# 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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### 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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### 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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### 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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### Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions?

Does $y^2 = x^6 - 3x^5 + 3x^4 + 10x^3 + 3x^2 - 3x + 1$ have any rational solutions? I have some reasonable pre/post graduate Math skills but no access to Magma etc. I suspect there are none other ...
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### Do distinct infinities exist?

We all know that there are the infinite sets $Q$ and $R$. I have been taught, and also read that there is one to one correspondence between the elements of both sets, and thus both are regarded as ...
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### Counter example for Baire's Theorem

Theorem: Let $(X,d)$ be a complete metric space, and let $D_n, n\in \mathbb N$ be open, dense subsets of $X$. Then also $\bigcap_{n\in\mathbb N} D_n$ is dense in $X$. This statement is false if $X$ ...
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### Which of the statements are false?

I have this statement: Let $a, b, c, d \in \mathbb{R} -$ {$0$}, with $\quad acd> 0$. If $– 1 < \frac{a}{b} < \frac{b}{c} < \frac{c}{d} < \frac{d}{a} < 1,$ Which of ...
Define the collection of sets $\left\{Q_n\right\}$ as follows: $$Q_1 = \{0\};\quad Q_2 = \left\{\frac{1}{2}\right\} \cup Q_1; \quad Q_3 = \left\{\frac{1}{3}, \frac{2}{3}\right\} \cup Q_2;$$ \left\{\...