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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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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88 views

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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92 views

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 ...
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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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38 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 ...
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3answers
148 views

Combinations of Transcendental Numbers are still transcendental numbers?

We know there are numbers like $\pi$, $e$, $\phi$ or also $\zeta(3)$ which are transcendental numbers. I was wondering if combinations of transcendental numbers are still transcendental numbers, like ...
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1answer
26 views

Finding transcendental roots to an algebraic equation

So for equations with rational roots, there's a theorem that lists all the possible roots (Rational Root Theorem). If an equation has imaginary or irrational roots, their respective theorems say ...
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1answer
24 views

Sum of algebraicly independent transcendentals is transcendental?

We say the transcendental numbers $\xi_1,...,\xi_n$ are algebraicly independent if an algebraic combination of them satisfies: $$ \sum_{1\leq i\leq n}\psi_i\xi_i= 0 \iff \psi_i=0, \forall i=1,...,n. ...
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30 views

How to prove $1/ \log a + 1/ \log b$ for rational $a$ and $b$ is a transcendental number?

I know how to prove $\log a$ for rational $a$ is transcendental, because if it were algebraic it would imply $e$ is algebraic as well (namely if $\log a = b, e = a^{1/b}$), and I can prove $\log a + ...
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27 views

Number with non-constructive algebraicity proof

Is there a computable number which is known to be algebraic but no explicit polynomial of which it is a root is known?
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38 views

Is every normal number transcendental? [duplicate]

This question is related to question http://math.stackexchange.com/q/197507 but not quite the same (or is it?): is every normal number transcendental? Can it be proved or is there a counterexample to ...
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1answer
29 views

Is this number a Liouville number?

Suppose I have a binary constant $q = 0.1010000000000000000000000000000000001001..._2$. In base 10 this number is $q $~$ .6250000000077325..$ and is defined as $$q = \sum_{\rho}^{\infty} ...
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A binary irrational with bits defined by primes

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to ...
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55 views

Is $\cos\log a$ a transcendental for all nonzero algebraic $a$?

Is this known? (excluding a=1 as was corrected in comments)
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If $a$ is a transcendental number, then is $a^n$ also a transcendental number? [closed]

If $a$ is a transcendental number (i.e., a number s.t. there does not exist a polynomial $P(x)$ s.t. $P(a) = 0$), is $a^n$ also transcendental? It would seem to me that it should be, but I can't ...
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Algebraic values of the Gamma function using $\pi$

Is there any $x\in(0,1)\cap\mathbb Q$ different from $1/2$ such that $\Gamma(x)$ is algebraic over $\mathbb Q(\pi)$?
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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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52 views

Other Algebraically Independent Transcendentals

I was thinking about a incomplete answer I gave earlier today to a interesting question made by the user lurker. The question was about wheter or not the sum or the product of two transcendental ...
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56 views

Adding or Multiplying Transcendentals

Is it possible to add or multiply (no subtraction) only positive transcendental numbers and yield a solution that is algebraic? Exponential manipulation is excluded from this question, as $e^{\ln2} = ...
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2answers
152 views

Is there a function whose limit approaches Pi?

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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To prove that element $\frac{3}{n}+i\frac{4}{5}$ has an infinite order in $\mathbb{C}$ for any $n\in\mathbb{Z}\backslash \{0\}$

The problem is to prove that element $z=\frac{3}{n}+i\frac{4}{5}$ has an infinite order in the group $(\mathbb{C},\, \cdot\, )$ for any non-zero integer $n$. Let's consider the case $|n|\neq 5$. ...
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Determine a number is transcendental/algebraic

Determine: $(0.064)^{\frac{1}{3}}$ is transcendental or algebraic To show a number is transcendental/algebraic do I need to show there is a monic polynomial with integer coefficients such that ...
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For which values of $\sin(\theta)$ is the function algebraic?

Earlier today I stumbled upon a very long formula for the sine of 1 degree. (http://www.efnet-math.org/Meta/sine1.htm). When I reflected on this, it occurred to me that I could probably make a similar ...
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Power of transcendental number.

There are some results that i found $1.$ if $a$ is an algebraic number other than $0$ and $1$ and $b$ is irrational algebraic then $a^{b}$ is transcendental like $2^{\sqrt{5}},3^{\sqrt{7}}$etc. $2.$ ...
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Are there any known transcendental which measures something in the natural world except pi and e? [closed]

For pi, it measures the ratio of the circumference and diameter of a circle, etc. And e also means many special thing(mesuring growth, Prime Number Theorem, etc.).
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Is the series: $\frac{\pi}{p_{1}!}+\frac{\pi}{p_{2}!}+…+\frac{\pi}{p_{n}!}$ convergent or divergent, where $p_n$ is the $n$-th odd prime?

Is the series: $$\frac{\pi}{p_{1}!}+\frac{\pi}{p_{2}!}+...+\frac{\pi}{p_{n}!}$$ convergent or divergent, where $p_n$ is the $n$th odd prime? And also why it is (the partial sums) ...
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257 views

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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1answer
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Let $a$ and $n$ be integers, such that $a,n>1$ and $n$ is not a perfect square; show that: $a^{\sqrt{n}}$ is a transcendental number.

