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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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
12 votes
1 answer
539 views

On Bailey–Borwein–Plouffe formula for irrational numbers

A BBP-type formula for an irrational number $\alpha$ in the integer base $b\geq 2$ is a formula in the form $\alpha=\Sigma_{k=0}^{\infty}\frac{1}{b^k}\frac{p(k)}{q(k)}$ ($p, q$ are polynomials in ...
Amit Sing Mukerjee's user avatar
9 votes
1 answer
720 views

Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?

In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...
Conifold's user avatar
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6 votes
2 answers
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Conjecture on irrational algebraic numbers

Conjecture: For every irrational algebraic number $q$ and natural number $b$, the representation of $q$ on base $b$ contains all the digits $[0,\dots,b-1]$. Questions: Has this conjecture been ...
barak manos's user avatar
65 votes
2 answers
5k views

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
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48 votes
2 answers
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Irrationality of $ \pi e, \pi^{\pi}$ and $e^{\pi^2}$

What is known about irrationality of $\pi e$, $\pi^\pi$ and $e^{\pi^2}$?
Vladimir Reshetnikov's user avatar
35 votes
1 answer
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Proving the irrationality of $\pi e$ and $\pi / e$

Rather than relying on the consequences of Schanuel's conjecture, I set about using the same ideas Apery had used to construct integer arguments converging fast enough to show $\zeta(3)$ is irrational ...
Brian's user avatar
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27 votes
4 answers
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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 ...
Klangen's user avatar
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14 votes
1 answer
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Transcendence of $\Gamma(1/3), \Gamma(1/4)$

This is a re-post from MSE as I did not get even a single comment there. Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a ...
Paramanand Singh's user avatar
10 votes
1 answer
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Measure of a set of irrational numbers

Let $A$ be a set of all irrational numbers $\rho \in (0, 1)$ represented as a continued fraction $\rho=[a_{1}, a_{2},...,a_{n},...],$ such that $a_{n}\leq \text{const}\cdot n^{\epsilon}$ for some $\...
sokho's user avatar
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8 votes
2 answers
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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 ...
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
7 votes
3 answers
595 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}...
bhbr's user avatar
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6 votes
1 answer
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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 ...
Vincent Granville'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
3 votes
1 answer
204 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 $\...
joro's user avatar
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1 vote
1 answer
224 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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