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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Visualizing shapes of initial segments of $\mathbb Q$?

A posted answer to this earlier question of mine caused me to think that I had not asked the particular question that I actually wondered about, so that will appear here. A nonempty proper initial ...
Michael Hardy's user avatar
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Rational solutions to $\cos(\lambda \pi) = \cos^2(a\pi) - \cos(b\pi) \sin^2(a\pi) $, with $a,b \in \mathbb{Q}$

In a similar vein to this question, I am trying to understand the occurrence of rational solutions $\lambda$ to the following equation $$\cos(\lambda \pi) = \cos^2 (a\pi) - \cos ( b\pi ) \sin^2 \left(...
Mary_Smith's user avatar
3 votes
1 answer
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Evaluating the generalized continued fraction obtained from the factorization of a bivariate polynomial equation

Happy New Year, MO community! We need someone expert in Generalized Continued Fractions (GCFs), with a deep knowledge of the GCFs’ convergence properties, to solve the following problem. PROBLEM ...
Monk's user avatar
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21 votes
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Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?

Is it known whether $$\int_0^1 e^{-x^2} \, dx$$ is irrational? It is well-known that $\int_0^\infty e^{-x^2} \, dx=\frac{\sqrt{\pi}}{2}$ which is irrational, but what about the prior integral? Also, I ...
Matthew Albano's user avatar
1 vote
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Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$

This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted. Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
Steven Clark's user avatar
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Sobolev-type estimate for irrational winding on a torus

Let $\mathbb{T} = \{ (x, y) \in \mathbb{R}^2 \}/_{x \mapsto x + 1, y \mapsto y + 1}$ be a real 2-torus. Let $\mathscr{C}^{\infty}_0(\mathbb{T})$ be the subset of $\mathscr{C}^{\infty}(\mathbb{T})$ of ...
user197284's user avatar
10 votes
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Behavior of $\sum_{n=1}^{\infty} (\{n \xi \} - \frac{1}{2})$ for irrational number $\xi$

Consider the series $$ \sum_{n=1}^{\infty} ( \{ n \xi \} - \frac{1}{2}) $$ where $\{ \}$ denotes the fractional part and $\xi$ is some irrational number. It clearly does not converge but can we show ...
Hampus Nyberg's user avatar
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Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed

The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
Vincent Granville's user avatar
7 votes
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Can you identify this irrational number?

There is a certain number, say $v$. I can prove it is irrational. That would be more interesting if it is expressible in terms of known values ... zeta functions, Catalan's number, L-functions, etc. ...
Gerald Edgar's user avatar
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17 votes
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Fractional part power

Does a irrational number $x > 1$ exist such that $\{x^n \} \le \frac{1}{2}$ for all positive integers $n$ ? $x=1+ \sqrt 2$ holds for $n$ odd, but not in even
Yessir03's user avatar
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To prove irrationality, why integrate?

I have been reading David Angell's lovely book, Irrationality and Transcendence in Number Theory, which has given me some fresh insights even with some of the easier proofs. But the book reminds me of ...
Timothy Chow's user avatar
8 votes
2 answers
337 views

Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?

Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$? I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am ...
Vincent Granville's user avatar
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287 views

Is $\sum\limits_{k=0}^{\infty}\frac{1}{(mk)!^{n+1}}$ irrational?

I was using Wolfram Alpha for things, and I came across $I_{0}(2)$. For fun I tried asking Wolfram Alpha if the number was irrational, but said it's unknown. I believe this is an error, as its ...
Frax's user avatar
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11 votes
4 answers
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Compilation of strategies to show that some constant is irrational

I'm looking into expanding my knowledge in ways to show that some constant is irrational. I'm gonna give some examples of irrationality proofs, and I'm interested in learning what strategies you guys ...
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Is $e^{{e^{\ \dots\ }}^n}$ ever an integer?

