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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4answers
28 views

Can a rational sequence and an irrational sequence have same limit?

The original question: If we have a sequence of real numbers and on that, we define a sequence of $n^{th}$ rational numbers and another of irrational numbers, then they are both subsequences of the ...
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3answers
31 views

Rational Numbers and Sequences

Can the rational numbers be arranged in a sequence? If so, consider any such sequence of all the rational numbers. Show that every real number is a subsequential limit of this sequence. Since ...
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Unique polynomial of degree at most 4 with rational coefficients

I'm working on the following problem: Prove that there is a unique polynomial $f$ of degree at most $4$ with rational coefficients such that $f(1) = 1, f(2) = 2, f(3) = 4, f(4) = 8, $ and $f(5) = 16$...
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Is $\mathbb{Q}$ the smallest ordered field up to isomorphism?

Pretty simple question. Does there exist a ordered field smaller than (i.e. is a strict subset of) $\mathbb{Q}$? It seems like we can't go any smaller than $\mathbb{Q}$. Is this true? Why?
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4answers
331 views

Does there exist a nonzero ring homomorphism from the ring of square rational matrices to the ring of rational numbers?

I am wondering if it is possible to construct a nonzero ring homomorphism from $M_n(\mathbb{Q})$ to $\mathbb{Q}$. So far, I've been unsuccessful in constructing such a nonzero ring homomorphism. Is ...
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2answers
56 views

Prove square root of 16 is rational [closed]

I am student and i understand that the square root of any perfect square is a rational number but i'am trying to prove it (e.g for 16).
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0answers
65 views

how to prove $p$-adic numbers are complete? [closed]

''' I'm new to $p$-adic numbers and am trying to prove they're complete, can someone please show me how to do this? I have that for $a_1, ..., a_n$ a Cauchy sequence of elements of $Q_p$, how I do ...
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0answers
41 views

Find all positive integers k so that…

Let $$ P = \sqrt{\frac{3\times10^{n}}{k}} $$ Find all $k$ positive integers so that $P$ is rational and belongs to $(0,1)$ for all $n$ positive integers.
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2answers
39 views

Do everywhere discontinuous functions like these one described exist?

These are the properties that such functions should (could?) have: 1) $f(\mathbb R)=\mathbb R$ 2) $f$ is everywhere discontinuous 3) $\mathbb Q \subseteq f( \mathbb I)$ 4) $f(\mathbb Q) \subset \...
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2answers
30 views

Infimum and supremum of the set A\subset \mathbb Q

Find the infimum and supremum of the set: $$A=\{\frac{m-n-1}{mn+4m+3n+12}:m,n\in \mathbb N\}$$ My mind was drifting and I entered a circle by factorizing the algebraic fraction: $\frac{m-n-1}{mn+4m+3n+...
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1answer
24 views

Is solving a convex optimisation for variables in $\mathbb{Q}$ just as hard as $\mathbb{Z}$?

If $\mathcal{F}^\mathcal{D}$ is taken to be the feasible set in domain $\mathcal{D}$, a convex optimisation problem of the form: \begin{align*} \text{minimise} \quad & f(\textbf{x})\\ \text{...
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50 views

A solution for $A-A=\mathbb{Q}\setminus \{\pm1,\pm 2,\cdots,\pm m\}$

Let $m$ be a given natural number and consider the additive group of rational numbers $\mathbb{Q}$. We are looking for a subset $A\subseteq \mathbb{Q}$ such that $A-A=\mathbb{Q}\setminus \{\pm1,\pm 2,\...
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1answer
31 views

Group Isomorphism of Rational numbers under addition

Is $(\mathbb{Q} \times \mathbb{Q}, +)$ isomorphic to the group $(\mathbb{Q}, +)$? I have already found that neither groups are cyclic, however am unsure how to prove or disprove an isomorphism. I ...
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1answer
23 views

Put a vector to base of Vectors in Rational number

Base: $$((1,0,2), (2,1,1), (1,1,1)) $$ I found out, all of vectors are linear independent. So it creates base of $\mathbb Q^3$. So I need to put this vector: $$(3,2,-3) $$ And put this vector to the ...
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4answers
63 views

Find $x$ so that rational function is an integer

Find all rational values of $x$ such that $$\frac{x^2-4x+4}{x^2+x-6}$$ is an integer. How I attempt to solve this: rewrite as $x^2-4x+4=q(x)(x^2+x-6)+r(x)$. If we require that $r(x)$ be an integer ...
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2answers
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Compatibility in Real numbers?

