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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5answers
341 views

Why is $[0, 1] \cap \mathbb{Q}$ not compact in $\mathbb{Q}$?

Statement: $[a, b] \cap \mathbb{Q}$ in $\mathbb{Q}$ is not compact. Thus the interior of all compact subsets of $\mathbb{Q}$ is $\emptyset$. I am trying to understand the first sentence. I read that ...
4
votes
1answer
39 views

Rationals $(\mathbb{Q},<)$ are isomorphic to a part of a finite partition

I believe the following statement is true but I can't find or figure out a proof: For any partition of the set $\mathbb{Q}$ of rationals into a finite number of parts, there is a part containing an ...
1
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1answer
37 views

The minimum cardinal of a geometrical set

Let $S$ be a set of points in a plane $P$, having the following property: for any point $X \in P$ there is at least one point $M \in S$ so that the distance $|XM|$ is rational. Find the minimum ...
6
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5answers
539 views

Is there a bijection between $\mathbb{Q}$ and $\mathbb{Q}_{>0}$?

Is there a bijection between $\mathbb{Q}$ and $\mathbb{Q}_{>0}$? For $\mathbb{R}$, we have the exponential function. Is there also a bijection $f: \mathbb{Q} \to \mathbb{Q}_{>0}$ or to $\...
3
votes
1answer
68 views

Coloring rational numbers

Here is my problem. Fix a color for the number $1$, for example yellow. Choose another color, for example green. Now, for a positive rational denoted $x$, there are two rules : $x$ and $1/x$ have ...
2
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4answers
17k views

Is a Whole Number A Rational Number [duplicate]

Is a Whole Number part of A Rational Number or a whole number??
2
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3answers
342 views
1
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1answer
30 views

a dense set in (0,1)

Define for $\epsilon > 0 $ $$V_\epsilon = \left( \bigcup_{j \in \mathbb{N}} (x_n - \frac{\epsilon}{2^{n+1}} , x_n + \frac{\epsilon}{2^{n+1}}) \right) \ \bigcap \ (0,1)$$ where $x_n$ stems from ...
0
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2answers
40 views

Equivalence classes and rational numbers

We defined $\mathbb{Q}$ as the set of equivalence classes for the relation $\sim$. Tentatively define operations $+,\cdot:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}$ by $[(a,b)]+[(c,d)] = [(ad+...
0
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1answer
34 views

Given a rational number produce a larger rational number

Given Dedekind cuts $A|B$ and $C|D$ in $\mathbb{Q}$, let $E=\{a+c:a\in A,c\in C \}$ Prove that E has no largest element. If I understand the first statement $A|B$ and $C|D$ are real numbers, but $A\...
2
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2answers
90 views

Dedekind cuts in Rudin's PMA

I'm working on Appendix to chapter I of Rudin's Principles of mathematical analysis and I have the following problem: Given a positive cut $\alpha$ and a rational $x>1,$ how can I prove that there ...
2
votes
2answers
88 views

Prove or give a counterexample If $a \in \Bbb R$\ $\Bbb Q$ exists $n \in \Bbb N$ such that $a^n \in \Bbb Q$

1)If $a \in \Bbb R$\ $\Bbb Q$ exists $n \in \Bbb N$ such that $a^n \in \Bbb Q$ 2)If $a \in \Bbb R$\ $\Bbb Q$ , $_n\sqrt a \in \Bbb R$ \ $\Bbb Q $ $\forall n \in \Bbb N$ For the second one: [by ...
0
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3answers
299 views

Proof Involving Rational Numbers

I asked this same question last night and got some answers but still can't make sense of this, normally I'd move on but since I know how to do everything else for the test I'm going to try to get this ...
-1
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3answers
267 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
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1answer
51 views

Deduce that $\mathbb Q^n$ is countable for any integer $n \in \mathbb N$ [duplicate]

How do I start this? Do I follow the same proof on why rational numbers are countable?
0
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0answers
23 views

Gaussian rationals with rational norm

Looking for information on Gaussian rationals with rational norm. A gaussian rational is a complex number of the form z = p + qi where p and q are rationals. Taking only those that have |z| = $\...
3
votes
2answers
78 views

Solving $3x^{4}-7x^{3}+2x^{2}=950$ over the rationals

I am asked to find only rational solutions. Factoring by $x^{2}$, I get: $$x^{2}(3x^{2}-7x+2)=950$$ By applying the quadratic formula, I have: $$x^{2}(3x-1)(x-2)=950$$ I don't know how to proceed ...
4
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1answer
96 views

Is $\int_0 ^1 \frac{1-x^p}{1-x} $ ever rational for rational non-integer values of $p$?