Let $n$ be a positive integer. It is clear that $e^n$ is not integer because $e$ is transcendental (not algebraic). Now for each positive integer $k$ let $F^k(n)$ denote the $k$-fold composition of $F(...
D.S. Lipham's user avatar
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-3 votes
1 answer
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Are mantissas of irrationals provably unique, at a given precision? [closed]

Many thanks to all responders! Is there any research as to the uniqueness of mantissas of irrationals? It's easy to see that the mantissa of the square root of 5 (0.236067977...) and the mantissa of ...
Joe C's user avatar
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3 votes
0 answers
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Help with this irrationality proof

I have a real number, that is quite messy so I'll just call it $x$. I want to prove it's irrational. It's a proof by contradiction. The contradiction will rise if I assume $x$ is a rational number $p/...
Pinteco's user avatar
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Irrational combination of rationally independent polynomials

Let $p_1,\dotsc,p_k: \mathbb{N} \to \mathbb{Z}$ be rationally independent polynomials with zero constant term. If $t_1,\dotsc,t_k \in [0,1)$ are not all rational, is it true that the polynomial $$p(n)...
User's user avatar
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13 votes
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Convergence of the series $\sum_{n=1}^\infty \frac{(2+\sin n)^n}{3^n n^a}$ for $a\in(0,1)$

This is inspired by this Math.SE question, for $a=1$. Borwein, Bailey, and Girgensohn pose in their book ([1,Problem 35]) as an open problem the convergence of the series $$\sum_{n=1}^\infty \frac{(2+\...
Clement C.'s user avatar
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5 votes
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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 ...
BPP's user avatar
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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, ...
Arc's user avatar
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2 votes
1 answer
204 views

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. ...
Dominik Kwietniak's user avatar
16 votes
1 answer
2k views

Extending Apéry's proof to Catalan's constant?

I've been looking into Apéry's irrationality proof of $\zeta (3)$, and one of the first questions I instantly had, was how did he derive the following continued fraction? $$\begin{equation*} \zeta (3)=...
KStarGamer's user avatar
4 votes
1 answer
324 views

Irrationality of this trigonometric function

I'd like to prove the following conjecture. Let $x = \frac{p}{q}\pi$ be a rational angle ($p,q$ integers, $q \geq 1$). Then $f(x) = \frac{2}{\pi} \arccos{\left(2\cos^4(2x)-1 \right)}$ is irrational if ...
nervxxx's user avatar
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Must a continuous $\varphi:\mathbb R^n\to\mathbb R^n$ with $\mathbb Q^n \subseteq \varphi[\mathbb Q^n]$ be surjective?

Let $\varphi:\mathbb R^n \to \mathbb R^n$ be just some continuous function. If the image of $\varphi$ happens to contain $\mathbb Q^n$, does it follow that in fact all of $\mathbb R^n$ is contained in ...
Louis Deaett's user avatar
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6 votes
1 answer
334 views

The square root of natural number expressed by an infinite series

Can you prove or disprove the following claim: Let $U(n,P,Q)$ be the nth generalized Lucas number of the first kind and let $m$ be a natural number. Then, $$\sqrt{m}=1+\displaystyle\sum_{n=1}^{\infty}...
Pedja's user avatar
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4 votes
1 answer
1k views

The constant $\pi$ expressed by an infinite series

I am looking for the proof of the following claim: First, define the function $\operatorname{sgn_1}(n)$ as follows: $$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \...
Pedja's user avatar
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1 vote
0 answers
225 views

Ergodic Theory and Euler-Mascheroni Constant

I am highly interested in doing research on proving irrationality of some specific numbers like Euler-Mascheroni Constant or $\zeta(5)$. A professor guided me that arithmetic nature of constants are a ...
user avatar
8 votes
2 answers
382 views

Are we able to estimate the fraction of the domain where $\cos (ax)+2\cos (b x)$ with $\frac ab \notin\mathbb{Q}$ is positive?

We know that the two functions $\{\;\cos (ax),\;2\cos (b x)\;\}$ where $\frac ab \notin \mathbb{Q}$ are independently positive (and negative) over $\frac 12$ of the domain. Is it possible to estimate ...
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1 vote
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On the degree of irrationality of two irrational numbers and their rational (in)dependence

Let $x$ and $y$ be some irrational numbers. If the degree of irrationality of $x$ is the same as that of $y$, is it necessarily the case that $x$ and $y$ are rationally dependent ? ADDENDUM: What if $...
PRIMES is in P.'s user avatar
6 votes
0 answers
280 views

Is the arithmetic-geometric mean of 1 and 2 rational?

