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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Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using the Gel'fond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental ...
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Are $\pi$ and $e$ algebraically independent?

Update Edit : Title of this question formerly was "Is there a polynomial relation between $e$ and $\pi$?" Is there a polynomial relation (with algebraic numbers as coefficients) between $e$ or $\pi$ ...
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$\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$ Are there any known ...
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Transcendental proofs vs. Irrational proofs

Why are proofs of the transcendence of certain numbers usually harder than irrationality proofs of those same numbers (for example, Lindemann's proof of the transcendence of pi vs. Niven's proof of ...
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Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can ...
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Numbers with no finite representation on paper

It occurred to me that there must be a lot of numbers without any form of finite representation on paper. Is there a name for these numbers? For example... Integers and rationals have a very simple ...
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The set $\{\frac{3^m}{\alpha^n}:\;m,n\in\mathbb Z\}$ is dense in $\mathbb R_+$

This seems that the set $$\left\{\frac{3^m}{\alpha^n}:\;m,n\in\mathbb Z\right\}$$ is dense in $\mathbb R_+$ (the set of positive real numbers), but I can not find the proof. How to prove this? Edit ...
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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 ...
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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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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 ...
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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...
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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?
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Proving that $\frac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\frac{(-1)^{(p_{k}-1)/2}}{p_{k}} \right )^{-1}$ 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 ...
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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 ...
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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 ...
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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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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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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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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 ...
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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 ...
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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!
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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 ...
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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 ...
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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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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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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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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 ...
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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 ...
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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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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 $0.1$ ?
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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)
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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 ...
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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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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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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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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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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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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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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 ...
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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} ...
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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 $\frac1{\log n}$ ...
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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. ...
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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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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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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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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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“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 ...
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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 ...