# 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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### Discretized Distributions on Rationals?

Consider the measure space $(\mathbb{Q}, 2^{\mathbb{Q}}, \nu)$, with $\nu$ being the counting measure. The space is then $\sigma$-finite. Is there any attempts made to define analogues of continuous ...
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### Closed form for the area under $f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)}$

Define a function $f:\Bbb Q \to \Bbb Q$ by the following $$f(x):=\lim_{N \to \infty}\frac{\pi(Nx)}{\pi(N)}$$ where $\pi(\cdot)$ is the prime counting function and $N\in \Bbb N.$ I would like to find ...
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### Proving, using Dedekind cuts, that $C(0)$ is an additive identity for addition on cuts.

My source is Franck Ayres' Modern Algebra. The author states the fact under discussion (Chapter 7, "Real numbers") but does not provide a proof. My question is about the second part of the ...
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### Inquiries Regarding the Submodule Structure of $_{\mathbb{Z}}\mathbb{Q}$

As is well known, any cyclic submodules of $_{\mathbb{Z}}\mathbb{Q}$ are superfluous. Here is a proof. This raises the following questions: Are all proper submodules of $_{\mathbb{Z}}\mathbb{Q}$ ...
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### Does the set ${\displaystyle A =\{q\in \mathbb {Q} |q<a\}}$ have a maximum element?

Does the set A have a maximum element? ${\displaystyle A =\{q\in \mathbb {Q} |q<a\}}$ Thoughts: I don't think it does. We can suppose that it has a maximum element, q. Then q is a real number and ...
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