It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
Rick Does Math's user avatar
7 votes
1 answer
367 views

Upper bounds on the irrationality measure of the arctan of an algebraic number

Let $x$ be an algebraic number. Must $\arctan(x)/\pi$ have finite irrationality measure? Are there any useful upper bounds?
Matt Hastings's user avatar
31 votes
1 answer
1k views

How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?

I was asked by an high school student if there is an algebraic way to find the exact value of the solution to the equation \begin{equation}\label{eq} x^{x+1}=(x+1)^x \end{equation} Let us define that ...
gigi's user avatar
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1 vote
1 answer
130 views

Rational linear subspace corresponding to an irrational vector

Given a vector $v = (v_1, \ldots, v_n) \in \mathbb{R}^n$, we can associate a rational linear subspace with this vector: assume $\{1, v_i \text{ for }i \in I\}$ is a linear basis of $\{1, v_1, \ldots, ...
HAORAN ZHU's user avatar
3 votes
1 answer
303 views

Distribution of the first occurrence of a maximum (record) run of zeros in the digits of a normal number (say $\pi$)

If the question was stated to appeal to the general public, it would be something like this. For a number such as $\pi$ or $\sqrt{2}$, the digits in base $b$ appear to be randomly distributed. We are ...
Vincent Granville's user avatar
5 votes
1 answer
326 views

Irrationality of $e^{x/y}$

How to prove the following continued fraction of $e^{x/y}$ $${\displaystyle e^{x/y}=1+{\cfrac {2x}{2y-x+{\cfrac {x^{2}}{6y+{\cfrac {x^{2}}{10y+{\cfrac {x^{2}}{14y+{\cfrac {x^{2}}{18y+\ddots }}}}}}}}}}...
Sourangshu Ghosh's user avatar
6 votes
1 answer
638 views

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 ...
Vincent Granville's user avatar
1 vote
0 answers
264 views

Looking for a proof that $\sqrt2 + \sqrt5 + \sqrt[3]3$ is irrational [closed]

What is the easiest way to prove that $\sqrt2+\sqrt5+\sqrt[3]3$ is irrational?
Mr. Habibi's user avatar
1 vote
0 answers
51 views

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 ...
John Wiltshire-Gordon's user avatar
8 votes
1 answer
714 views

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, ...
Sebastien Palcoux's user avatar
8 votes
1 answer
720 views

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 ...
Mert K. Denizciler's user avatar
17 votes
0 answers
702 views

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 ...
Timothy Chow's user avatar
-3 votes
1 answer
277 views

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?
Joe Joe's user avatar
4 votes
1 answer
337 views

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 $G_\delta$-...
D.S. Lipham's user avatar
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0 votes
0 answers
126 views

What numbers (irrational) in radicals are expressible as trigonometric rational fraction with only rational multiplies of $\pi$?

What irrational expressions $A$ with radicals can be expressed as trigonometric rational fraction (not a series) with only rational multiplies of $\pi$. Example: $ \frac{1}{\sqrt5} = \frac{\sin\frac{\...
Leonid Dworzanski's user avatar
3 votes
0 answers
190 views

Irrationality or transcendence of $i^{i\Omega}$ and $2^\Omega$, with $\Omega=W(1)$ and $W(x)$ being the main branch of Lambert $W$ function

In this post we denote the main (or principal) branch of the Lambert $W$ function as $W(x)$, I add as reference that Wikipedia has the article Lambert $W$ function. The particular value $W(1)=\Omega$ ...
user142929's user avatar
3 votes
0 answers
166 views

What is known about the irrationality of ratios and products of logarithms of integers?

Let $a,b,c,d$ be positive natural numbers such that $\{a,b\} \neq \{c,d\}$ and such that none are perfect powers. Is it true that $$\frac{\log a \log b}{\log c \log d} \notin \mathbb{Q} ?$$ The ...
Mark Lewko's user avatar
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1 vote
0 answers
289 views

Question about proof of irrationality of $\zeta(3)$ [closed]

I'm reading this article of Henri Cohen about Apery's proof of the irrationality of $\zeta(3)$ but I don't really get the details of "THEOREME 1". My first doubt is about the relation $a_n \sim A \...
Eparoh's user avatar
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8 votes
2 answers
583 views

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: ...
user765118's user avatar
1 vote
1 answer
223 views

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 ...
D.S. Lipham's user avatar
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