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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A sequence of rational numbers having a limit that is an irrational number

Let $T = \{ t \in \mathbb{Q}: 0 \leq t \leq 1 \}$. With the parameterization of the unit circle given by $x(t) = \frac{1-t^2}{1+t^2}$ and $y(t) = \frac{2t}{1+t^2}$. With an indexing of the set $T$ by ...
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Construction in Ciesliski's proof of Sierpinski’s Topological Characterization of Q

An elementary proof of the fact that any countable metric space $(X,d),$ without isolated points is homeomorphic to $\mathbb Q$ uses a construction that I thought I was familiar with, but I'm having ...
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Why the proof of given two rational numbers their sum is rational involves the sum of both numbers. Wouldn't this be a contradiction?

In this question why doesn't proof of sum of two rational number is rational not proving the irreducibility of fraction $\frac{ad+bc}{bd}$?, the author gives a proof of why given two rationals the ...
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Are the rational numbers in order of size a sequence?

I am struggling through Abbott's Understanding Analysis and have been asked if there is a sequence that contains subsequences that converge, severally, to each term of the harmonic sequence. The ...
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Infinite irreducible polynomial over Q[x] using Eisenstein

Like the title describes, I know that over Q for each number n ≥ 1, one can easily construct infinitely many irreducible polynomials of degree n. But I want to prove using Eisenstein's criterion this ...
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For which $f: \mathbb{Q}\rightarrow\mathbb{Q}^+$ does the sum $\sum_{q\in\mathbb{Q}} f(q)$ converge?

BACKGROUND: This is not a homework question -- I am not even in school. I am purely interested in the question itself. I actually asked this question in my second semester of real analysis during my ...
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How do you find "good" rational approximations to a decimal number?

When presented with real number as a decimal, are there any methods to finding "good" rational approximations $a/b$ to that number? By "good" I mean that $a$ and $b$ are reasonably ...
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Is there a proof that logarithm whose anti-logarithm is not a natural power of the base is not a rational number?

To write it simplier (at least for me): Is there a proof that $\forall a,b\in\mathbb{N}_+\backslash{\{1\}},b>a, \text{a is prime}, \forall k\in\mathbb{N}, b \neq a^k: \\ \log_a b\notin \mathbb{Q}$ ...
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Prove or disprove a claim regarding irrational numbers

I am trying to prove the following claim: Let $0\leq n \in \Bbb Z$ and suppose that there exists a $k \in \Bbb Z$ such that $n=4k+3$. Prove or disprove: $\sqrt n \notin \Bbb Q$ . The problem I am ...
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Does the range of the correlation coefficient or the covariance belongs to $\mathbb{Q}$

I have a question that seems to me logic, but I haven't seen anywhere such a claim. Could we claim that the range of the correlation coefficient or the covariance belongs to $\mathbb{Q}$, namely in ...
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What is the reason for the existence of real numbers? Is it an artifact/side effect of our thought process?

The observation of lengths that can not be represented by rational numbers was noticed if I recall correctly by some Pythagorean disciple over applying the Pythagorean theorem on a triangle with side ...
Is $\frac{0}{\pi}$ rational?
In math lesson, our teacher showed this formula. $Q = \{\frac{a}b\}\land (a\land b\in Q)\land (b\neq 0)$ According to this formula, $\frac{0}\pi$ is... strange. You know, $\frac{0}\pi$ is 0 and 0 is ...