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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1answer
39 views

Countability of $\mathbb Q\cap[0,1]$ [closed]

I do not understand why this set is countable $$\mathbb Q\cap[0,1]$$
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1answer
37 views

Can the set of positive rational numbers be decomposed into two non-empty, disjoint parts such that closed by the addition for? [closed]

I can't get started, and I'm pushing for a deadline. I can't start, Thank you! Can the set of positive rational numbers be decomposed into two non-empty, disjoint parts such that closed by the ...
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1answer
17 views

Formula for transforming periodical decimal expansion into fraction

How to prove $0.a_1a_2...a_n\dot{b_1}b_2...\dot{b_k}$ is equal to $\frac{a_1a_2...a_nb_1b_2...b_k - a_1a_2...a_n}{99...9 \cdot 10^n}$ ($\text{exatcly}$ $k$ $\text{nines}$) for any $a_i, b_j \in \...
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0answers
21 views

Identify the rational equation, rational inequality, rational function

whether the given is a rational function, rational equation and rational inequality. 1.) y = 5x2 – 2x + 1 2.) – 8 = 3.) = 4 4.) = x2 5.) g(x) = 6.) 6x - Instruction: Solve the following rational ...
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1answer
10 views

Simplification of complex rational?

I was looking at a problem in a textbook and found the following simplification: $$\frac{1}{x+i\omega} * \frac{2y}{y^2 + \omega^2} = \frac{2}{xy} * \frac{1}{1+i\frac{\omega}{x}}*\frac{1}{1 + (\frac{\...
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1answer
56 views

Find rational $a,b$ with $a\ne b$ such that $x²-bx-\frac{a³-b³}{3b}=0$ has rational solutions

I was working on the generalization of a geometry problem, which resulted in the quadratic equation $$x^2 - bx - \frac{a^3 - b^3}{3b}= 0$$ Its solutions are $$-\frac{b}{2} \pm \sqrt{\frac{4a^3 - b^3}{...
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1answer
24 views

A Symmetric Numeric System of Multiplicative Inverses?

I was just thinking about how strange it is that multiplicative inversion bifurcates the rational numbers into such asymmetric segments. Additive inversion bifurcates the number line cleanly into a ...
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1answer
29 views

Do we come to a contradiction if we accept completeness axiom for rational numbers?

I found the following formulation of the completeness axiom: If two non-empty sets ($A$ and $B$) are such that for any two elements from the first and second sets ($a$ and $b$, respectively) it is ...
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0answers
19 views

Automorphisms of a structure of non-zero rational numbers with multiplication [duplicate]

I'm given the structure $\mathcal{M} = (\mathbb{Q} \setminus \{0\}; =; \cdot)$ and I need to prove that Aut $\mathcal{M}$ is continual. In other words, first of all, I want to find all (or at least ...
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1answer
62 views

Does $\operatorname{Re}(a+bi)^{n}=\operatorname{Re}(a+bi)^{n+1}$ have at least one solution $a,b\in\mathbb{Q}$, for all $n\in\mathbb{N}$?

Does $$\operatorname{Re}((a+bi)^{n})=\operatorname{Re}((a+bi)^{n+1})$$ have a solution $a,b\in\mathbb{Q},|a|\neq |b|\neq0$, for all $n\in\mathbb{N}$? I was working on my previous question, $\...
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0answers
25 views

Finding small rational solution to under-defined linear equations.

I suspect this is a well studied problem with multiple solution, and I just don't know what terms to search for. If I knew what to search for, I suspect I could figure out how to apply those known ...
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2answers
67 views

How to show that $\big\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\mid a,b,c,d\in \mathbb{Q}\big\}$ is a field. [closed]

How to show that $A=\big\{a+b\sqrt{2}+c\sqrt{3}+d\sqrt{6}\mid a,b,c,d\in \mathbb{Q}\big\} \subset \mathbb{R}$ is a field? In particular, I am wondering how to show that $A^{*} = A \setminus \{0\}$, i....
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1answer
41 views

Prove that there is no rational root of $x^3+x+1=0$ by contradiction

I have found that this equation only has one root between $-1$ and $0$. I try to let this root=$p/q$ where $gcd(p,q)=1$ and get $p^3+pq^2+q^3=0$. I want to know how to prove it by the method of ...
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5answers
45 views

$a \in \mathbb{R}$ and $a^7, a^{12} \in \mathbb{Q}$. Prove that $a \in \mathbb{Q}$.

