Transcendental numbers are numbers that cannot be the root of a nonzero polynomial with rational coefficients (i.e., not an algebraic number). Examples of such numbers are $\pi$ and $e$.
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Can a finitely generated $\mathbb{Z}$-algebra contain $\mathbb{Q}$?
Is there a ring between $\mathbb{Q}$ and $\mathbb{R}$ that is finitely generated as an algebra over $\mathbb{Z}$? My guess is there isn't.
I can see that it would have to be finitely generated over ...
3
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0answers
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The Tribonacci constant and the Dragon
Let $x = \frac{\ln T}{\ln 2} = 0.879146\dots$ where $T$ is the tribonacci constant, then x solves the transcendental equation,
$$4^x(2^x-1)=(2^x+1)$$
Let $x = \frac{\ln y}{\ln 2} = 1.523627\dots$ ...
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3answers
691 views
What are examples of unexpected algebraic numbers of high degree occured in some math problems?
Recently I asked a question about a possible transcendence of the number ...
23
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1answer
414 views
What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?
I got the number
$$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$
in the ...
8
votes
2answers
82 views
Irrational numbers, decimal representation
Can this even be proved? (Or disproved?)
Any irrational number without a 0 (zero) in its decimal representation is transcendental.
Not sure where to start on this one...
13
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1answer
163 views
Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?
I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
6
votes
1answer
105 views
Proving that $\frac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\frac{(-1)^{\frac{p_{{k}}-1}{2}}}{p_{k}} \right )^{-1}$ an identity of Euler's.
This is another identity of Euler's relating $\pi$ to the prime numbers, available here
\begin{align*}
\dfrac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\dfrac{(-1)^{\dfrac{p_{{k}}-1}{2}}}{p_{k}} \right ...
4
votes
2answers
103 views
Why is the concept of transcendental numbers linked with rational coefficients? Why not real nor complex coefficients?
I've read this:
In mathematics, a transcendental number is a (possibly complex) number
that is not algebraic—that is, it is not a root of a non-zero
polynomial equation with rational ...
3
votes
4answers
119 views
Non-existence of irrational numbers?
I realize the title of my question will probably cause the raising of some eyebrows, so let me explain. Not sure whether to file this under "math" or "philosophy". This also might be able to be ...
14
votes
1answer
227 views
Why is an irrational number's algebraic complexity the opposite of its Diophantine complexity?
Definition 1. Given $x \in \Bbb{R}$, the algebraic degree of $x$ is the degree of the minimal polynomial of $x$ over $\Bbb{Q}$. If $x$ is transcendental, we will define its algebraic degree to be ...
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1answer
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Is $e^{\pi \alpha}$ known to be transcendent for all real algebraic $\alpha$?
The MathWorld article Transcendental Number contains a reference to Yu. V. Nesterenko proof of transcendence of $e^{\pi \sqrt{2}}$. Is there a more general result about transcendence of $e^{\pi ...
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votes
2answers
204 views
Proving that $\pi=\sum\limits_{k=0}^{\infty}(-1)^{k}\left(\frac{2^{2k+1}+(-1)^{k}}{(4k+1)2^{4k}}+ \frac{2^{2k+2}+(-1)^{k+1}}{(4k+3)2^{4k+2}}\right)$
Long time ago I've been playng with formulas for $\pi$ and found that one above in the title which can also be expressed as
\begin{align*}
...
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1answer
67 views
What's the name of this class of transcendental numbers?
I'm considering the set $$\left\{\sin(k)\mid k\in\Bbb Z\backslash \left\{0\right\}\right\}.$$ All of its members are transcendental numbers, but together they don't form the complete set of all ...
4
votes
1answer
91 views
Prove or disprove transcendence of numbers
I have two in one question:
1) Let $\{p_n\}_{n\in \mathbb{N}}$ be sequence of all prime numbers. Is number $\displaystyle\alpha = \sum_{n=1}^{\infty} 10^{-p_n}$ transcendental number?
2) Let ...
4
votes
2answers
120 views
How to show $e^{2 \pi i \theta}$ is not algebraic.
I was wondering if someone could possibly help
me figure out how to show $e^{2 \pi i \theta}$
is not algebraic when $\theta$ is irrational.
Thanks!
17
votes
2answers
702 views
Does $\sin(x)=y$ have a solution in $\mathbb{Q}$ beside $x=y=0$
Is there a way to show, that the only solution of
$$\sin(x)=y$$
is $x=y=0$ with $x,y\in \mathbb{Q}$.
I am seaching a way to prove it with the things you learn in linear algebra and analysis 1+2 ...
