# 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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### 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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### 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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### 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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### 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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### 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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### $a \in \mathbb{R}$ 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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### 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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### Prove that $\sqrt{2}+\sqrt{4}$ is irrational [duplicate]

As the question says, how can I prove that $\sqrt{2}+\sqrt{4}$ is irrational? I have tried setting it to be equal to $a$, and $\sqrt{4}$ equal to $a^2$, but I haven't gotten anywhere. The ...
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### 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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### 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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### 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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### 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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### 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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### 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 $\Bbb Q^{\times}$ ...
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### 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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### 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 ...