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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algebraic dependence over Q

Are numbers $\sqrt{2}$ and $e$ algebraically dependent over $\mathbb{Q}$? If yes, they belong to the same Mahler class. However, $\sqrt{2}$ is A-number, while $e$ is S-number. On the other hand, if ...
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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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Proof that ${\pi}$ can(not) be expressed as a root or as a root in combination with a fraction

I was doing some math for a programming project of myself and ran into decimal numbers and how to define them without losing precision while calculating an expression, so I tried writing them down as ...
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Algebraic numbers that cannot be expressed using integers and elementary functions

Can we give an explicit${^*}$ example of a real algebraic number that provably cannot be represented as an expression built from integers and elementary${^{**}}$ functions only? ${^*}$ explicit ...
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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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Where is the least upper bound property used in transcendence proofs?

The second-order theory of real numbers is what you get when you take the axioms for ordered fields and add one more axiom, the least upper bond property, also known as Dedekind completeness: that ...
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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: $$c_{\small d}x^{\small ...
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Proving that $\frac{\pi}{4}$$=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 ...
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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 ...
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Given $\log(p(x)) = q(x)$ are $p$ and $q$ algebraically independent?

Since $e^x$ and $\log y$ are transcendental functions, does $$\log p(x) = q(x)$$ mean that polynomials $p$ and $q$ (of finite degree $n$ and $m$ respectively) are algebraically independent? What ...
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How find $\Gamma{\left(\frac{8}{9}\right)}=\frac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$

show that $$\Gamma{\left(\dfrac{8}{9}\right)}=\dfrac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$$ where Gamma function:http://en.wikipedia.org/wiki/Gamma_function I found this problem is ...
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284 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 ...
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476 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 ...
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Are elements of the range of a transcendental function themselves transcendental, excepting a “few” special cases?

Let $f(x)$ be a transcendental function with $x\in\mathbb{C}$. Then are the values $f(x)$ themselves transcendental, except perhaps for a "few" exceptions? For example, it is known that $f(x)=e^x$ is ...
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Is $a^b$ transcendental when $a$ and $b$ are?

I am being asked this question as an exercise in Garling's "A course in Galois theory". But isn't this an open question in math?
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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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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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Gelfond-Schneider Constant $2^{\sqrt{2}}$

Someone knows a proof (books , articles) that $2^{\sqrt{2}}$ is irrational ? Without using that $2^{\sqrt{2}}$ is transcendent. Any hints would be appreciated.
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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? ...
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The origin of $\pi$

How was $\pi$ originally found? Was it originally found using the ratio of the circumference to diameter of a circle of was it found using trigonometric functions? I am trying to find a way to find ...
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115 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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Please help me understand this proof that $e$ is transcendental

This started with my question "Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?". Kunnysan suggested that I model a proof on the standard proof that $e$ is transcendental. I ...
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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 ...
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Is $\large \frac {\pi}{e}$ rational, irrational, or trandescendal?

Is there an argument for why $\large \frac {\pi}{e}$ is rational, irrational, or trandescendal? Can the quotient of any two transcendental numbers (which are not rational multiples of each other) be ...
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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 ...
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number of the form $\frac{a_0+a_1\pi +\text{…}+a_n\pi ^n}{b_0+b_1\pi +\text{…}+b_m\pi ^m}$ [closed]

all numbers of the form $$\begin{align*}\frac{a_0+a_1\pi +\text{...}+a_n\pi ^n}{b_0+b_1\pi +\text{...}+b_m\pi ^m}\end{align*}$$ form an number field. $n,m$ are arbitrary non-negative itegers, ...
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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 ...