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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71 views

$e+\pi$ 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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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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1k views

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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39 views

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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11 views

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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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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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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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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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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196 views

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
106 views

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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87 views

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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40 views

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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10 views

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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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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3answers
76 views

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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14 views

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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128 views

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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19 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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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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56 views

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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44 views

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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1answer
74 views

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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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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64 views

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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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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34 views

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 question....
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66 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 e^{\...
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98 views

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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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\ ]=a_0+\...
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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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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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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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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 ...
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Logarithm of a transcendental number

Can anything be said about the nature of the number $\log y $ where $y $ is a transcendental number not of the form $y=e^x $ or written trivially in that form using $x=\log w $ for some $w $ ...
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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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Integration/Fundamental Calculus/Transcendental Numbers

Given $\int\sqrt{1 + \frac{-x}{\sqrt{4-x^2}}} dx$, how would this be integrated? On a universally-calculus side of things, do all functions have integrals? And is it easier to evaluate definite ...
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62 views

Is the sum of transcendental and algebraic number transcendental number?

I know almost nothing about transcendental numbers, I know the definition of them and maybe few results about them and that is all. But the question in the title somehow naturally arises when ...