Let $a \in \mathbb{R}$ and $a^7, a^{12} \in \mathbb{Q}$. Prove that $a \in \mathbb{Q}$. my solution: $a^7 \in \mathbb{Q} \Rightarrow a^{14} \in \mathbb{Q}$ $a^{14}=a^{12}\cdot a^{2}\in \mathbb{Q}$, ...
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0answers
93 views

The orthogonal behaviour of a subset of vector spaces

Notations. Let $n\ge 4$ be an integer, $d,e\in\{1,\ldots,n-1\}$ and $j\in\{1,\ldots,\min(d,e)\}$. We say that a subspace of $\mathbb R^n$ is rational if it admits a rational basis. We denote by $\...
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3answers
101 views

If A is dense in (0, 1), then A is dense in R [closed]

Is it true or false? Explain please. Thanks. Dense definition: Let $∅ ≠ A, B ⊆ R$. We say that $A$ is dense in $B$, if $A ⊆ B$, and, in addition, if for every $x, y ∈ B$ such that $x < y$ there ...
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0answers
44 views

Does the group of rationals under addition have a basis?

I'm trying to determine whether $\mathbb Q$ under addition has a basis. My naive idea, based on what I have covered so far, would be to try and show that if we assume $\mathbb Q$ has a basis then it ...
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3answers
67 views

Irrational Number $\pi$ and $\dfrac{22}{7}$ [closed]

If some no. after decimal is NRNT then it may be irrational the condition being it should not be near any other irrational no. Leaving the condition of expressing in $\frac{p}{q}$ formate (which is ...
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2answers
69 views

Proof that $\mathbb Q$ is not cyclic. [duplicate]

I tried to show that $\mathbb Q$ with addition is not a cyclic group. Here is the proof: If possible assume that there is $a\in \mathbb Q$ such that $\mathbb Q=\langle a \rangle$. Note that $a\neq 0$. ...
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2answers
80 views

Does the definition of rational numbers already assumes an understanding of them?

I mean, how do we know that the correct equivalence relation is the equality of the crossed products? Does it already assume an intuitive informal understanding of rationals? If yes, what would that ...
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1answer
26 views

Give an open cover of $[0,1]\cap\mathbb{Q}$ that has no finite subcover

I'm stuck on this one. We know that $[0,1]\cap\mathbb{Q}=\{\frac{a}{b}, a,b \in \mathbb{N}:\frac{a}{b}\in[0,1]\}$ Thus, they look something like $\{0,\frac{a_1}{b_1},\frac{a_2}{b_2},...,1\}$ Pick $x\...
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1answer
31 views

Proof a function is injective - (intermediary step to proving the rationals are countable)

I am proving the rationals (greater than 0) are countable. I have to prove a function is injective and I just wanted another opinion on whether my proof is correct or if there is an even better method ...
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2answers
94 views

Prove that $\sqrt[3]{2}+\sqrt[3]{4}$ is irrational [duplicate]

As the question says, how can I prove that $\sqrt[3]{2}+\sqrt[3]{4}$ is irrational? I have tried setting it to be equal to $a$, and $\sqrt[3]{4}$ equal to $a^2$, but I haven't gotten anywhere. The ...
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2answers
84 views

Some alternative examples to the question “Are there two irrational numbers $x$ and $y$ such that $x^y$ is rational?”

This question is a classic and is on Stack Exchange several times, but I am looking for some atypical answers. The basic question, as you all already know is, "Find two irrational numbers $a$ and ...
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1answer
26 views

Proving that rational numbers cannot have denominator zero (without referring to division by zero)

Given the set $S$ of all pairs $(a, b),$ with $a, b \in \mathbb{Z},$ the relation $Q$ on $S$ is defined by $(a,b)Q(c,d) \iff ad=bc$. How can I prove that $b$ cannot be equal to zero, without using the ...
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3answers
46 views

From $3$ chocolate bars we can make at most $5$ chocolate rabbits. What is the greatest number of rabbits that can be made from $16$ bars?