Although it is very hard to determine if a number is transcendental, I could appreciate any basic or simple insight or opinion concerning the statement, whether it is true or false. Regards
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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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66 views

Is the golden ratio a transcedental number?

I have been studying the concept of transcedental numbers. Till now, I had taken it for granted that all important numbers like pi and e were transcedental. I have no reason for assuming this or for ...
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50 views

A simple(?) query about algebraic independence

Suppose $x,y,z\neq 0$, that $(x,y,z)$ is a point in $\mathbb{R}^3$ and that $td[\mathbb{Q}(x,y,z):\mathbb{Q}]=2$ (where $td[,]$ denotes the transcendence degree of the field extension). Is it true ...
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180 views

Random irrational number generator?

Is it possible to create a algorithm that will generate irrational numbers $0<x<1$ with a density that is uniform down a specified resolution? Would such an algorithm be necessarily limited to ...
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Is $2^e$ in the field extension $\mathbb{Q}(e)$?

As the title says, is $2^e$ in the field $\mathbb{Q}(e)$? I mostly study analysis, but this came up trying to answer someone else's question. So far, my idea has been to suppose it's true and use the ...
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Constructive proof of transcendence of $e$ and $\pi$?

Someone asks to me that can we prove the transcendence of $\pi$ without using proof by contradiction. I find some proofs of transcendence of $\pi$ and $e$ and I found that all of proof I found starts ...
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55 views

Rationality or irrationality of $\log$ function

Can this be proved that $\log(n)$ is irrational for every $n=1,2,3,\dots$ ? I find that question in my mind in searching for if $\log(x)$ is irrational for every rational number $x\gt0$.
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Is this true :${(a+ib)}^{(k+ij)}=0$ iff $0<a=k<1$ and $b<j$?

let $z=a+ib ,s=k+ij$ are two complex numbers and let $f(z,s)$ be a complex function defined as follow :$$f(z,s)=z^s={(a+ib)}^{(k+ij)}$$ and $a,b,j, k$ are non -nul real numbers . .After some ...
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When $\cos(\theta) = 1/8$ it's easy to show $\theta$ is an irrational angle. Is it algebraic?

Along the lines of my lines of my previous question about irrational angles "$45^\circ$ Rubik's Cube: proving $\arccos ( \frac{\sqrt{2}}{2} - \frac{1}{4} )$ is an irrational angle?", I was working on ...
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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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Does there exist a $z\in \Bbb R$ such that $\sin z=t \in \Bbb T$?

Does there exist a $z\in \Bbb R$ such that $\sin z=t \in \Bbb T$: the set of transcendental numbers? I've had this doubt and I didn't know how to tackle it... Edit: Changed my domain to reals only, ...
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Does the sequence $\{\sin(en)\}$ converge or diverge?

Is it known if $\{\sin(en)\}$ converges or diverges? Also, I have a more general question. For almost every rational $r$, I think we can say that $\{\sin(rn)\}$ diverges. Does that statement hold ...
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Prove $\log(x)$ is transcendental

What is a proof that $\ln(\alpha)$ is transcendental for $\alpha$. I believe I heard somewhere that the natural logarithm of any rational number is transcendental. Do you guys have any proofs of that ...
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Are there more transcendental numbers or irrational numbers that are not transcendental?

This is not a question of counting (obviously), but more of a question of bigger vs. smaller infinities. I really don't know where to even start with this one whatsoever. Any help? Or is it ...
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Is :$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$ irrational or transcendental or real number?

Is there someone who can show me if :$$\sqrt{i\pi+\sqrt{i\pi+\sqrt{i\pi+\sqrt\cdots}}}$$ is irrational or real or transcendental number ? Thank you for any help
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46 views

Transcendence of Values of Beta Function

Wikipedia mentions that the number $$a = \dfrac{\Gamma\left(\dfrac{1}{4}\right)}{\pi^{1/4}}$$ is transcendental. Since $\Gamma(1/2) = \sqrt{\pi}$ the above number $a$ seems to connected to a ...
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$\log \log (\alpha)$ transcendental??

$\log \log (\alpha)$ transcendental?? ($\alpha$ algebraic $\neq 0$ and $1$) I supposed $\log \log (\alpha)=\beta$ , $\beta$ transcendental. Then $\log(\alpha)=e^{\beta}$ and it is know $e^{\beta}$ is ...
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$\sum_{n=1}^{\infty} a^{-n!}$ is transcendental ??

Is $\sum_{n=1}^\infty a^{-n !}$ transcendental for any positive integer a ? I know $\epsilon =\sum_{n=1}^{\infty} 10^{-n!}$ is transcendental, for Liouville´s Theorem, ($p_k=10^{k!} \sum_{n=1}^k ...
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Transcendence of $\Gamma(1/3), \Gamma(1/4)$

Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a reference to that proof? Or maybe some details on the essential ideas ...
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Wrong proof…But where is the mistake?

So I've just watched this wonderful Numberphile video about transcendental numbers. In the video, the guy shows that ...