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 $x^{x^{x^x}}$ 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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What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? ...
31
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661 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 ...
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504 views

Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly ...
25
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865 views

Is this number transcendental?

My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer. After tiring of ...
23
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686 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)
21
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558 views

Is $0.1010010001000010000010000001 \ldots$ transcendental?

Does anyone know if this number is algebraic or transcendental, and why? $$\sum\limits_{n = 1}^\infty {10}^{ - n(n + 1)/2} = 0.1010010001000010000010000001 \ldots $$
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450 views

Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
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4answers
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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 ...
19
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376 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?
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Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the ...
17
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804 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 ...
17
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529 views

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 $\pi$ ...
15
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296 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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Irrationality of $\pi$ another proof

Proposition. Let $\alpha\in\mathbb{R}$. If there is a sequence of integers $a_n,b_n$ such that $0<|b_n\alpha-a_n|\longrightarrow 0^+$ as $n\longrightarrow \infty$, then $\alpha$ is irrational. ...
12
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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 ...
12
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438 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*} ...
12
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Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

It's easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational. However, can it be shown whether it is algebraic or transcendental? My hunch is that it's transcendental but I don't know ...
12
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262 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...
11
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562 views

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 ...
11
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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 ...
11
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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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348 views

Linear independence of the numbers $\{1,\pi,{\pi}^2\}$

Does someone know a proof that $\{1,\pi,{\pi}^2\}$ is linearly independent over $\mathbb{Q}$ ? The proof should not use that $\pi$ is transcendental. $\{1,e,e^2,e^3\}$ is linearly independent over ...
11
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181 views

Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?

This question is inspired by my answer to the question "How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?". The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer ...
10
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439 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} ...
10
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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 ...
9
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Can $x^x$ be a natural number for non-integer $x$?

Does some real non-integral $x$ exist such that $x^x$ equals a natural number? Thanks, Tom
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255 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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On Bailey and Crandall's sum for $\sum_{n=0}^\infty \frac{1}{5^{5n}}\left(\frac{5}{5n+2}+\frac{1}{5n+3}\right)$

On page 20 of "On the Random Character of Fundamental Constant Expansions", Bailey and Crandall gave the rather unusual sum, $$u_2 = \sum_{n=0}^\infty ...
8
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Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
8
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507 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 ...
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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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Transcendentality of the $\log$ of the golden mean

We know that $\phi$, the golden ratio, is algebraic. Is it known whether $\log(\phi)$ is algebraic? Thank you! PS. I am not in number theory, so I apologize in advance if this is obvious.
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Prove that $\pi$ is a transcendental number

Does anyone has a link to a site that confirms that $\pi$ is a transcendental number? Or, can anyone show how to prove that $\pi$ is a transcendental number? Thank you in anticipation!
7
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1answer
304 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 ...
7
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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 ...
7
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165 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 ...
7
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Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Transcendental?

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Clearly this series is convergent (compare to geometric series with ratio 1/2). I'm sure it's irrational since a rational number written in base 2 ...
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What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental. My question is what are the transcendentals ...
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Is the solution of $e^x \log(x)=1$ transcendental?

Let $u$ be the solution of the equation $$e^x \log(x)=1$$ Is $u$ rational, irrational algebraic or transcendental? $u$ seems to be transcendental, but I cannot prove it. Perhaps, someone has an ...
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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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How do we prove the existence of uncountably many transcendental numbers?

I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that ...
6
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Can $\pi$ be a root of a polynomial under special cases?

What if we consider polynomials whose coefficients are either rational or $e$, that is, a polynomial in $\mathbb{Q} \cup \{e\}$ with $\pi$ as a root. Can this happen? Does it matter if we change ...
6
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What is $\sum\limits_{n=0}^{\infty} r^{an^2 + bn + c}$ ? or: is $0.0100100010000100001…$ transcendental?

The idea is a more convenient form for $N = 0.01001000100001000001...$ in base $r$, hopefully to show whether it is transcendental. Sorry for brevity.
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Is $\log 2\pi$ rational?

Is it known whether $\log 2\pi$ is rational (where the base of the logarithm is $e$)? Or algebraic?
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Does the number $2.3\,5\,7\,11\,13\ldots$ exist and, if so, is it rational or irrational &/or transcendental? [duplicate]

Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$ if so, will it be rational or irrational &/or transcendental?
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Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: $$c_{\small d}x^{\small ...
5
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Is product of algebraic and transcendental number transcendental?

Let $\alpha \in \mathbb{A}$, and $\gamma \in \mathbb{T}$. I know that the reciprocal of a transcendental number is transcendental. Question: Is $\alpha\cdot \gamma \in \mathbb{T}$?
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For integer $k > 1$, is $\sum_{i=0}^{\infty} 1/k^{2^i}$ transcendental or algebraic, or unknown?

Title says it all, I have an itch about series like this that seem to fall in the gray area where classical proofs that rational partial sums that converge too quickly must converge to transcendental ...