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 $\pi+e$ or $\pi e$ is transcendental. [on hold]

I understand to prove at least one of them irrational you would compose a function by which $\pi$ and $e$ are roots $((x-\pi)(x-e))$, and show that at least one coefficient cannot be rational because ...
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How was the difference of the Fransén–Robinson constant and Euler's number found?

I recently ran across the following integral: $$ \int_{0}^{\infty}\frac{1}{\Gamma(x)}dx $$ Which I learned is equal to the Fransén-Robinson constant. On the linked wikipedia page for the ...
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57 views

Proof of transcendence of $\ln (\pi)$

From Wikipedia $\ln (\pi) $ is unknown to be transcendental. $e^{(ie^{(\ln(\pi)})}=-1$ $i(e^{(\ln(\pi)})=i\pi$ is transcendental. Due to the Lindemann–Weierstrass theorem any transcendental ...
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Complicated series converges to $\pi$.

How do I get this result? $$\frac {426880 \sqrt {10005}}{\large \sum_{k = 0}^{\infty}\frac {(6k)!(545140134k + 13591409)}{(k!)^3 (3k)! (-640320)^{3k}}} = \pi$$ It seems formidable. Context: I came ...
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66 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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Why do transcendental numbers exist?

(This is a revision of the below question, which was not clear. If I have used incorrect terminology, please offer corrections.) Given the sets $A$ and $B$, $B$ contains transcendental elements ...
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Help complete this proof on transcendentalism

Proof $\pi*e$ is transcendental. either $\pi + e$ or $\pi*e$ is transcendental to see take $(x-\pi)(x-e)=x^2-(\pi+e)x+\pi*e$. Case 1 assume $\pi$ and $e$ are algebraically independent. It follows ...
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Algebraic dependent summation of transcendental numbers.

Question if $a$ and $b$ are both transcendental but algebraically dependent over Q. what do we know about $(a+b)$? In particular is there a way to bring $(xa^y+nb^m)$ where $x,y,n,m$ are rational and ...
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Can $\pi$ or $e$ be a root of a polynomial with algebraic coefficients?

Since $\pi$ and $e$ are transcendental, neither can be the root of a polynomial with rational coefficients. However, it is easy to construct a polynomial transcendental coefficients (with $\pi$ or $e$ ...
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Is there a proof that says that an operation that can take a transcendental number and make it an integer cannot exist?

Motivation: To get an integer to become a different integer, you have to add or subtract another integer, e.g. $1+2=3$ To get a rational number to become an integer, you have to multiply by the ...
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Does successive application of $\sin$ function on some nonzero algebraic number ever yields a sequence of transcendental numbers?

On the Wikipedia page about Transcendental numbers there is a section about numbers that are proven to be transcendental and there you can read that transcendental numbers are sin(a), cos(a) and ...
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Independent Transcendental Numbers

I've been thinking about numbers which have not yet been proven nor disproven to be transcendental, such as $e + \pi,\, \pi - e,\, \frac{\pi}{e},\, \gamma,\,\zeta(3),$ etc. Some of these numbers ...
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104 views

When did Liouville come up with the first transcendental numbers?

There are some conflicting sources regarding this. It is a matter of fact that Liouville defined what it was for a number to be approximated to degree $n$ by rational numbers. He then effectively ...
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Connection between prime numbers and transcendental numbers

I think there may be a strong connection between prime numbers and transcendental numbers. I am unable to prove what I have in mind by myself, so I am seeking help. My hypothetic theorem would be: ...
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Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
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16 views

Spigot algorithms for transcendental numbers

I'm trying to write a program that will compute digits of transcendental numbers using a spigot algorithm. While researching, I found the BBP Formula, and a Compendium of BBP-Type Formulas, alas, I ...
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41 views

Algebraic values of sine function

Are there algebraic inputs to the sine function that produce algebraic outputs? Other than zero, that is? This is assuming the sine function in radians.
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Is there an explicit irrational number which is not known to be either algebraic or transcendental?

There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be ...
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relation betwn ln and e

If $f(x) = ln(x)$ and $f^-1(x) = e^x$ then is $e^x = 1/ln(x)$??? because I see $e^9 = 8103$ but $1/ln9 = .455$ How are they reverse? I don't understand!
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Is this a well defined problem in terms of Euclidean Geometry?

I am trying to construct an example of a geometric problem, stated in terms of Euclidean Geometry, that is not Machine Provable (or in an equivalent definition Automatically Provable)-i.e no computer ...
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The number $\sum\limits_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$ is transcendental

Prove that the number: $$\sum_{n=-\infty}^{\infty} \frac{1}{2^{n^2}}$$ is transcendental. I don't have a direct proof but a round one. The series can be expressed in terms of $\vartheta_3$ ...
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Why do we care about the Champernowne constant?

I was browsing code golf and I came across this challenge: http://codegolf.stackexchange.com/questions/68685/the-rien-number It caught my interest and I wanted to learn a bit more about the ...
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Is $(-2)^{\sqrt{2}}$ a real number?

Is $(-2)^{\sqrt{2}}$ a real number? Clarification: Is there some reason why $(-2)^{\sqrt{2}}$ is not a real number because it doesn't make sense why it shouldn't be a real number. Mathematically ...
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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.
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In simple English, what does it mean to be transcendental?

From Wikipedia A transcendental number is a real or complex number that is not algebraic A transcendental function is an analytic function that does not satisfy a polynomial equation However these ...
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If $\theta$ is a rational number, is $e^{i\pi\theta}$ algebraic?

