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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22
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843 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 + ...
19
votes
0answers
447 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
votes
0answers
140 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
votes
0answers
112 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
181 views

Are there integers $a, b$ such that $\pi^a = e^b$?

Is $\log \pi $ a rational number? That is, are there non-zero integers $a, b$ s.th. $\pi^a = e^b$ ?
6
votes
0answers
120 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
61 views

Elementary proof that finite sums of square roots of primes is irrational

It is relatively easy to show that if $p_1$, $p_2$ and $p_3$ are distinct primes then $\sqrt{p_1}+\sqrt{p_2}$ and $\sqrt{p_1}+\sqrt{p_2}+\sqrt{p_3}$ are irrational, but the only proof I can find that ...
5
votes
0answers
97 views

How to prove $e^{1/e}$ is irrational?

How do we prove $e^{\frac{1}{e}}$ is irrational ? Also how do we show it is transcendental ? The number $\eta = \exp(\exp(-1))$ occurs naturally in the context of tetration and power towers. Let ...
4
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0answers
99 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
88 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
205 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 ...
4
votes
0answers
193 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
85 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
votes
0answers
122 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
200 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
votes
0answers
71 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
votes
0answers
45 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
56 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 $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...
3
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0answers
59 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
votes
0answers
100 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 ...
3
votes
0answers
284 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
votes
0answers
173 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
votes
0answers
96 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
votes
0answers
114 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
votes
0answers
21 views

Is $\sum \frac{n+1}{b_{n+1}}$ irrational, when $b_1=2$ and $b_{k+1}=2^kb_k(b_k-1)+1$, $k\geq 1$?

Let the sequences of positive integers $$a_n=n$$ when $n\geq 1$, and $$b_{n+1}=2^nb_n(b_n-1)+1$$ for $n\geq 1$ taking $b_1=2$. I've computed with previous sequences to assert that satisfy the ...
2
votes
0answers
65 views

Paper of Paul Erdös

I'm trying to understand On Arithmetical Properties of Lambert Series by Erdös, but am stuck on the first page. He states: Put $k=\left[(\log n)^{1/10}\right]$ and let $p_1,p_2,\ldots$ be the ...
2
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0answers
51 views

Irrationality of $\pi+c$

How to prove that $\pi+c$ is irrational? where $c$ is the Champernowne Constant.
2
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0answers
58 views

Alternative prrof of $\sqrt 2 $ (and $\sqrt {\text {of any non square integer}}$ ) not being rational

Long time ago for an assignment, I submitted an alternative proof of $\sqrt 2$ not being rational along the following lines: suppose $\frac{a^2}{b^2}=2$ then we must also have $a=b+n$ for $a,b,n$ ...
2
votes
0answers
59 views

Bijection between the set of irrational numbers from 0 to 1 and the set of infinite integers? (Actually, 10-adic integers)

It seems to me we can assign to every irrational from 0 to 1 an infinite whole number in the following way: $$ 0.a_1a_2a_3a_4... \rightarrow ...a_4a_3a_2a_1 $$ For example: $$ 0.14159265... ...
2
votes
0answers
29 views

Smallest number of workers in factory, Diophantine approximation

Q. In a factory, the percentage of male workers was $53.7802\%$ (rounding to nearest fourth decimal place) last year. What is the smallest number of female workers working there? Hint: Diophantine ...
2
votes
0answers
90 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
104 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
235 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 ...
2
votes
0answers
80 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
60 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
100 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
63 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
votes
0answers
45 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
156 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
25 views

the summation goes to the integration

Let $f$ be a continuous function on $\mathbb R$ with period $1$. Show that for any irrational number $x$, $\frac { \sum_{k =1 }^ N f (k x ) } { N} \to \int _ 0 ^1 f(t) \, d t $ as $N \to \infty$. I ...
1
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0answers
35 views

How do I know if this will be rational or irrational? ($a^b$)

Usually, when I have $a^b$ when $a$ and $b$ are both irrational, I assume that it will be irrational. But that is not always true, I assume, so when is the result irrational? How will I know? Take ...
1
vote
0answers
64 views

Interesting facts/ proofs about rational and irrational numbers

We got set some work to find some interesting facts or proofs regarding rational and irrational numbers. I wonder if anyone could offer some insight or recommend a good book/ website to look at.
1
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0answers
64 views

Am I pretty close to proving that e is irrational?

Show that $e=1+1/1!+1/2!+1/3!+…$ is an irrational number. Hint: show that, for all positive integers $p$, $0<p![e−(1+1/1!+…+1/p!)]<1$. Then conclude that $e$ cannot be a ratio of two integers ...
1
vote
0answers
17 views

A binary irrational with bits defined by primes

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to ...
1
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0answers
14 views

irrationality of $\sum_{k=1}^{n} k^{\frac{1}{m}}$

For arbitrary $n \geq 2$ and $m \geq 2$, is $$\sum_{k=1}^{n} k^{\frac{1}{m}}$$ an irrational number?
1
vote
0answers
67 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
55 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
67 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 ...
1
vote
0answers
41 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.
1
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0answers
713 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 ...