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 change of basis modify irrationality/transcendence?

Fix a real number $x$. We can consider its binary expansion, for instance $x = (0.01101001100101101001011\ldots)_2$. Now we consider the real number $y = (0.01101001100101101001011\ldots)_{10}$ : we ...
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$e+\pi$ and $e\cdot\pi$ Transcendental number or what?

$e+\pi$ and $e\cdot\pi$ Transcendental number or what? I know what $e$ and $\pi$ transcrendental and $e\cdot\pi$ posible be rational and posible to be irrational, ($\sqrt2\cdot\sqrt2=rational$,$...
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Example of a complex transcendental number?

Researching transcendental numbers I have only come across ones with a transcendental real part. I can't think of any which are pure imaginary or are not based on a real transendental number, t, of ...
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140 views

Complex transcendentals not known in component form?

Are there any transcendentals whose real or imaginary components have not been found in exact form?
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220 views

Is there a function whose limit approaches Pi? [closed]

I don't think my knowledge of Pi, irrationality, and transcendental numbers in general is complete. I've Googled for a day before posting this question. Intuitively, I understand why the ratio of ...
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Book that helps recognizing underlying mathematical expression of numerical results

I strongly remember having seen a couple of years ago a huge book that only consisted out of an enormous table over hundreds of pages containing approx. 10⁵ numerical results in ascending order with ...
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Are there any irrational numbers that have a difference of a rational number?

Are there any irrational numbers that have a difference of a rational number? For example, if you take $\pi - e$, it looks like it will be irrational ($0.423310\ldots$) - however, are there any ...
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643 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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Quantitative Statements in Transcendental Number Theory

I would like to know the purpose of having quantitative bounds in transcendental number theory. In particular, with the help of examples, I would like to know how to interpret these statements as ...
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67 views

Can e be expressed in pi's digits? (or vice-versa)

Given that both pi and e's decimal places are completely random and infinite... I was wondering about how people say that "Every phrase ever uttered can be expressed in the digits of pi" -- which, for ...
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136 views

$a$ transcendental $\implies a^a$ is transcendental?

Suppose $a\in \mathbb{C}$ is not a algebraic number. Then is $a^{a}$ also transcendental number ? I've not idea about how to do it. I got motivation for asking this question from the fact that $e^...
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9answers
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Calculating pi manually

Hypothetically you are put in math jail and the jailer says he will let you out only if you can give him 707 digits of pi. You can have a ream of paper and a couple pens, no computer, books, previous ...
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67 views

Non-algebraic structures?

We call group, ring, field,... "algebraic structures". Do we have similar analogue for transcendental numbers? If not, then how do we study interactions between various transcendental numbers? Also, ...
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28 views

Deducing Lindemann-Weierstrass from Baker's theorem

I'm aware that Baker's theorem with $n=1$ (for one algebraic number only) follows from that of Lindemann-Weierstrass. It is also often mentioned that Baker's result is a generalization of Lindemann-...
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12 views

On a Corollary of Liouville's Theorem

I want to prove, using Liouville's Theorem that: Let $\theta$ be an irrational algebraic number of degree $n$. Then, given any $\epsilon > 0$ there exist only a finite number of pairs of integers $...
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1answer
53 views

Five exponentials theorem

The six exponentials theorem is proved in most textbooks on transcendental number theory, and the four exponent conjecture is an open problem. Is there any good/accessible exposition of the five ...
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64 views

Cardinality of the set of algebraically independent transcendental numbers

It is not difficult to see that there must be an uncountable infinity of transcendental numbers – but has anybody yet proved that there is an uncountable infinity of algebraically independent ...
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If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ necessarily transcendental over $\mathbb{Q}$?

If $a,b \in \mathbb{C}$ are transcendental over $\mathbb{Q}$ then is $a^b$ is necessarily transcendental over $\mathbb{Q}$ ? In Wiki I found answer is no but I can't cook up an counter example. ...
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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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Prove $\pi+e$ or $\pi e$ is transcendental. [closed]

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 $...
6
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1answer
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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 Fransén-...
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71 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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1answer
67 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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422 views

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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376 views

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 tan(a)...
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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 haven'...
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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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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 $\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)/\left(\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\...
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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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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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42 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 we ...
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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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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 e^{\...
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40 views

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?