I'm reading Good Math by Mark Chu-Carroll and it says, "$\le$" is compatiable with "+" and "*": ... if x $\le$ y, then for all z where 0 $\le$ z, (x * z) $\le$ (y * z). if x $\le$ y, ...
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2answers
182 views

Is there a monotone bijection between the rationals of two intervals? [closed]

Given two intervals $I$ and $J$ (both open or both closed), do there always exist a monotone bijection between $I\cap \mathbb{Q}$ and $J\cap \mathbb{Q}$?
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1answer
36 views

Prove that there are no rational solutions

Prove that $2r^4 +20r^2 = 15r^3 + 15r - 6$ has no rational solutions without solving for $r$. My first thought was using remainders upon division, but I'm not sure how to apply this with variables.
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Linear dependent/independence on polynomial ring

On polynomial ring over R, decide about linear dependent/independence. $$(x^2+x+1,x^2+2x,x^2+2)⊂R[x]$$ In this case I dont really know how to start... I know what is linear dependent - but I cant see ...
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1answer
94 views

Rational with minimal denominator between two rationals [duplicate]

My question from an easy problem. $p,q$ are positive integers such that $$ \frac{5}{9}<\frac{p}{q}<\frac{4}{7} $$ find $p,q$ such that $q$ is the smallest number that satisfies this ...
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The least upper bound of $\{ x \in \mathbf{Q} \mid x^2 < 2 \}$ is non-rational [duplicate]

Is it possible to prove that the set $\{ x \in \mathbb{Q} \mid x^2 < 2 \}$ has no least upper bound in the rational numbers, but without assuming the existence of real numbers? It seems like I ...
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1answer
72 views

Prove that between any two irrational numbers x<y there is a rational number [closed]

This question provides answers using theorems/methods I have not learnt and dont understand. Is there a rational number between any two irrationals? I completely stuck at the step: pick n ∈ N large ...
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1answer
61 views

relation between rational and irrational, non-transcendental numbers?

a long time ago, when watching a video about continued fractions, I saw something interesting, all continued fractions in that video (all that were non-transcendental) had a rational-looking fraction. ...
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3answers
834 views

Probability of a number being rational

If $x \in [0, 1]$, what is $\text{P}(x\in \mathbb Q)$? In other words, what is the probability that $x$ is rational? This is what I tried: $$\begin{array}{rcl}\text{P}(x \in \mathbb Q) &=&...
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4answers
174 views

Intuition behind Diophantine approximation: why do we express the bound as function of denominators?

The field $\mathbb Q$ is dense in $\mathbb R$, so we can approximate a real number $\alpha$ by an arbitrarly close rational number $r=\frac{a}{b}$. The purpose of Diophantine approximation is to find ...
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3answers
130 views

Proving that the set of rational numbers between 0 and 1 is disconnected.

Show that the set of rational numbers between 0 and 1 $(A = \mathbb{Q} \cap [0,1])$ is disconnected. Note that $A \subseteq \mathbb{R}$ is a subspace topology. Definition of disconnectedness: ...
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1answer
21 views

Need help solving word problem with Negative integers involving descent - would greatly appreciate it.

This is a question from my son's test. He got (a) correct but (b)wrong. He doesn't understand why. Would appreciate an explanation that can help him understand his mistake. A submarine is at −750 ...
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1answer
34 views

Circular definition of rationals.

If we define rational numbers as A rational number is any number that can be fraction $\frac pq$ of two integers $p$ and $q$, with the denominator $q$ not equal to zero. But integers themselves ...
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1answer
35 views

Significant figures problems

Between my first assessments I met some exercices that I don't know how to do them because I did not understand well the rules or maybe I applied in a wrong manner. For instance: The following ...
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1answer
60 views

Proving an inequality between the difference of $\sqrt2$ and any rational number. [duplicate]

Let $a = \sqrt{2}$ Prove that for every $m,n\in N$ $|a - \frac{m}{n}| \gt \frac{1}{(2\sqrt2+1)n^2}$ Hint: Consider $|a - \frac{m}{n}|\geq 1$ and $|a - \frac{m}{n}|\leq 1$ as separate cases and ...
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1answer
38 views

Absolute value in rational numbers

We define the absolute value in $\mathbb{Q}$ as an application $||\, \cdot \, || : \mathbb{Q}\rightarrow [0,\infty )$ that fulfills the properties: $||x||=0$ if and only if $x=0$. $||xy||=||x||\, ||y|...
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1answer
84 views

For integer $n>1$, can $\sum_{k=1}^{n}\sqrt{k}$ be a rational number? Can it be an integer? [duplicate]

I know that the sum of two (or more) irrational numbers can be rational. For example, both $\sqrt{2}$ and $1-\sqrt{2}$ are irrational numbers, but their sum is rational. Also I know that $\sqrt{m}$ ...
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2answers
47 views

Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$.

Let the rational number $p/q$ be closest to but not equal to $22/7$ among all rational numbers with denominator $< 100$. What is the value of $p − 3q$ ? My approach: $22/7=3.14$, therefore, $p/q=...
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2answers
70 views

Prove the sum of two rational number is equal to $\frac{e}{lcm(b,d)}$ for some integer $e$.

