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Questions tagged [rational-numbers]

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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Given $x^2=2$ prove for any rational number $\frac{p}{q} < x$,there exists $\frac{m}{n}$ such that $\frac{p}{q}<\frac{m}{n}<x$

Without using limits or the definition of irrational numbers, how do you solve this? I was thinking proof by contradiction, but I keep running into problems.
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0answers
32 views

Visualizing rational numbers as multiplication graphs

It's an interesting fact, that there's a straight forward way to visualize rational numbers. To each rational number – given as two integers $n<m$ – there corresponds a multiplication ...
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4answers
60 views

Square root of 6 proof rationality

I was proving $\sqrt 6 \notin \Bbb Q$, by assuming its negation and stating that: $\exists (p,q) \in \Bbb Z \times \Bbb Z^*/ \gcd(p,q) = 1$, and $\sqrt 6 = (p/q)$. $\implies p^2 = 2 \times 3q^2 \...
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1answer
30 views

I don't understand why this axiom makes the difference between a completely ordered field and an ordered field

I've been reading about ordered fields and completely ordered fields, and I'm stuck on the difference between the rational and real numbers that makes the rational numbers an ordered field and the ...
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2answers
39 views

A course of Pure mathematics incorrect

I thought I'd go through 'A Course Of Pure Mathematics' by G H Hardy And I came to this part: If the reader will mark off on the line all the points corresponding to the rational numbers whose ...
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1answer
8 views

Show that if $x$ has a terminating decimal expansion then $x=p/q$ for integers $p,q$ where the only prime factors of q are 2's and 5's.

I've proven the converse statement but don't know where to start for this statement. Would you suggest doing a proof by contradiction?
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2answers
30 views

The set {${m^m}/{n^n}: m,n\in \mathbb{N}$} density in ${\mathbb{Q}_+}$

I missed this topic (density) , so need help. Is the set {${m^m}/{n^n}: m,n\in \mathbb{N}$} everywhere dense in positive rational numbers set? I think maybe first of all here we must try with, for ...
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Evaluating $\sum_{(a,b,c)\in T}\frac{2^a}{3^b 5^c}$, for $T$ the set of all positive integer triples $(a,b,c)$ forming a triangle

Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a$, $b$, $c$. Express $$\sum\limits_{(a,b,c)\,\in\,T}\;\frac{2^a}{3^b 5^c}$$ as ...
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0answers
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Urgent: Rational Functions Word problems related to speed?

For example, a guy rides his bike for 2 days, 60km each day. On day 2, he doubles his speed from day 1. His total riding time was 6 hours. What was his average speed? I sort of came up with the ...
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2answers
33 views

Abelian Group of Rationals over addition - axiomatic or derived? [closed]

Sorry if this is a daft question, but if we consider the set of rationals under the 'addition' operator we can form an Abelian Group. I'm curious: in that situation, is the definition of addition (...
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Show that rational numbers have the Archimedean property

By Archimedean property I mean: For any positive rational numbers $x = \frac{a}b$ and $y = \frac{c}d$, there is an integer $n$ such that $nx > y$, namely, $nx \equiv (x+x+ \ldots+x)$ with $n$ ...
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1answer
35 views

$\mathbb{Z}\leq G \leq \mathbb{Q}$ and $\varphi:G\rightarrow H$ a map which behaves nicely on fractions.

Let $(H,+)$ be any abelian group, and let $\mathbb{Z}\leq G\leq \mathbb{Q}$ a subgroup of the rational numbers with respect to addition. Let $\varphi:G\rightarrow H$ be a map with the following ...
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1answer
29 views

Decimal expansions of rational numbers.

One can use modular arithmetic to find the decimal expansion of a rational number. (see 1: https://i.stack.imgur.com/kw4Gk.png). Using the same method I have run into a couple of problems. $x = \...
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1answer
57 views

Are sets everywhere dense in rational numbers set?

I'm new in MSE and in functional analysis. Really need explanation with this topic and exercise: Is the set {${m^2}/{n^2}: m,n\in \mathbb{N}$} everywhere dense in positive rational numbers set? And is ...
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2answers
163 views

Prove the set of all dyadic numbers is countable

A number is said to be dyadic if it has the form $\frac{k}{2^n}$ for some integers $k$ and $n$ in $\mathbb Z^+ $. Show that the set of all dyadic numbers is countable. Here is a proof I have (but ...
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1answer
23 views

Find real numbers satisfying some conditions of rational dependance

Does there exist $\xi_1,\ldots,\xi_4\in\mathbb R$ such that $$\forall (\alpha,\beta)\in\mathbb Q^2\setminus\{0,0\},\quad \begin{cases}\dim_{\mathbb Q}(\alpha \xi_1+\beta\xi_3,\,\alpha\xi_2+\beta\...
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3answers
70 views

Proof that every rational number is between 2 consecutive integers

I need to prove that $\forall$ $p \in \mathbb{Q}$, $\exists n \in \mathbb{Z}$ such that $n \leq p < n+1$. What I have done is assume that such a $p$ does not exist. Which implies that either $p>...
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1answer
141 views

Show that $\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x=0$ [duplicate]

The Dirichlet function is defined as the indicator function of rational numbers. I have also seen this function described by: $$f(x)=\lim_{k\to\infty}\lim_{j\to\infty}\cos^{2j}k!\pi x$$ How does this ...
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1answer
40 views

Interior and closure of $\mathbb{Q}\cap (0,1)$.