6
votes
1answer
97 views
Prime elements in $\mathbb{Q}[[X,Y,Z]]$ whose status as an infinite series is unchanged by arbitrary multiplication
Let's suppose $R$ is the ring $\mathbb{Q}[[X,Y,Z]]$. I'm interested in finding power series $f(x,y,z) \in R \setminus \mathbb{Q}[X,Y,Z]$ which are, first of all, prime elements in $R$, but also ...
1
vote
2answers
81 views
Can every transcendental number be expressed as an infinite continued fraction?
Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction?
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2answers
68 views
Is this a transcendental number?
A complex number that has transcendental real part is always transcendental?
How about in the case of imaginary part?
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1answer
136 views
Liouville's proof of the existence of transcendental numbers
The existence of transcendental numbers can be shown easily by considering the cardinality of the set of solutions to polynomials with integer cofficents and the cardinality of the real numbers.
It ...
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votes
2answers
147 views
Algebraic and Transcendental Numbers - Set Theory
Denote $\mathbb Q$$[x]$ = set of polynomials with coefficients $c_1$, $c_2$, $...$ ,$c_n$ in $\mathbb Q$.
A number $a$ is algebraic if there exists a polynomial $f(x)$ in $\mathbb Q$[x] such that ...
3
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0answers
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Schneider's theorem about the transcendence of values of the $j$-function
It is known that the $j$-function takes algebraic values when evaluated at imaginary quadratic integers. This is a result that was proved by Schneider in 1937 apparently. To be precise, Schneider ...
6
votes
1answer
142 views
Is every complex number the root of a polynomial? (Converse to fundamental theorem of algebra.)
For every polynomial with complex coefficients, the fundamental theorem of algebra guarantees the existence of complex numbers which happen to be roots of it. But is this everything? i.e. is the ...
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votes
2answers
140 views
Are the digits of irrational/transdental numbers random?
If I were to look at the decimal representation of some irrational
or even transdental number,
and if I choose a natural number at random
can I expect that it is some digit with probability $0.1$ ?
20
votes
1answer
539 views
Is $0.23571113171923293137\dots$ transcendental?
Is the following number transcendental?
$$0.23571113171923293137\dots$$(Obtained by writing prime numbers consecutively from left to right, in the decimal expansion)
7
votes
1answer
252 views
Is it possible to express $e$ in terms of $\pi$ algebraically and vice-versa?
Am I right in thinking this is not possible since both are known to be transcendental?
Also, $e^{i\pi}+1=0$ suggests this is not possible - we can not isolate $e$ or $\pi$ from this since it involves ...
4
votes
1answer
88 views
Is the Glaisher–Kinkelin constant transcendental?
As the title says, is it known whether or not the Glaisher constant is a transcendental number?
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3answers
139 views
Does the Abel-Ruffini Theorem contradict the Fundamental Theorem of Algebra?
It is my understanding that the Abel-Ruffini Theorem implies that certain polynomial equations $(x^5-x+1=0$, for instance) have transcendental roots. However, the Fundamental Theorem of Algebra states ...
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1answer
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proof that $e^x$ is a transcendental function of $x$?
Let a function $f(x)$ be algebraic if it satisfies an equation of the form $$c_n(x)(f(x))^n + c_{n-1}(x)(f(x))^{n-1} + \cdots + c_0(x)=0,$$ for $c_k(x)$ rational functions of $x$, and let $f$ be ...
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2answers
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which representation/fraction approximates the value of $\pi$ in a better way. one which is most widely used is $\frac{22}{7}$ [duplicate]
Possible Duplicate:
Why is $22/7$ a better approximation for $\pi$ than $3.14$?
Approximating $\pi$ with least digits
do we know any representation/fraction other that $\frac{22}{7}$ ...
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votes
1answer
53 views
Polynomials with roots having the same module and linear dependent arguments
Is it possible for a polynomial with integer coefficients to have some of its roots:
$$m_1e^{i\theta_1 \pi}, m_2e^{i\theta_2 \pi}, \ldots, m_ke^{i\theta_k \pi}$$
such that there exist nonzero integers ...
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1answer
127 views
matrix exponential is a rational or not?
I want to know whether following are true or false:
for any given natural number $n$, $T>0$ a rational, suppose that $Q_1, \cdots, Q_n$ are $m\times m$ matrices with rational entries, $t_1, ...
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vote
3answers
180 views
Is there a general way to solve transcendental equations?
To make things definite, let's narrow them and call transcendental equation of the form
$$f(x) = 0$$
where $f$ is a real elementary function in the usual sense. For example
$$\cos(\pi x) + x^2 = ...