As title state: $\frac{a}{b} + \frac{c}{d}=\frac{e}{lcm(b,d)}$ for some integer $e$. Here is what I tried: $\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}$ Since $gcd(b,d)lcm(b,d)=bd$, so I got $\...
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1answer
69 views

Ordering of rationals

Let $(\mathbb{Q},<)$ be the usual ordering of rationals. Show that there is a family $\mathcal F$ of subsets of $\mathbb{Q}$ such that $|\mathcal F|=2^\omega$ and for every $A, B \in \mathcal F, (A,...
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2answers
38 views

Generalising a problem when two fields $F \ncong K$

I am trying to generalize a problem that I came across previously. $\mathbf{ Problem:}$ Are the fields $\mathbb{Q}$ and $\mathbb{Q[\sqrt2]}$ isomorphic? $\mathbf{Generalisation:}$ Let $F$ and $K$ ...
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1answer
41 views

To prove that $\mathbb{Q}$ is the smallest subfield of $\mathbb{C}$

Assumption: There exsits $F$ which is a subfield of $\mathbb{C}$ such that $F\subsetneq \mathbb{Q}$. Claim: $\mathbb{Z}\subset F$. Proof: Let $m \in \mathbb{Z^+ }$. We know, that $1 \in F$. Taking $...
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2answers
41 views

Proof that the set of rationals is countable with finite preimages?

I'm working through the proof that the set of Rational numbers is countable and the proof says in order to do this you just have to show every rational number can be mapped to the set of natural ...
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3answers
52 views

Fractions that have interesting, fun or noteworthy decimal expansions

I'm looking to discover more fractions that have interesting* decimal expansions. (I'm asking out of curiosity, there is no particular academic reason as far as I'm concerned). Here are a few ...
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2answers
117 views

Optimal division on $\mathbb{Z} $

I am trying to understand the construction of $\mathbb{R}$ with slopes / quasi-isomorphism, as shown here at some point, the following property is used : $$\forall p \in \mathbb{Z},\forall q \in \...
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1answer
46 views

Showing that $\mathbb{Q}(\sqrt{2}) \subseteq \mathbb{Q}(\sqrt{2}+\sqrt[3]{2})$

I am doing some early study in field theory and am stuck on the following problem. Show that $\mathbb{Q}(\sqrt{2}) \subseteq \mathbb{Q}(\sqrt{2}+\sqrt[3]{2})$ and that $\mathbb{Q}(\sqrt[3]{2}) \...
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1answer
34 views

Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion?

Why $\mathbb{Q}$-vector space has no $\mathbb{Q}$-torsion ? What does the notion of $\mathbb{Q}$-torsion technically mean ?
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0answers
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Proof that $(1+\frac{1}{n})^n$ can't converge to a rational number

One of my colleagues challenged me (and his students) with the following: "Assume you don't know that $\lim_{n\to +\infty}(1+\frac{1}{n})^n=e$. Prove the sequence $u_n=(1+\frac{1}{n})^n$ ...
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1answer
34 views

Why does the definition for the multiplication of dedkind cuts explicitly include the negative rationals?

If A and B are both dedekind cuts. Then $A \times B=\{ab \mid a \in A, b \in B, a \geq 0, b \geq 0 \} \cup \{q \in \mathbb{Q} \mid q <0 \}$. Can someone explain why this definition doesn't work: $...
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2answers
41 views

The smallest positive integer vector from a positive rational vector

Suppose $\mathbf{q} = \left[\begin{array}{cccc}q_1 & q_2& \dots &q_n\end{array}\right]\in \mathbb{Q}_{>0}^n$ is a vector of positive rational numbers with relatively prime numerator and ...
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1answer
68 views

Constructing a non-empty perfect set of real numbers that does not contain rationals.

Duplicate: Perfect set without rationals My approach: We consider the set $[e, \pi]$. I am trying to "cover" the rationals by enclosing each one of them by open intervals with irrational endpoints, ...
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0answers
163 views

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial?

Is $z=e^{\frac{1}{\log(x)}}$ a solution to a polynomial? $x\in\Bbb Q~\cap(0,1).$ Wolfram is telling me $z$ is algebraic but I'm not sure that I believe this. I believe it would follow from Schanuel'...
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3answers
74 views

Is even divided by even a rational or irrational number? [closed]

For any rational number, $\frac{p}{q}$ , $p$ and $q$ should be integers, $q\neq0$ and $p,q$ should not have any common factors. Now, if we have two even numbers, say $2m$ and $2n$ where $m$ and $n$ ...
0
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1answer
52 views

Rational topological basis for Euclidean topology - topology without tears 2.2.3

Context: self-studying topology without tears, now at question 2.2.3. Question: Let $\mathcal{B}$ be the collection of all open intervals $(a,b) \in \mathbb{R}$ with $a \lt b$ and $a, b$ rational ...
8
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2answers
315 views

Collection of intervals covers $[0,1]$?

For each $n=1,2,3,...$ and each $m=0,1,2,...,n-1$, let $$K^n_m = \left[ \frac{3m-n+1}{n} , \frac{3m-n+2}{n} \right] \subset (-1,2). $$ I am struggling these with two questions for quite some time: (...