Determine the interior and closure of the set $A=\mathbb{Q}\cap (0,1)$. My approach: First note that the set of rationals $\mathbb{Q}$ and irrational $\mathbb{I}=\mathbb{R}\setminus\mathbb{Q}$ are ...
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1answer
59 views

Prove that $\mathbb{Q}_+$ can be enumerated as $(q_n)$ such that $\lim\sqrt[n]{q_n}$ exists.

Prove that the set of positive rationals can be enumerated as $(q_n),$ such that $\lim\sqrt[n]{q_n}$ exists. Comment. I don't know if I should be looking for a certain "formula" on $q_n's$, or a way, ...
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1answer
48 views

$P(x)$ doesn't have a rational root?

Let $P(x)$ be a polynomial with integer coefficients. In what conditions that $P(x)$ doesn't have a rational root? From https://en.wikipedia.org/wiki/Rational_root_theorem, if $P(x)=a_nx^n+\cdots +...
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1answer
23 views

A family of subspaces of $\mathbb R^4$ which does not intersect non-trivially a rational subspace

This question is somehow linked to this previous question I asked earlier. Let's consider the following family $F$ of subspaces of dimension $2$ of $\mathbb R^4$. A subspace $A$ is in $F$ if it ...
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1answer
29 views

Is this particular subspace of $\mathbb R^4$ intersecting non-trivially a rational subspace?

Let's consider $A$ the subspace of $\mathbb R^4$ given by $A=\mathrm{Span}(Y_1,Y_2)$ where $$\begin{cases} Y_1=(0,1,\sqrt 2,\sqrt 3), \\ Y_2=(1,0,\sqrt 3,\sqrt 2).\end{cases}.$$ I was wondering ...
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4answers
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Are rational points dense on every circle in the coordinate plane?

Are rational points dense on every circle in the coordinate plane? First thing first I know that rational points are dense on the unit circle. However, I am not so sure how to show that rational ...
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1answer
33 views

Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist?

Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist? If not, prove it. I don't think the supremum exist, but I don't know how to prove it.
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1answer
20 views

Rational numbers in irrational bases

If you take the base-$b$ expansion of a rational number where $b$ is irrational, do you get a non-terminating sequence of digits (assuming you pick the right(?) digits)? More informally, do rational ...
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2answers
34 views

Minkowski Set Addition [closed]

What is the Minkowski sum of Q to itself, where Q is the set of all rational numbers? I can't find a way to solve this with freshman knowledge. Thank you in advance.
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2answers
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Proof by contradiction - am I correct?

I am trying to learn some discrete mathematics alongside my arts course to try and expand my knowledge. I have this question: Prove that if $x$ is irrational then $\frac{x+1}{x-1}$ is irrational. My ...
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2answers
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Does $f(x)=x^3+x^2-8x+1$ have rational roots?

Problem Does $f(x)=x^3+x^2-8x+1$ have rational roots ? Attempt to solve A citation from our lecturer Possible rational roots are in form of: $$ \frac{\text{factor of constant}}{\text{factor of ...
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2answers
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Rational fraction expression for triangular powers of 2

Why does the following pattern hold? $$1=1=2^0$$ $$\frac{2(2^2-1)}{3}=2=2^1$$ $$\frac{3^2(3^2-1)(3^2-2^2)}{3^2 \times 5}=8=2^3$$ $$\frac{4^2(4^2-1)^2(4^2-2^2)(4^2-3^2)}{3^3 \times 5^2\times 7}=64=...
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1answer
37 views

Irrationality of $r\sqrt{5}$ for rational number $r$? [duplicate]

As $5$ is a prime number, thus $\sqrt{5}$ is an irrational number. Now I am thinking about how to prove - If $r$ is a rational number, then how do we prove $r\sqrt{5}$ is an irrational number? I ...
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50 views

Why is R* “smaller” than Q* [closed]

Why is the dual space of $\mathbb{R}$ a subset of the dual space of $\mathbb{Q}$? For the dual space I am looking at every linear functional of the form $$L:X\rightarrow\mathbb{Q}$$ for $X$ either $\...
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2answers
28 views

Measure of a relatively small subset of $\mathbb R^4$

Context. Let's denote by $\lambda$ Lebesgue's measure on $\mathbb R^4$. I have a subset of $\mathbb R^4$, let's call it $E$ of full Lebesgue measure. I have another set $F$, and I would like $E\cap F$...
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52 views

Show that the set of rational numbers is $\textbf{dense}$ in $\mathbb{R}$

Show that the set of rational numbers is $\textbf{dense}$ in $\mathbb{R}$, meaning that every real number is a limit of rational numbers. Wrote a proof, wondering if it's correct/rigorous enough and ...
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1answer
62 views

Set of Propositional Formulae with no equivalent independent subset

A set of propositional formula $X$ is said to be equivalent to $Y$ if for any formula $\alpha$, $X \models \alpha$ iff $Y \models \alpha$. Also, $X$ is said to be dependent if there exists $\alpha \in ...
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1answer
28 views

When is the denominator of the sum of two positive fractions not divisible by the denominator of either numbers?