I'm using some questions from the 2019 International Math Contest to help my students prepare for the Mathcounts competition. While the IMC gives me the answers, I like to give my students the worked ...
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5answers
46 views

Prove that there is no rational $x$ such that $x-1 = 1/x$

As said in title, I need to prove that there doesn't exist a rational number $x$ that satisfies $x-1 = 1/x$. I remember doing something like this a while back at school but I can't recall how to do it....
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1answer
16 views

Measure of the set $x\in [0,1]$ such that $|x-p/q|>(Aq^2)^{-1}$ for any rational $p/q$

Is there a constant $A$ such that the measure of the set $x\in [0,1]$ such that $|x-p/q|>(Aq^2)^{-1}$ for any rational $p/q$ is equal to 1? While trying to find the answer to this question, I found ...
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0answers
45 views

Number of group homomorphism from the group $D_8$ to $\Bbb Q^{\times} $

How does one determine the number of group homomorphism from $ D_{8}$ to $\Bbb Q^{\times}$? For any homomorphism $f$, $o(f(x))$ divides $ o(x) $, in case of both being finite. Here $\Bbb Q^{\times}$ ...
2
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1answer
49 views

Prove that $\mathbb{Q}^n$ is a countable dense set in $\mathbb{R}^n$ [duplicate]

Prove that $\mathbb{Q}^n$ is a countable dense set in $\mathbb{R}^n$. The pre-requisites are $\mathbb{Q}$ is a dense subset of $\mathbb{R}$ and $\mathbb{Q}$ is countable. I need to show $\mathbb{Q}^...
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4answers
81 views

Prove that there is no rational lowest upper bound for $\sqrt{3}$.

I am trying to prove this and have looked at similar questions to gauge how to approach this. I have: Suppose that there exists a smallest rational number greater than $\sqrt{3}$. We shall call that ...
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0answers
28 views

Show that a set of open intervals can be formed by the countable union of sequences of rationals.

I recently decided to get into probability theory. I am following Jacod and Protter, "Probability Essentials". One of the first theorems (T.2.1) uses the following : Let $C$ denote all open ...
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1answer
58 views

Find the values for which positive integer $n$ makes $A=\sqrt{n(n+182)}$ a rational number

Find the values for which positive integer $n$ makes $A=\sqrt{n(n+182)}$ a rational number I tried to solve it in the following way: $n(n+182)=k^2$ where k is an integer $n^2+182n=k^2$ $(k-n)(k+n)=...
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2answers
35 views

If $b \in \mathbb{R}\setminus \mathbb{Q}$ then are there $h$ very close to $0$ such that $b+h \in \mathbb{Q}$?

This question occured to me when I was solving a problem in analysis. Rephrased in a different way: Is it true that given an $b \in \mathbb{R}\setminus \mathbb{Q}$, for all $\varepsilon >0$ there ...
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0answers
64 views

Prove/disprove that irrational - irrational is always irrational

could you help me? I am trying to prove / disprove that the subtraction of two irrational numbers is irrational Let $a,b\in\mathbb{R}\setminus\mathbb{Q}$ i want to prove that exists some $i\in\mathbb{...
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0answers
63 views

For real $x$, prove that $x\in\mathbb{Q}\iff-x\in\mathbb{Q}$

Let $x \in \mathbb{R}$. Prove that $x\in \mathbb{Q}\iff-x\in\mathbb{Q}$. So I am having trouble coming up with an approach to figure out this proof. I feel as if I have the basics covered. I was ...
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2answers
38 views

Prove that $2\times3 = 6$ using Dededkind cuts

I'm reading Classic Set Theory by Goldrei, and in Exercise 2.10, after defining real multiplication using Dedekind cuts, I'm asked to prove: Show that $2 +_{\mathbb{R}} 3 = 5$ and $2 \cdot_{\mathbb{R}...
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0answers
17 views

Values of positive integer $n $ such that we would have a polynomial bijection between $ n^{\mathbb{Q}\times\mathbb{Q}} \rightarrow n^\mathbb{Q}$?