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0answers
47 views
Free module, $\mathbb{Z}[a]$ over $\mathbb{Z}[(a+1)^2]$ for transcendental number a
I'm trying to prove that for a transcendental number $a$ the module $\mathbb{Z}[a]$ over $\mathbb{Z}[(a+1)^2]$ is free. For $\mathbb{Z}[a+1]$ over $\mathbb{Z}[(a+1)^2]$, the basis is $\{1,a+1\}$. What ...
8
votes
1answer
320 views
Does this show that the Apery Constant is transcendental?
Last August I posted this on mathoverflow: http://mathoverflow.net/questions/71856/a-serendipitous-riemann-identity. I show the (slightly revised) equation below:
$$\zeta (3)=\frac{2\pi^4}{315} ...
2
votes
0answers
63 views
Linear independence of reciprocals of logarithms
I would like to ask whether there is a proof of the following statement:
Let $p$, $q$ be primes and $n$ positive integer coprime with $pq$. Then $\frac1{\log p}$, $\frac1{\log q}$ and $\frac1{\log n}$ ...
2
votes
2answers
133 views
The definition of “algebraically independent”
In Lang's Algebra, he gives a definition that
Elements $x_1, \cdots, x_n\in B$ are called algebraically independent over $A$[a subring of $B$] if the evaluation map $$f\mapsto f(x)$$is injective. ...
4
votes
1answer
131 views
Infinitely many transcendental numbers over Q
My previous question was not well-framed so I will ask again:
Can you explicitly produce an infinite set of real numbers which is algebraically independent over $\mathbb Q$?
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vote
1answer
94 views
Producing infinite family of transcendental numbers
Weierstrass proved the result [Lindemann-Weierstrass theorem] that if $a_1, \cdots, a_n$ are reals linearly independent over the rationals, then $e^{a_1}, \cdots, e^{a_n}$ are algebraically ...
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votes
2answers
156 views
Are some numbers more computable than others?
As I understand it (layman alert), the definition of computable numbers is binary: either a number is or is not computable.
Is it meaningful to imagine a function telling how computable (or ...
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votes
3answers
204 views
Numbers which are “Provably Difficult to Compute”?
We recall that a computable number $\alpha \in \mathbb{R}$ satisfies the following: there exists a computable function $f$ such that, given any positive rational error bound, $f$ outputs a rational ...
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0answers
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“The Galois group of $\pi$ is $\mathbb{Z}$”
Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question:
The Galois group of $\pi$ is $\mathbb{Z}$.
In what sense/framework is ...
18
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4answers
857 views
Uncountable set of irrational numbers closed under addition and multiplication?
Is such a thing even possible?
There's not much to say really. Obviously if there was a set it would be full of transcendental numbers. This led me to think of a function generating transcendental ...
0
votes
1answer
105 views
A question on transcendental numbers
Transcendental numbers are numbers that are not the solution to any algebraic equation.
But what about $x-\pi=0$? I am guessing that it's not algebraic but I don't know why not. Polynomials are over ...
5
votes
0answers
291 views
Is ${^5\pi}$ an integer? [duplicate]
Possible Duplicate:
How to show $e^{e^{e^{79}}}$ is not an integer
Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it?
Here ${^5\pi}$ means the result of tetration ...
9
votes
2answers
201 views
Erdős: Sum of rational function of positive integers is either rational or transcendental
I am trying to find a conjecture apparently made by Erdős and Straus. I say apparently because I have had so much trouble finding anything information about it that I'm beginning to doubt its ...
9
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1answer
151 views
Is there a dense subset of $\mathbb{R}^2$ with all distances being incommensurable?
Is there a set $S$ of points on the real plane $\mathbb{R}^2$ such that:
there is a point belonging to $S$ in any neighborhood of every point of $\mathbb{R}^2$ (so, $S$ is dense) and
ratio of any ...
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vote
1answer
170 views
Are numbers : $(-1)^{i} , 1^{-i} , 1^{i} $ transcendental numbers? [duplicate]
Possible Duplicate:
What is the value of 1^i?
According to Euler's formula : $e^{ix}=\cos x + i\cdot \sin x$ we may write :
$$e^{i\cdot \frac{\pi}{2}}=i \Rightarrow \left(e^{i\cdot ...
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votes
0answers
408 views
Convergent sum with primes
If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
9
votes
3answers
335 views
Closed form for a pair of continued fractions
What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ?
What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ?
It does bear some resemblance to the continued fraction for $e$, which is ...
