# Questions tagged [irrational-numbers]

An irrational number is a real number that cannot be expressed in the form $\frac{n}{m}$ where $n$ and $m$ are integers.

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### Irrational numbers [closed]

1.There exist $\alpha_1,..., \alpha_n$ irrational numbers greater that $1$, such that $\{\alpha_1^k\}+...+\{\alpha_n^k\} \ge cn$ for $c>0$ and $n, k$ positive integers sufficiently large 2.there ...
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### What is known about constructively irrational numbers?

Intuitively, a constructively irrational number is one for which we can effectively separate it from any rational number in terms of the latter's denominator. More formally, a constructively ...
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### The power of irrationality: ${\rm sin}(x) + {\rm sin}(\pi x)$ [closed]

I'm a physicist studying undulatory phenomena. Reducing the problem, I find the issue lies on the relative irrationality of the angular frequencies of two superimposed waves. To state it simply, ...
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### System of two delayed functional equations with dilatations

Consider the operator $S:L^{2}(0,s+r)\longrightarrow L^{2}(0,1)^{2}$ defined by \begin{equation*} Su(x)=\left( au(xs)+bu(xs+r),cu(xr)+du(xr+s)\right) , \end{equation*} where $a,b,c,d$ are positive ...
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### Irrational rotations are rank 2 by intervals without spacers

Let $\alpha$ be an irrational number, and $R_\alpha$ be the rotation by $\alpha$, that is $R_\alpha(x)=x+\alpha\bmod 1$. S. Ferenczi in his survey [Systems of finite rank. Colloq. Math. 73 (1997), no. ...
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### Algebraic and rational parts of a real number

Let $\alpha$ be a positive real number. Does it make sense to define the closest rational to $\alpha$ as the number $R(\alpha)=\frac{p_1}{p_2}$ such that $p_1,p_2$ are positive co-prime integers ...
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### Looking for a proof that $\sqrt2 + \sqrt5 + \sqrt3$ is irrational [closed]

What is the easiest way to prove that $\sqrt2+\sqrt5+\sqrt3$ is irrational?
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### Illuminating a just-barely irrational polygon

As has been discussed earlier on MO,1,2 recently an impressive advance was proved concerning internally illuminating a mirrored polygon. Here is the result: Corollary 3. Let $P$ be a rational polygon....
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### When does the set of possible walk lengths start being $\varepsilon$-dense?

Let $\Gamma$ be a finite directed graph, and suppose each directed edge $e \colon a \to b$ has a positive real length. Suppose given vertices $x, y \in \Gamma$, and suppose there are infinitely many ...
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### An alternative to continued fraction and applications

This post is inspired by the Numberphile video 2.920050977316, advertising the paper A Prime-Representing Constant by Dylan Fridman, Juli Garbulsky, Bruno Glecer, James Grime and Massi Tron Florentin, ...
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### Is there a real valued function whose limit exists only on irrational numbers?

I have been trying to find a function $f : \mathbb R \to \mathbb R$ such that $\lim_{x \to c} f(x)$ exists when $c$ is irrational and the limit doesn't exist when $c$ is rational. I tried variations ...
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### Picture of Lambert's proof that $\pi$ is irrational?

With a suitably generous notion of "picture proof" or "proof without words" or "geometric proof," there do exist such proofs of the irrationality of square roots and even ...
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### Is the super square root of $2$ irrational? [closed]

The super square root of $n$ is the solution/solutions to $x^x=n$. Is the super square root of $2$ irrational?
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### Mapping $\mathbb P$ onto $\mathbb Q ^\omega$

Let $\mathbb P$ denote the space of irrationals. Is there a continuous bijection (one-to-one and onto) $f:\mathbb P\to \mathbb Q ^\omega$ that maps each closed subset of $\mathbb P$ to a $G_\delta$-...
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### Irrationality measure of arctan(1/3)

I recently came across the concept of the irrationality measure. It really fascinated me and when I was looking for known values $\mu(x)$ for mathematical constants $x$, I also came across this paper: ...
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### Quotients of the irrationals

Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
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### Possible Birkhoff spectra for irrational rotations

Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
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### Examples for a Golomb's result, and rationals as $\sum_{n\geq 1}\frac{|G_n|}{P(n)}$, where $G_n$ are Gregory coefficients and $P(x)$ a polynomial

After I was stuying the first pages of a chapter of the book , in particular the statement of Corollary 10.3 and its proof, I wondered what can be interesting examples of irrational numbers that ...
### Looking for a proof that $\pi$ is irrational using a series representation for it
This have been asked on MSE but got no answers. I'm searching for a proof that $\pi$ is irrational using a series representation for $\pi$, but can't find it. However, on this wikipedia page show'...