It is well known that the $n$-th harmonic number $H_n$ has the integral representation $\int_0^1 \frac{1-x^n}{1-x}$. If we replace $n$ with rational non-integer $p$, do we ever get a rational outcome? ...
5
votes
2answers
104 views

Condition implying rationality of $u^n+v^n$

$Given :\ u+v \ is \ rational, \ u^2 + v^2 =1 \ , prove \ v^n + u^n \ is \ rational$. What I have done so far is proving that $uv$ is rational by expanding $(u+v)^2$. I expanded $(u+v)^n$ using ...
1
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2answers
100 views

Intuitive explanation of the Dirichlet function and rationality

The Dirichlet function is defined by $f(x)=\begin{cases} c &\text{ if } x\in \mathbb{Q}\\d &\text{ if } x\notin \mathbb{Q}.\end{cases}, c\neq d$ See MathWorld's page for the full definition. ...
2
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0answers
33 views

Smallest number of workers in factory, Diophantine approximation

Q. In a factory, the percentage of male workers was $53.7802\%$ (rounding to nearest fourth decimal place) last year. What is the smallest number of female workers working there? Hint: Diophantine ...
3
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2answers
74 views

Enumerating the positive rationals without repetition

From Makarov's Selected problems in real analysis Let $r_n$ be defined as follows\begin{cases} r_1=1 & \\ r_{2k}=1+r_k \\ r_{2k+1}=\frac{1}{r_{2k}} & \end{cases} ...
0
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0answers
43 views

using long division to find the oblique asymptote of rational function

To find the oblique asymptote of a rational function, the book I'm reading says to divide the denominator of a fraction into the numerator. The example rational function it gives is $$\frac{x^2 - 9} {...
3
votes
4answers
1k views

If $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational [closed]

Prove that if $x$ and $y$ are irrational but $x + y$ is rational then $x - y$ is irrational. I can understand how it works in my head, I don't know how to prove it though.
2
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1answer
135 views

Prove that the quotient of a nonzero rational number and an irrational number is irrational

$0\neq a\in \mathbb Q, b\in \mathbb R \setminus\mathbb Q \text{ (b is irrational)}$ Prove that $\frac a b$ is irrational. From defintion $a=\frac m n$ such that $m,n\in \mathbb Z, n\neq 0$. Take ...
0
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0answers
83 views

A simple question about rational numbers without a simple proof?

As in this question, study the quasigroup $(Q_+,/)$ of positive rational numbers under division. There are two obvious identities: $a/(b/c)=c/(b/a)$, for all $a,b,c\in Q_+$ $(a/b)/c=(a/c)/b$, for ...
1
vote
2answers
44 views

Is this proof about the countability of $\Bbb Q \times \Bbb Q \times \cdots \times \Bbb Q$ sound?

If $\Bbb{Q}$ is countable, prove that the set $\Bbb{Q}^n$ for $n = 2,3,...$ is countable. Base case: $n = 2 \rightarrow \Bbb{Q}^2 = \Bbb{Q}\times\Bbb{Q}$ which, by Proposition 4.5 (see bottom of ...
0
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2answers
54 views

division of fraction simplification

The expression is this: ${{y^2 - y} \over 1 {}} \div {{y^2 - 1} \over 3}$ The first step is to swap the second expression round to: ${{y^2 - y} \over 1 {}} \div {3 \over {y^2 - 1}}$ The answer ...
2
votes
3answers
208 views

There is no largest rational number $p$ such that $p^2 < 2$

In Rudin's analysis example 1.1, he tried to show the following Let $A$ be the set of all positive rationals $p$ such that $p^2<2$ and let $B$ consist of all positive rationals $p$ such that $p^2 &...
4
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2answers
6k views

Can we ever get an irrational number by dividing two rational numbers?

If we try to divide any two random arbitrarily long rational numbers like 103850.2387209375029375092730958297836958623986868349693868398659825528365... and <...
3
votes
1answer
95 views

Prove that $\mathbb{Q}[\sqrt{p}]$ is a field between $\mathbb{Q}$ and $\mathbb{R}$

I really have trouble understanding a task. We've got $p\in$ P, while P are all prime numbers. Now we construct a field $$\mathbb{Q}[\sqrt{p}]:=\{x+y\sqrt{p}:x,y \in \mathbb{Q}\}$$ The Task is to ...
11
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3answers
211 views

Determine all functions satisfying $f\left ( f(x)^{2}y \right )=x^{3}f(xy)$

Denote by $\mathbb{Q}^{+} $ the set of all positive rational numbers. Determine all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the following equation for all $x,y \in \...
4
votes
1answer
971 views

