Questions about real numbers not expressible as the quotient of two integers. For questions on determining whether a number is irrational, use the (rationality-testing) tag instead.

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19
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581 views

Irrationality of e

I have a question about the irrationality of $e$: In proving the irrationality of $e$, one can prove the irrationality of $e^{-1}$ by using the series $$e^x = 1+x+\frac{x^2}{2!} + \cdots + ...
18
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0answers
402 views

If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb Q$?

Question : For every even $k\ge 4$, is the following $(\star)$ true? $$\begin{align}\text{If $\beta=0.{a_1}^{k}{a_2}^{k}{a_3}^{k}\cdots\in\mathbb Q$, then $\alpha=0.a_1a_2a_3\cdots\in\mathbb ...
8
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0answers
74 views

Is there any known application for normal numbers?

Background: I am writing a master thesis on the complexity of the expansions of algebraic numbers in some complex basis $\beta$ with $|\beta| > 1$. This is a very small step towards proving the ...
8
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0answers
132 views

The “trick” functions in the “$\pi$ is transcendental” proofs

I was reading this paper and I wondered how did Hermite decide to define a function $$f(x)=\frac{x^{p-1}(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ Are these functions only tricks or there is a deeper meaning?
7
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0answers
99 views

The sum $\sum_{n=1}^\infty \min_{k\le n}\{\alpha k\}$ for irrational $\alpha$

Let $\alpha$ be an irrational number. For every $n$ let $z_n$ be the integer closest to the number $\alpha n$. Then we can define $$A(\alpha):= \sum_{n=1}^\infty |\alpha n - z_n|.$$ We can also ...
7
votes
0answers
292 views

How does one prove that $\zeta(3)$ is irrational?

How does one prove that $\zeta(3)$ is irrational ? I would like to know how Apery did it. In particular how a recursion gives rise to irrationality !?
6
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0answers
110 views

Irrationality proof by fast converging series?

I read here http://www.mathpages.com/home/kmath455.htm that $\sum_{n=1}^\infty \frac{1}{d_n}$ is irrational if $d_{n+1} > d_{n}^2$ for all $n > N_0$. Can we prove $\pi$, $e$ or some other ...
5
votes
0answers
153 views

Are there integers $a, b$ s.th. $\pi^a = e^b$?

Is $\log \pi $ a rational number? That is, are there non-zero integers $a, b$ s.th. $\pi^a = e^b$ ?
4
votes
0answers
78 views

Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
4
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0answers
72 views

Extention of Euclid's GCD Algorithm. (The Art of Computer Programming, Volume 1, Edition 3, Section 1.2.1, Exercise 12)

Euclid's GCD algorithm which is used to find GCD of two input numbers, say, $c$ and $d$, needs the inputs to be positive integers. Exercise 12 provides an extension to this algorithm and allows $c$ ...
4
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0answers
73 views

Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
4
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0answers
178 views

Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
4
votes
0answers
78 views

Digits of two irrational numbers, given their power with fixed number of digits

I have $a, b \in \mathbb{R} \setminus \mathbb{Q}$, I want to know the result of $a^b$, but I don't know exact $a, b$ because I write them in numeric form. My question is how many digits of $a, b$ have ...
4
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0answers
116 views

Finding a closed expression for a calculated value.

Sometimes, when getting some numerical results when investigating number theory sequences with a computer, I find myself suspecting that a decimal value ($a$) I have found might be a quadratic ...
4
votes
0answers
179 views

Is the maximal temperature of the curlicue fractal acheived by $e\times\gamma$?

The Curlicue Fractal is defined as follows: Choose an irrational number $s$ and a horizontal unit segment with angle $\phi_0 = 0$. Define $\theta_{n+1} = \theta_{n} + 2 \pi s \pmod{2 \pi}$, with ...
3
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0answers
58 views

Transcendence of $\Gamma(1/3), \Gamma(1/4)$

Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a reference to that proof? Or maybe some details on the essential ideas ...
3
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0answers
40 views

Do the second-last-digits of the primes $\ge 11$ form a transcendental number?

