# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ?
### 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 $u_n=(1+\frac{1}{n})^n$ ...