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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0
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1answer
34 views

Can reflection across a line segment be done using the rational field?

Assume that I have a point and a line segment, all specified using rational coordinates. Can I compute the reflection of the point across the line segment using only rational numbers? This previous ...
4
votes
1answer
207 views

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number.

Find all $\theta$ such that sin$\theta$ and cos$\theta$ are both rational number. I thought this question might have been asked by someone else, but I couldn't find any. Currently I'm studying ...
0
votes
3answers
65 views

How Can I calculate this expression?

I have this repeating expression $5+\dfrac {6} {5+\dfrac {6} {5+..}}$ I saw a solution on a book. which is: $5+\dfrac {6} {5+\dfrac {6} {5+..}}=x$ $5+\dfrac {6} {x}=x$ $x^2-5x-6=0$ $x=6 $ or ...
3
votes
3answers
164 views

Find all cluster points for the sequence $x_{n}$ = The $n$-th rational number

Find all cluster points for the sequence $x_{n}$ = The $n$-th rational number Note: In this problem a labeling of rational numbers by positive integers is used. Such labellings do exist because ...
2
votes
1answer
46 views

How to express $15.3\dot{9}$ in fractional form

In the number $15.3\dot{9}$, $9$ is repeated forever. If the number is rational then it can be expressed as a fraction (i suppose it is rational since it's an exercise for me to find it's rational ...
6
votes
1answer
138 views

What are some of the implications of $\pi + e$ being rational?

Whether or not $\pi + e$ is rational is an open question. If it were rational, what would some of the implications be?
1
vote
1answer
163 views

Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational? [duplicate]

I know the famous proof that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to ...
1
vote
5answers
276 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 ...
4
votes
1answer
38 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
vote
1answer
34 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
votes
5answers
528 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
65 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
votes
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
votes
3answers
324 views
1
vote
1answer
28 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
votes
2answers
39 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)] = ...
0
votes
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 ...
2
votes
2answers
89 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
84 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
votes
3answers
297 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
votes
3answers
236 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
votes
1answer
47 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
votes
0answers
22 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
76 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
votes
1answer
95 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
101 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
vote
2answers
94 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. ...
4
votes
3answers
173 views

When can a set have an upper bound but no least upper bound?

So I'm taking real analysis and have noted that one of the benefits of the Dedekind cut is that 'if one of the sets made has an upper bound it also has a least upper bound'. I don't understand how a ...
2
votes
0answers
29 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
votes
2answers
72 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} ...
2
votes
1answer
56 views

A proof that $\frac{(2\phi)^n-(-1)^n}{\phi^{2n}-(-1)^n}\cdot\left(2^n-\phi^n\right)\cdot\sqrt5\in\mathbb Q$ for all $n\in\mathbb Z$

During computation of some series (with help of a CAS), at an intermediate step I encountered an expression, that after dropping non-essential parts looks like this:$$\mathcal ...
0
votes
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
votes
1answer
108 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 $m,n\in \mathbb Z, n\neq 0$. ...
0
votes
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
votes
2answers
52 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
201 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
votes
2answers
5k 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
votes
3answers
206 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
852 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
149 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
696 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
114 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...
-6
votes
4answers
230 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
votes
0answers
39 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
271 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 ...
2
votes
1answer
95 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 ...