Suppose, the number $x$ is constructed from the second-last-digits from the primes $\ge 11$ The first $1996$ digits of $x\ =\ 0.11112...$ after the decimal point are : ...
3
votes
0answers
57 views

When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ ...
3
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0answers
58 views

There are infinitely many $\frac{m}{n}\in\mathbb Q$ such that $|x- \frac{m}{n} |<\frac{1}{n^2}$

Let $x$ be an irrational number. Prove that there are infinitely many rational numbers $\frac{m}{n}$ such that $$|x- \frac{m}{n} |<\frac{1}{n^2}$$ It's clear that $-1<1/n^2 <1$ with ...
3
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0answers
188 views

Irrational numbers to the power of other irrational numbers: A beautiful proof question

The following theorem has a very beautiful proof. Theorem: There exist two irrational numbers $x$ and $y$ such that $x^y$ is rational. Proof: If $\sqrt{2}^{\sqrt{2}}$ is rational then we ...
3
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0answers
199 views

Minkowski's question mark function iterations

The Minkowski's question mark function (we use the sign $?$ to note this function) was designed in 1904 by Minkowski. It can be defined as an increasing bijection between $\mathbb Q$ and the set of ...
3
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0answers
153 views

Must be rational number

Let $a$, $b$ positive rational number. Suppose that there exist two odd positive integers $p$, $q$ such that $\sqrt[p]{a}+\sqrt[q]{b}$ is rational. Prove that both $\sqrt[p]{a}$ and $\sqrt[q]{b}$ are ...
3
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0answers
94 views

how many numbers of irrationality measure $x$

Does there exist $x>2$ such that uncountably many reals have irrationality measure x? Must there exist at least one number of irrationality measure $x$? related question on sets of constant ...
3
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0answers
104 views

Linear independence of reciprocals of logarithms

I would like to ask whether there is a proof of the following statement: Let $p$, $q$ be primes and $n$ positive integer coprime with $pq$. Then $\frac1{\log p}$, $\frac1{\log q}$ and $\frac1{\log n}$ ...
2
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0answers
65 views

What is the Best Introduction to Dedekind Cuts?

I'm looking for a clear, thorough, and easy-to-follow introduction to Dedekind cuts that is specifically geared towards those with an interest in foundational issues. So far, the discussions that I ...
2
votes
0answers
59 views

Conway's box function generalized as 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 ...
2
votes
0answers
40 views

Prove that $E_0$ is transcendental

Consider the non-negative natural numbers: $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$ Encode the primes as $1$, the rest as $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...
2
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0answers
71 views

What is the name of this irrational math constant and is there a compact way to write it? 0.10110111011110…

I think this number is a transcendental number and I've tried looking online to see who first made it, I'm not sure if it's a Liouville Number or if there is a more common or better name for it. Does ...
2
votes
0answers
57 views

Is this irrationality proof correct?

Consider a non-square integer $n$. If its square root was rational, then we would have $$\sqrt n=\frac{a}{b}$$ for some $a,b\in\mathbb{Z}$ and so $a^2=nb^2$. But this is impossible, because $n$ is ...
2
votes
0answers
88 views

Question about $e^{e^{e^e}}$

Is there a proof that the power tower of length $4$ of $e$ is irrational? Is it known whether or not $$e^{e^{e^e}}$$ is transcendental?
2
votes
0answers
52 views

Apery's constant

I read that it is unknown if $\zeta (3)$ is algebraic but it is known to be irrational. Has anyone proved anything of the form $\zeta (3)$ is not a root of a polynomial of degree $12345$ with integer ...
2
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0answers
43 views

Experimental calculation and $\mathbb{Q}$

I have been reading this article and have a question about the first line of the second paragraph on the first page. It says: The basis for this suggestion is the simple fact that all experimental ...
2
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0answers
151 views

Irrationality of $\pi$ from the spigot algorithm?

The spigot algorithm for BPP formula gives hexadecimal digits of $\pi$ one at a time. Is it possible to prove directly that this algorithm cannot be computed with bounded-memory? (From R J Lipton It ...
1
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0answers
195 views

Very simple proof that $\sqrt{2}$ is irrational.

