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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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 $...
4
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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 $a_{n+1} = \frac{a_n + g_n}{2}$, ...
7
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1answer
195 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?
28
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1answer
873 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 ...
1
vote
1answer
75 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, ...
3
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1answer
239 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 ...
5
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1answer
266 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 }}}}}}}}}}...
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1answer
520 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 ...
1
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0answers
223 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?
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77 views
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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40 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 ...
8
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1answer
481 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, ...
7
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1answer
326 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 ...
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496 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 ...
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1answer
179 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?
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1answer
274 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$-...
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118 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{\...
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0answers
109 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$ ...
3
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0answers
152 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} ?$$
The ...
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225 views
Question about proof of irrationality of $\zeta(3)$ [closed]
I'm reading this article of Henri Cohen about Apery's proof of the irrationality of $\zeta(3)$ but I don't really get the details of "THEOREME 1".
My first doubt is about the relation $a_n \sim A \...
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2answers
448 views
Irrationality measure of arctan(1/3)
I recently came across the concept of the irrationality measure. It really fascinated me and when I was looking for known values $\mu(x)$ for mathematical constants $x$, I also came across this paper: ...
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1answer
183 views
Quotients of the irrationals
Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
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115 views
Possible Birkhoff spectra for irrational rotations
Let $\mathbb{T}=\mathbb{R}/\mathbb{Z}$ be the unit circle (think of it as of the interval $[0,1)$ with endpoints identified). Assume that $\alpha$ is irrational and consider the rotation by $\alpha$, ...
2
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1answer
85 views
O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval
O(1) or o(logn) discrepancy for multiples of an irrational for at least one sub interval.
Using $\{x\}$ to denote the fraction part of $x$ we can define for any $I\subset [0,1]$,
$$E(n,\theta, I) ={ ...
1
vote
1answer
127 views
Rational Peano curves
An rr function (i.e. rational rational function) is a quotient
$$ \frac fg\,:\, \Bbb Q\ \to\ \Bbb Q\cup\{\infty\} $$
such that $\ f,g\,\in\,\Bbb Z[X],\ $ where $\ g\ne 0.$
QUESTION Do there exist ...
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1answer
554 views
If $x^x=2$ then is $x$ expressible using elementary functions?
I have a curious question. Let $x∈\mathbb{R}^+$ such that $x^x=2$. I am aware that the Gelfond–Schneider theorem implies that $x$ cannot be algebraic. However, is it still possible that $x$ can be ...
2
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0answers
304 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 $n$ ...
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1answer
174 views
Real number which is different from all rationals [closed]
By diagonalization, it is possible to construct a real number $r \in [0,1]$ such that for every rational $q \in [0,1]$, there exists an index $i \in \mathbb{N}$ such that $r_i \neq q_i$ (where $x_i$ ...
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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 ...
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3answers
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Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
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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 [1], in particular the statement of Corollary 10.3 and its proof, I wondered what can be interesting examples of irrational numbers that ...
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1answer
205 views
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'...
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1answer
113 views
infinite set of mutually irrational numbers which odd linear combinations approximate 0 badly
I'm looking for a set of real numbers $\{\lambda_i;i\geq 1\}$ such that for each odd $n$, one can control $\delta_n:=\inf| \sum_i \pm n_i \lambda_i|$ where the $n_i$ are natural integers that sum to $...
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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 ...
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346 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 $\zeta(3)$, ...
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1answer
176 views
$\psi(2,1/6),\psi(4,1/6)$ in terms of zeta and pi only and another closed form for zeta
Let $\psi(n,x)$ denote the polygamma function.
In this answer Lucia gave linear relations for $\psi(m,1/3),\psi(m,1/6),\zeta(m+1)$.
The computer managed to find closed form for $\psi(2,1/6)$ and $\...
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0answers
215 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{...
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2answers
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Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?
Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case ...
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0answers
128 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,
$$
S=\{(m_1,m_2)\in\mathbb{Z}\...
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2answers
461 views
Are rationals everywhere equally dense? [closed]
I would like to know is there any notion of density over the rationals with which we could determine are rationals everywhere equally dense on the real line, because, for example, I am not sure would ...
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2answers
708 views
Irrational number with known probability distribution on digits
Is there any irrational number that is known the probability distribution of digits?
Something like 0 appears 10% of time, 1 appears 10% of time, etc.
Probably irrational numbers that are defined ...
3
votes
2answers
343 views
Example of irrational number with a pattern in digits [closed]
Suppose I created the following random number generator.
A trusted person choose a irrational number. That can easily defined and computed by a computer. Like square root of a prime.
Every time the ...
8
votes
1answer
904 views
Why is the Euler-Mascheroni constant not a Liouville number?
Let $\gamma$ be the Euler-Mascheroni constant. Why is $\gamma$ not a Liouville number? Are there any upper bounds for the irrationality measure of $\gamma$ known?
Any pointers to the literature are ...
11
votes
1answer
444 views
“Transcendental tilings”: Do they exist?
Let $T$ be a tiling of the plane.
Fix an origin and shoot a ray $r$ from the origin.
Mark off points $p_i$ along $r$ separated by unit distance.
Compute from $r$ a binary number $0 < b(r) < 1$ ...
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3answers
403 views
Irrationality of generalized continued fractions
An infinite simple continued fraction
$$\frac{1}{b_1 + \frac{1}{b_2 + \frac{1}{b_3+\dots}}} (b_i\in\mathbb Z)$$
is irrational. Now for a generalized continued fraction:
$$\frac{a_1}{b_1 + \frac{a_2}...
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1answer
987 views
Chudnovsky algorithm and Pi precision
What are the precision/ number of correct Pi digits after N iterations of Chudnovsky algorithm. Looking for a formula (rather than a table) and reference.
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2answers
169 views
Is $x=\frac{1}{2}$ the solution of this equation $\zeta(2)= 1+{{{{x}^{x}}^{x}}^{x}}^{\cdots } $?
I would like to study the irrationality of ${{{{x}^{x}}^{x}}^{x}}^{\cdots } $
for $x=\frac{1}{2} $ using the irrationality of $\zeta(2)$ .
Some computations in wolfram alpha show to me that :
$${...
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1answer
398 views
Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? [closed]
I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\...
4
votes
2answers
1k views
Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function?
I'm interested in knowing more about the question if $f(\pi)$ is rational or not, where $f$ is some well-known function. For example, $\cos(\pi) =-1$ is rational, while ${e}^{\pi}$ is irrational as ...
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vote
0answers
178 views
Researching the irrationality of a number [closed]
I am conducting a little research on checking if a number, written in positional numeral system is irrational.
Let $h^p_n$ be the most right non-zero digit of number $n!$ written in numeral system ...
