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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9
votes
2answers
136 views

How to show this cover of $\mathbb{Q}$ doesn't cover $\mathbb{R}$?

Let $\{q_n : n \in \mathbb{N}\}$ be an enumeration of $\mathbb{Q}$ and define $\mathcal{O} = \{I_n : n \in \mathbb{N}\}$ being $$I_n = \left(q_n - \frac{1}{2^n}, q_n + \frac{1}{2^n}\right).$$ It is ...
0
votes
5answers
102 views

Prove rational sum and product of two irrational numbers

I need to prove that $$\exists a,b \in \mathbb{R} \setminus \mathbb{Q} : a + b, ab \in \mathbb{Q}$$ Any ideas? I, unfortunately, don't have one yet. The most obvious way with equations in integers (...
1
vote
4answers
26 views

Convert rational number in $\frac {p}{q}$ form

Convert rational number in $\frac {p}{q}$ form $0.40\bar 7$ (here bar is over $7$). solution: By solving I got the answer $367/900$ by multiplying by $10$ My friends are getting answer $4037/9900$ ...
2
votes
1answer
59 views

Difference between $\mathbb{Q}$ and $\mathbb{R}$ - countability proof

We know $\mathbb{Q}$, the rational numbers, is countable; the real numbers is not. My professor in the course of real analysis proved the title by showing $\{0,1\}^{\mathbb{N}}$ is not countable, ...
0
votes
1answer
31 views

Simplifying Rational Expressions

Simplify the following rational expression: $5/(x+3) - 7x/(x-1)$ I came across this question in my homework and because it is a fraction, I decided that I needed to establish a common denominator of:...
0
votes
1answer
38 views

Is it possible to construct a maximal set with irrational distance between elements?

As part of my algebra homework a few weeks ago, I was asked to prove some things about the relation $R$, defined by $(x,y) \in R$ if $x - y \in \mathbb{Q}$. The homework problem itself wasn't ...
1
vote
1answer
3k views

Prove that the difference between two rational numbers is rational

This is a terribly simple question I'm sure, but I can't find a work-around in my proof. I must prove that the difference between two rational numbers is thus rational. Here is my attempt: Let $a$ ...
1
vote
1answer
242 views

How do I write the opposite of a rational number? [closed]

Write the opposite of each rational number A)$ 9$ B)$-17.6$ C) $6.12 $ D) $-7 \frac{5}{7 }$ Some one please help! I am not doing very good in Math I'm in grade 9 and I'm struggling I would ...
-1
votes
3answers
244 views

What decimal is between 0.5 and 0.625 [closed]

I would really appreciate some help with this. I have been literally stumped with it for an hour. So if you know the answer please comment below! Thank you for your time:)
-2
votes
1answer
1k views

The difference between two rational numbers always is a rational number [duplicate]

Claim: The difference between two rational numbers always is a rational number Proof: You have a/b - c/d with a,b,c,d being integers and b,d not equal to 0. Then: a/b - c/d ----> ad/bd - bc/bd ---...
3
votes
2answers
145 views

Length of digits before the period in decimal expansion for rational numbers

I'm a newbie with number theory and I've been reading this page and trying to figure out how to calculate the length of the digits before the period and digits of the period of a rational number of ...
2
votes
2answers
543 views

Prove that $x\in\mathbb Q$

Let $a\in\mathbb Q$ and $a>\dfrac43$. Let $x\in\mathbb R$ and $x^2-ax,x^3-ax\in\mathbb Q$. Prove that $x\in\mathbb Q$. EDIT: Thsi is my attempt: Let $x^2-ax=b$ and $x^3-ax=q$ for some $b,q\in\...
0
votes
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
212 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 $x=...
3
votes
3answers
201 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
48 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
158 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
187 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
373 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
vote
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
votes
5answers
542 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
69 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
345 views
1
vote
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
votes
2answers
42 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
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 $A\...
2
votes
2answers
91 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
votes
3answers
300 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
276 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
52 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
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
79 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
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
vote
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
votes
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
votes
2answers
75 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
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
144 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
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
55 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
211 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
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 ...