate primitive solutions to the equation $a^3 + b^3 = c^2$

The solution for this is that we are supposed to pick numbers x and y, then we can substitute them in the equation and obtain some z, which we then multiply the left side of the equation with to ...
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10answers
823 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 ...
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votes
1answer
142 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 ...
4
votes
1answer
85 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 ...
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3answers
145 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 ...
7
votes
1answer
179 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 $$p\cdot \pi = q\cdot e$$
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1answer
50 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!
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2answers
77 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 ...
12
votes
0answers
122 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
1answer
51 views

How to prove that $\bar{\mathbb{Q}}=\mathbb{R}$?

How to proof that $\bar{\mathbb{Q}}=\mathbb{R}$, where $\bar{\mathbb{Q}}=\mathbb{Q}\cup\mathbb{Q}^{\prime}$ and $\mathbb{Q}^{\prime}$ are the limit points of $\mathbb{Q}$?
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0answers
17 views

Continuity - approximating an irrational number via rationals [duplicate]

If $x=p/q$, where $(p,q)=1$ are integers, then $f(x)=1/q$. If x is irrational then f(x)=0. Prove that: a) f is continuous for all irrationals b) f is not continuous for all rationals. I think ...
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2answers
469 views

What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
1
vote
1answer
51 views

Determine all positive rational numbers $r \neq 1$ such that $r^{\frac{1}{r-1}}$ is rational?

Here's what I've got so far: Let $r = \frac{a}{b}$, where $a$ and $b$ are integers. We then have $$r^{\frac{1}{r-1}} = \frac{a^{\frac{b}{a-b}}}{b^{\frac{b}{a-b}}}$$ Clearly, $a-b=1$ and $a-b=-1$ ...
2
votes
2answers
35 views

Proving $\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$

I need to prove that: $$\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$$ Well, $ \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} $ is the set of ...
3
votes
1answer
91 views

Find the maximum number of rational points on the circle with center $(0,\sqrt3)$

Find the maximum number of rational points on the circle with center $(0,\sqrt3)$ Let the equation of the circle be $x^2+(y-\sqrt3)^2=r^2$ Let $(a,b)$ be any rational point on the circle ...
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0answers
37 views

How do I know if this will be rational or irrational? ($a^b$)

Usually, when I have $a^b$ when $a$ and $b$ are both irrational, I assume that it will be irrational. But that is not always true, I assume, so when is the result irrational? How will I know? Take ...
18
votes
7answers
1k views

How many sequences of rational numbers converging to 1 are there?

I have a problem with this exercise: How many sequences of rational numbers converging to 1 are there? I know that the number of all sequences of rational numbers is $\mathfrak{c}$. But here ...