I want to know if $\theta$ is a rational number, is $e^{i\pi\theta}$ an algebraic number or not? For the first step I tried to write it $(e^{i\pi})^\theta$, that equals $(-1)^\theta$, but I think ...
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Where are the transcendental numbers?

This question is motivated from an exercise from Rudin. The exercise says that prove that set of all algebraic numbers is countable. Proof: We know that a number $z$ is called algebraic if it is the ...
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Values of Eisenstein Series

I'm trying to prove the algebraic independence of $\pi, e^\pi$ and $ \Gamma(1/4)$ while using Nesterenko's Theorem ($\{q, P(q), Q(q), R(q)\}$ contains at least three algebraically independent numbers ...
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64 views

How to prove the following formula is a constant?

For any $\rho\in\mathbb{R}^+$, prove that the following formula equals a constant: $$\dfrac{1}{\rho^2}{\int_{-\rho}^\rho x^2 e^{\left(\tfrac{\rho^2}{x^2-\rho^2}\right)}dx}\left({\int_{-\rho}^\rho ...
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Is there an easy enough way to show that between two algebraic numbers there is an infinite number of transcendental numbers?

We know that between two different rational numbers there is an infinite number of irrational numbers and that between two different irrational numbers there is an infinite number of rational numbers. ...
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Do transcendental numbers contain any string of digits?

It is often said that $\pi$ contains any string of digits. But does the property "transcendental" imply "contains any string of digits?
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Liouville numbers and continued fractions

First, let me summarize continued fractions and Liouville numbers. Continued fractions. We can represent each irrational number as a (simple) continued fraction by $$[a_0;a_1,a_2,\cdots\ ...
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Finite powers of fixed transcendentals

I was thinking about the complex unit circle $S_1\subset\mathbb{C}$ as a group under multiplication and how, if possible, an element $z$ of infinite order could generate the circle itself. A friend ...
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Can I construct a line segment with the length $e$ or $\pi$? [closed]

What I really mean is that without restriction(only circle and ruler),can we construct it with geometric method or something else. If we can or not,how or why?I am just interested in this ...
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Why is Gelfond's constant transcendental?

I have seen a proof of $\pi$ being transcendental by conclude that transcendental number powered by algebraic number must be transcendental and algebraic number powered by algebraic number must be ...
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Convert PI to base 4. Does my unique human genome exist in the sequence of digits?

The human genome consists of sequences of BASE Pairs A G C T Convert the number PI to base 4. Does my unique human genome exist in the sequence of digits?
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Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
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Is $0.248163264128…$ a transcendental number?

My question is in the title: Is $a=0.248163264128…$ a transcendental number? The number $a$ is defined by concatenating the powers of $2$ (in base $10$). It is possible to express $a$ as a ...
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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 ...
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Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $ a=\sqrt 2 = 1.41421356237309504880...$, and define $$b=f(a)=1.14243165323790058408...$$ by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to ...
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Proof of irrationality of $\zeta(2)$ without explicitly calculating it

Question is pretty much the title. It is pretty easy to show that $\zeta(2n)$ is irrational for all $n$ once you know that $\zeta(2n)$ is a rational multiple of $\pi^{2n}$ (and then also use the fact ...
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Calculating Closed form of Basel type problem.

I want to find the sum of $\displaystyle\sum_{n=1}^\infty \frac{1}{n^{\phi(n)}}$. where $\phi$ is Euler's totient. So for example $\frac{1}{1^1} + \frac{1}{2^1} + \frac{1}{3^2} +\frac{1}{4^2} + ...
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Do the Liouville Numbers form a field?

The Liouville numbers are those which are better-than-polynomially approximated by rationals. More precisely, we say $x\in\mathbb{R}$ is Liouville when for all $n\in\mathbb{N}$ there is a $\tfrac ...
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Strange Algebraic Number

We call a number algebraic if and only if it is the solution of a polynomial with integer coefficients. A number (complex or real) is transcendental if and only if it is not algebraic. A while back ...
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Analytic forms of $\frac{\sinh(2\pi/7)}{\sinh^{2}(3\pi/7)} - \frac{\sinh(\pi/7)}{\sinh^{2}(2\pi/7)} + \frac{\sinh(3\pi/7)}{\sinh^{2}(\pi/7)}$

On page 183 of Berndt's Ramanujan's Notebooks Vol. 4, eq. 32.34 reads: $$ \frac{\sin(2\pi/7)} {\sin^{2}(3\pi/7)} - \frac{\sin(\pi/7)}{\sin^{2}(2\pi/7)} + \frac{\sin(3\pi/7)} {\sin^{2}(\pi/7)} = ...
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Is there a pythagorean triple such that all angles of the corresponding triangle are simple fractions of $\pi$?

Obviously, the most interesting pythagorean triple $(a, b, c)$ would be one for which the corresponding triangle (with integer side lengths $a, b, c$) has angles 90°, 60° and 30° ($\frac{\pi}{2}, ...
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Is $\pi^0$ actually rational? How about $\pi^i$? [duplicate]

Is there a rational argument that a transcendental or irrational number raised to zero should magically turn it into an integer, beyond obtuse convention? How about $\pi^i$? Is there a reasonable ...
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Real numbers that are not the roots of any polynomial equation with algebraic coefficients

An algebraic number is a number which is a root of some non-zero polynomial equation with rational coefficients. A transcendental number is a number which is not a root of any non-zero polynomial ...
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Is $\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+…$ algebraic or transcendental?

Inspired by this question, the series $\dfrac{1}{2^{2^{0}}}+\dfrac{1}{2^{2^{1}}}+\dfrac{1}{2^{2^{2}}}+\dfrac{1}{2^{2^{3}}}+\dots$ is clearly irrational. But is it algebraic or transcendental? I ...