How to find all rational points on the elliptic curves like $y^2=x^3-2$

Reading the book by Diophantus, one may be led to consider the curves like: $y^2=x^3+1$, $y^2=x^3-1$, $y^2=x^3-2$, the first two of which are easy (after calculating some eight curves to be solved ...
8
votes
1answer
150 views

Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
1
vote
3answers
724 views

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $x/y$ is a rational number [duplicate]

Prove that if $x$ and $y$ are rational numbers and $y\ne 0$, then $\frac{x}{y}$ is a rational number. How do I prove this, and also which proving method would I use? I'm confused between that and ...
0
votes
2answers
115 views

Is $\Bbb Q^n$ dense in $\mathbb{R}^n$ for $n>1$? [closed]

It is well known that the rational numbers, $\mathbb{Q}$, are dense in $\mathbb{R}$. My question here :Is $\ \mathbb{Q}^n$ dense in $\mathbb{R}^n$ for $n>1$ ? Edit : I edited the question as it ...
6
votes
5answers
1k views

Question on compact metric space.

Is the set of rationals in $[0,1]$ compact? I seems like every open covering should have a finite sub-covering, yet I have read that compact metric spaces are complete...
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votes
4answers
234 views

Difference between ${2\over 9}$ and ${22\over 99}$? [closed]

The fractions ${2\over 9}$ and ${22\over 99}$ both have the same decimal value $0.22222\ldots$ But obviously they are not equal. What causes this situation? And also, what is the correct rational ...
0
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0answers
42 views

What functions stem from Fourier Series with rational-only coefficients?

Given the Fourier series $$f(z) = \sum_{k=-\infty}^\infty c_k e^{ikz}$$ but with $c_k\in(\mathbb Q+ i\mathbb Q)$ instead of $\mathbb C$ (or even purely real), are the functions obtained this way in ...
6
votes
3answers
277 views

Why is the remainder uniformly distributed when 1,2,3,… are divided by an irrational number?

Let remainder $r$ be defined as $$ r = n - pq $$ where $n \in \mathbb{N}$ is the dividend , $q \in \mathbb{R}$ is the divisor, and $p = \mathrm{floor}(n/q)$. I calculated the remainders by dividing ...
3
votes
1answer
100 views

Simplified rational distance problem

① Is there a point on a square with sides of rational length that is a rational distance from each vertex? Note that this is a very specific case of the Rational Distance Problem, which can be ...
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2answers
166 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
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2answers
132 views

Why do the integers, rationals and any countable set have zero measure?

There is an exercise in my text that tells me to prove the "obvious and easy to see" fact that $\mathbb{Z}$ and $\mathbb{Q}$ have measure zero. Er...here is what I know so far. If I have an interval, ...
2
votes
1answer
461 views

Rational vs Irrational distribution

Imagine I draw a number line, and I took two points. What's the distribution of rational and irrational numbers between them? If I put it in a diagram where I color rational with a color and ...
1
vote
1answer
67 views

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$

Find all functions $f:\mathbb{Q}^+\rightarrow\mathbb{Q}^+$ such that $$f(x)+f(y)+2xyf(xy)=\frac{f(xy)}{f(x+y)}$$ for all $x,y\in\mathbb{Q}^+$. Before this problem, I have solved few similar ...
1
vote
2answers
81 views

If $a \in \mathbb{I}$ , how is $\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$

If $a \in \mathbb{I}$ , how is $$\overline{\mathbb{Z}+ a\mathbb{Z}}=\mathbb{R}$$ It says in my notebook that this set in dense in $\mathbb{R}.$ How do I prove this density? With say $\mathbb{Q}$ and $...
4
votes
5answers
599 views

H0w t0 prove that periodic decimal numbers are rational? $a_1…a_k(b_1b_2..b_l)={m \over n}$

Given $a_1...a_k(b_1b_2..b_l)={m \over n}$ how can I prove that periodic decimal numbers are rational? Where do I even begin?
2
votes
2answers
67 views

Can a non-rational polynomial be rational at all integers?

Is there a polynomial $f \in \mathbb{R}[X]$ such that for every $x \in \mathbb Z,\>\> f(x)$ is rational but at least one of the coefficients of $f$ is irrational?
2
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0answers
110 views

What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
1
vote
1answer
130 views

(Ir)rationality of Real Numbers

I am working on this proof and this is what I got so far, can someone help me verify if what I have done is right? For all real numbers $x$ and $y$, if $x+y$ is rational and $x-y$ is irrational ...