Let $r_1 < r_2$ be two positive rational fractions, both $> 1$ and both in their lowest terms i.e. the numerator and denominator of each fraction have no common factors. If $r_3 = r_1 + r_2$ is ...
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0answers
17 views

Checking for rational numbers with small denominators in an interval

I have been given two rational numbers $\frac{a}{b}$ and $\frac{c}{d}$, and I have been told that there is a unique rational number $\frac{p}{q}$ satisfying $|q| \leq n$ (where $n$ is some known ...
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1answer
27 views

Two whole numbers n and k . Print k decimal digits of 1 / n .

I have to print these decimal numbers in C++ . But first i need to understand this question mathematically .
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Is there a well-behaved finitely additive “measure” on $\mathbb{Q }$?

There exists no nontrivial measure on the set of rational numbers for which the measure of singletons is zero. That’s because the rational numbers are countable, so any set of rational numbers is a ...
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1answer
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Is $[0,1] ∩ Q$ open or closed in real numbers?

According to https://en.wikipedia.org/wiki/Closed_set, " $(0,1) ∩ Q^c $ is not closed in the real numbers". Why is it so? I think that its complement is $(-∞,0)∪ (1,∞) ∪ Q^c$ is open and all points ...
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Vector field in $\Bbb Q^2$ and $\Bbb R^2$

I'd like to define a weak vector field in $\Bbb R^2$ that is tangential to a family of sine curves at each point. I define the family of sine curves as: $A_n\sin(x)$; $n=1,2,3,...$ and $A_n$ is a ...
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4answers
74 views

$18a$ and $25a$ both integers, then so is $a$

Let $a\in \mathbb{Q}$ such that $18a$ and $25a$ are integers, then we wish to prove that $a$ must be an integer itself. What that means is that $a=\frac{p}{1}$ where $p \in \mathbb{Z}$. What we do ...
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0answers
91 views

Finding a minima for a linear form with integer coefficients

Some context. This question is aiming to fill gaps in a larger proof, so in a way, it is kind of related to this two other questions (this one and that one) that I asked earlier. But since the ...
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1answer
31 views

What is this intersection of dense open sets?

Let $(r_k)_{k \in \mathbb N}$ be an enumeration of $\mathbb Q$ and $U_{1/p} := \cup_{k \in \mathbb N}(r_k - \frac{1/p}{2^k}, r_k + \frac{1/p}{2^k}) \subset \mathbb R$ What is $\cap_{p \in\mathbb N^*}...
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2answers
108 views

A linear form can not be too small on integer points

Notations. Let $\xi=(\xi_1,\xi_2,\xi_3,\xi_4)\in\mathbb( R\setminus\mathbb Q)^4$ such that $\xi_1\xi_4-\xi_2\xi_3\ne 0$ and the $\xi_i$ are linearly independent over $\mathbb Q$. I have the ...
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0answers
27 views

Making both $a + \sqrt{b}$ and $1/a + \sqrt{b}$

For which positive integers $b$ is there a nonzero real number $a$ such that $a + \sqrt{b}$ and $1/a + \sqrt{b}$ are both rational? I know there are a few trivial cases, such as when $b$ is a perfect ...
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1answer
33 views

Tensor product of finite group with group of rational number.

The tensor product of the finitely generated abelian group G with a rational number is zero if and only if G is finite. is it true for any finite group? For example, we take the tensor product of the ...
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1answer
56 views

Prime and rational numbers

I came across the following question while studying that is stumping me. Can anyone please help me solve it? Let "$a$" be a prime number greater than $10,000$ and let $x=\sqrt{a}$. Which of the ...
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1answer
111 views

Proof for rationality of a number

Let $a, b, c\neq 0$ be rational numbers such that $\sqrt[3]{ab^{2}}+\sqrt[3]{bc^{2}}+\sqrt[3]{ca^{2}}\neq 0$ is a rational number. Prove that $$\sqrt[3]{\frac{1}{ab^{2}}}+\sqrt[3]{\frac{1}{bc^{2}...
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2answers
31 views

Question about the validity of a group of rational numbers

The question asks the following: I know that I need to prove the four criteria of a group (closure, associative, identity, and inverse), but I'm having a hard time with closure. When you input ...