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.

31 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
26 votes
0 answers
818 views

Is the Flajolet-Martin constant irrational? Is it transcendental?

Facebook has a new tool to estimate the average path length between you and any other person on Facebook. An interesting aspect of their method is the use of the Flajolet-Martin algorithm. In the ...
Jeffrey Shallit's user avatar
17 votes
0 answers
705 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
  • 78.1k
13 votes
0 answers
305 views

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
  • 1,342
8 votes
0 answers
189 views

Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?

"Universal" means that the construction steps are independent of the length of the given segment. In the Euclidean plane one can take the diagonal of a square built on it. Without the "universal" the ...
Conifold's user avatar
  • 1,599
7 votes
0 answers
254 views

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
  • 40.2k
7 votes
0 answers
282 views

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 $\alpha$, ...
Dominik Kwietniak's user avatar
7 votes
0 answers
550 views

Irrationality of the values of the prime zeta function

Preamble: I asked this question on Math.SE with a bounty of 100, but received no replies. The bounty has now ended, and it was suggested I post this here instead. Since Apéry we know that $\zeta(3)$, ...
Klangen's user avatar
  • 1,943
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
6 votes
0 answers
458 views

Conway's box function iterated to produce a hierarchy of nested sets of real numbers

Conway's box function is the inverse of Minkowski's question mark function. It maps the dyadic rationals on the unit interval to the rationals using the Stern-Brocot tree (Farey sequence). When the ...
Timothy J. Doyle's user avatar
6 votes
0 answers
301 views

Irrationality of the sum of the reciprocal of perfect powers

A couple of days ago I was trying to remember a classical exercise (which I now find out goes by the name of Goldbach-Euler theorem). Eventually I figured out that it asked to prove that $$\sum_{p\in\...
Marco Golla's user avatar
  • 10.4k
5 votes
0 answers
107 views

Approximation of an irrational point from a given direction

Taking norms to be maximal norm, then the simultaneous version of the Dirichlet's approximation theorem states that given real numbers ${\displaystyle \alpha _{1},\ldots ,\alpha _{d}} $,there are ...
hyyyyy's user avatar
  • 233
4 votes
0 answers
428 views

The irrational numbers α such that n odd and m=⌊nα⌋ odd implies ⌊mα⌋ odd

This post is the analogous of that one (about $\sqrt{2}$) but with a much stronger expectation here. We observed, and then this comment of Lucia proved, that for $\phi$ the golden ratio, if $n$ ...
Sebastien Palcoux's user avatar
4 votes
0 answers
235 views

A connection between basic hypergeometric series and number theory

I am studying functions given by the power series: $$f(z)=1+\sum_{n=1}^{\infty}\frac{z^n}{(1-q)(1-q^2)\cdots(1-q^{n})}.$$ The parameter $q$ is usually assumed to be such that $|q|<1$. Then it is ...
Twi's user avatar
  • 2,188
3 votes
0 answers
207 views

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
  • 521
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
  • 11.7k
2 votes
0 answers
155 views

Subsets of particular values of $\zeta'(k)$ that contain irrational numbers

We consider the set of elements $\zeta'(2),\zeta'(3),\zeta'(4),\zeta'(5),\ldots$ where $\zeta(z)$ is the Riemann zeta function and $\zeta'(z)=\frac{d}{dz}\zeta(z)$ its derivative. Thus we consider ...
user142929's user avatar
2 votes
0 answers
139 views

Combination of irrationals

Fix a very small $\epsilon>0$; and irrationals $a_1,a_2>0$. Now suppose we look at all integer combinations of these irrationals which has a small norm; that is, $$ S=\{(m_1,m_2)\in\mathbb{Z}\...
hookah's user avatar
  • 1,096
2 votes
0 answers
148 views

Bitwise operation of two square roots

Let $\sqrt 2 = 1.a_1a_2\dots _2$, and $\sqrt 3 = 1.b_1b_2\dots _2$. What can one say about the number $n = 0.c_1c_2\dots$ where $c_i = 1$ if $a_i = b_i$ and $0$ otherwise? There is no reason to ...
Mayank Pandey's user avatar
1 vote
0 answers
136 views

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
1 vote
0 answers
106 views

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
  • 1,061
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
1 vote
0 answers
73 views

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
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
1 vote
0 answers
219 views

Two exponents being algebraic

Schanuel conjecture implies this, so likely it is true. Let $f(x),g(x)$ be polynomials with coefficient in $\mathbb{Z}[i]$. Assume that for some complex number $x_0$, both $\exp{f(x_0)}$ and $\exp{...
joro's user avatar
  • 24.2k
1 vote
0 answers
176 views

binomial coefficients and irrationals

The following, probably either currently impossible to deal with, or having a negative solution, arose from an ergodic theory question, presumably itself currently intractible. I am not a number ...
David Handelman's user avatar
0 votes
0 answers
70 views

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
0 votes
0 answers
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
  • 101
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
0 votes
0 answers
57 views

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 [1], in particular the statement of Corollary 10.3 and its proof, I wondered what can be interesting examples of irrational numbers that ...
user142929's user avatar
-2 votes
0 answers
151 views

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