# 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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### Produce an explicit bijection between rationals and naturals?

I remember my professor in college challenging me with this question, which I failed to answer satisfactorily: I know there exists a bijection between the rational numbers and the natural numbers, but ...
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### How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals?

I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly ...
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### Is there a rational number between any two irrationals?

Suppose $i_1$ and $i_2$ are distinct irrational numbers with $i_1 < i_2$. Is it necessarily the case that there is a rational number $r$ in the interval $[i_1, i_2]$? How would you construct such ...
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### System of linear equations having a real solution has also a rational solution.

I saw this question Let $A ∈ M_{m\times n}(\mathbb{Q})$ and $b ∈ \mathbb{Q}^m$. Suppose that the system of linear equations $Ax = b$ has a solution in $\mathbb{R}^n$. Does it necessarily have a ...
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Claim: If $m/n$ is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better. My attempt at the proof: Let d be the distance between $\sqrt{2}$ and some estimate, s. So we have $d=s-\sqrt{2}... 6answers 5k views ### Why is$[0, 1] \cap \mathbb{Q}$not compact in$\mathbb{Q}$? [duplicate] Statement:$[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 ... 3answers 563 views ### Equality of positive rational numbers. I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these$3$... 1answer 153 views ### Proof that Epicycloids are Algebraic Curves? Epicycloids are most commonly described by the parametric equations,$x(t) = (R + a)\cos(t) – a \cos \left(\frac{R + a}{a} t \right),y(t) = (R + a)\sin(t) – a \sin \left(\frac{R + a}{a} t \right)...
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 $a$ ...
Let $r$ be a root of the polynomial $p(x)=(\sqrt{3}-\sqrt{2})x^3 + \sqrt{2}x-\sqrt{3}+1$. Find another polynomial $q(x)$, with all integer coefficients, such that $q(r)=0$.