I came across a nice-looking proof that $\sqrt{2}$ is irrational here. It somehow seems to good to be true. What are the assumptions being made in the proof and if this proof is indeed correct, why is ...
1
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0answers
56 views

Natural bijection between $\mathbb{N}$ and algebraic numbers?

Q. Is there a canonical, explicit bijection between the natural numbers $\mathbb{N}$ and the algebraic numbers? The earlier MSE question, "Bijection for algebraic numbers," does not quite ...
1
vote
0answers
32 views

Proof Verification of Sum of Irrational Numbers

The chosen answer here claims that for rational $u$ and $v$, $u^{\frac{1}n} + v^{\frac{1}n}$ is rational iff $u^{\frac{1}n}$ and $v^{\frac{1}n}$ are both rational. However, the link to a proof seems ...
1
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0answers
49 views

Sum of 2 different irrational logarithms = Irrational?

I am having some problems proving that the following sum is irrational or rational: $\log_2(3)+\log_3(2)$ = irrational. This is all I've got for now: $\log_2(3)=\frac mn \iff 2^{\frac mn}=3 \iff ...
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0answers
42 views

What is the limit of $k^2|\pi-n(k)/k |$, where $k$ minimizes $|k\pi -n|$?

Let $k\in \mathbb N$ and for any such n, let $k=k(n)$ minimizes the distance $|k\pi-n|\leq 2 \pi$. It is clear that, by fixing the value of $n$, it is possible to choose $k$ (and vice versa). ...
1
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0answers
53 views

“Building blocks” for computable functions

In an (otherwise very enlightening) answer to another question of mine the question came up What functions are allowed as building blocks for computable functions? I was astonished that there ...
1
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0answers
64 views

Irrational numbers and series

Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational ...
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0answers
38 views

Proving irrationality i p,s and k are primes number?

Can you prove that $\frac{k^{\frac13}-p^{\frac13}}{s^{\frac13}-p^{\frac23}}$ is irrational if p, k and s are different prime numbers. I am certain it is but i dont know how to prove it.
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0answers
455 views

Third degree polynomial with integer coefficient and three irrational roots

There are some polynomial with the above characteristic, and real roots of such polynomials cannot be found using rational number theorem and irrational conjugate theorem. The example of such function ...
1
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0answers
116 views

Gelfond-Schneider Constant $2^{\sqrt{2}}$

Someone knows a proof (books , articles) that $2^{\sqrt{2}}$ is irrational ? Without using that $2^{\sqrt{2}}$ is transcendent. Any hints would be appreciated.
1
vote
0answers
105 views

Continued fraction of $\gamma+1$ using recursion

Number $\gamma,$ the Euler-Mascheroni constant, is defined as the value of $$\gamma = \lim_{n\to\infty} \sum_{k=1}^n \frac{1}{k} - \ln(n).$$ We know that $$\lim_{n\to\infty} ...
1
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0answers
120 views

Asymptotic behavior of $\sum_{j=1}^n \cos^p(\pi u j)$ for large $n$ and $p$?

Consider the sum $$S=\sum_{j=1}^n \cos^p(\pi u j),$$ where $n$ and $p$ are positive integers and $u$ is irrational. Let's say $p$ is even. I'm interested in the asymptotic behavior of this for $n$ ...
1
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0answers
453 views

What is the exact definition of a rational power?

I was taught in school that $$x^{a/b} = \sqrt[b]{x^a}$$ however, wolfram says this is not always true: $\sqrt[3]{x^2} \ne x^{2/3}$ ...
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0answers
80 views

Are limits on exponents in moduli possible, if the modulus is relatively prime?

I asked a similar question to this recently. Here, I consider an arbitrary, but fixed, modulus m, which is relatively prime to x and y. Can anybody extend the answer given in the previous question? ...
0
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0answers
44 views

Proving transcendental numbers

I'll warn now that this is probably a big question, but I am wondering if anyone can explain why it is so difficult to prove whether a number is transcendental or algebraic. For example, it is now ...
0
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0answers
30 views

On Diophantine approximation and irrationality proofs

This question is an offshoot from this previous MSE post. I have a ratio of two numbers $a$ and $b$ (presumably both positive integers), where $a$ and $b$ are determined by some arithmetic / ...
0
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0answers
55 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...