Inspired by this question ,I want to know if we can find such positive integer $n$ such that we would have $f\colon n^{\mathbb{Q}\times\mathbb{Q}} \rightarrow n^\mathbb{Q}$ is a bijection with $f(x,...
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0answers
37 views

function $f : \mathbb{R} \to \mathbb{R}$ that $\mathbb{lim}_{x \to c} f(x) $ exists only when $c$ is irrational [duplicate]

The following is an exercise in Real Analysis and Foundations(4th ed, Steven G. Krantz, 105p). Give an example of a function $f : \mathbb{R} \to \mathbb{R}$ so that $\mathbb{lim}_{x \to c} f(x) $ ...
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4answers
125 views

Is there any way to prove that $\sqrt {n-1} + \sqrt n + \sqrt {n+1}$ is irrational? [closed]

Before this is marked as a duplicate I just want to say that I've already read a similar thread, where the original poster asked how they would prove that $\sqrt 2 + \sqrt 5 + \sqrt 7$ is an ...
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2answers
87 views

Can any at most countable metric space be embedded topologically in $\mathbb{Q}$?

Theorem. (Sierpiński) Any countable metric space without isolated points is homeomorphic to $\mathbb{Q}$. Let $\mathbb{N} = \{1, 2, ...\}$, $\mathbb{Q}_- = \{r\in \mathbb{Q}: r<0\}$. If $X$ has a ...
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1answer
52 views

Find $a$ for $\sqrt{\frac {9a + 4}{a - 6}} = n$, with $a \in \Bbb Z$ and $n \in \Bbb Q$.

The problem: Find all values of $a$ such that $\sqrt{\frac {9a + 4}{a - 6}} = n$, with $a \in \Bbb Z$ and $n \in \Bbb Q$. What i tried: I arrived to this: $$ n^2 = \frac {58}{a - 6} + 9 $$ i tried ...
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1answer
27 views

Given a number between 0 & 1, knowing that the decimal expansion terminates, how could you find out the number of decimal places?

Pretend someone hands you a real number between 0 and 1 (not including 0 or 1). All you know is that its decimal expansion terminates. What could you do to determine the number of decimal places in ...
2
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1answer
33 views

Naive height of zero

The naive height function of a rational number $x=\cfrac{m}{n}$ (in lowest terms) is defined as $$H(x) = H\left(\frac{m}{n}\right) = \max\{|m|, |n|\} $$ However, $0$ can be denoted by $\frac{0}{1}, \...
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1answer
65 views

Proving that a number is rational $(210.12)?$

I am a little lost regarding how to prove that a number is rational. I am given a number $210.12$, and I have to prove that it is rational. Looking at the definition of rational numbers, it can be ...
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1answer
47 views

Describe the quotient group $\mathbb{Q}^{*}/ (\mathbb{Q}^{*})^{2}.$ [duplicate]

In an abelian group $G,$ let $G^{2} = \{g^2 \mid g \in G\},$ which is a subgroup. Let $\mathbb{Q}, \mathbb{R}$ and $\mathbb{C}$ denote the rational, real and complex fields, respectively. Let $\mathbb{...
1
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1answer
69 views

How to find out the type of automorphism of this field extension of Q?

Let $\alpha$ = $2^{1/5}$ $\in\mathbb R$ and $\xi$ = $e^{2\pi i/5}$. Let $K=\mathbb Q[\alpha , \xi]$. Pick the correct statements from below: There exists a field automorphism $\sigma$ of $\mathbb C$ ...
12
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2answers
341 views

Convergence of $\sum_{n}\frac{q_n}{n}$, where $(q_n)$ enumerates $\mathbb{Q}\cap[0,1]$?

This problem is from a book reviewed Sep. 2020 Notices of AMS. Rational numbers in $[0,1]$ are countable. Can they be ordered as $(q_n)_n$ so that $\sum_{n=1}^\infty \frac{q_n}{n}$ converges? My ...
2
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1answer
67 views

Group structure of this quotient group

I'm curious about the quotient group $\frac{\mathbb{Q}^* }{(\mathbb{Q}^*)^2}$. What do its elements look like? Is it isomorphic to some infinite group? ** For clarification on notation, $\mathbb{Q}^* =...
5
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1answer
106 views

Does there exist a right triangle such that all side lengths and angles in degrees are rational?

Note: I used degrees in the title for the sake of brevity, but am using radians in the body of the question for clarity. Sorry for any confusion this causes. Here's the question: are there any right ...

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