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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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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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 $h(...
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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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71 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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193 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 $|n|\neq 5$. The ...
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487 views

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 the ...
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32 views

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) transcendental? ...
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270 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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36 views

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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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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51 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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289 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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115 views

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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60 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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39 views

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 unsolvable?...
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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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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 10^{-...
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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 $$e=\sum_{n=0}^\infty\frac{1}{n!}=1+\frac{1}{1}+\frac{1}{1\cdot2}+\frac{1}{1\...
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Do the second-last-digits of the primes $\ge 11$ form a transcendental number?

Suppose, the number $x$ is constructed from the second-last-digits from the primes $\ge 11$ The first $1996$ digits of $x\ =\ 0.11112...$ after the decimal point are : ...
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Transcendental number

While reading on Wikipedia about transcendental numbers, i asked myself: Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e, \frac{\pi}{e}$ etc. are transcendental numbers? ...
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Simplest proof that some number is transcendental?

I tried googling for simple proofs that some number is transcendental, sadly I couldn't find any I could understand. Do any of you guys know a simple transcendentality (if that's a word) proof? E: ...
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Prove: $e^x$ is transcendental over the polynomials with coefficients in $\mathbb{R}$

I have to prove the following for my math study: Prove: $e^x$ is transcendental over the polynomials with coefficients in $\mathbb{R}$. So far, I've done this: It's enough to prove that if $a_n(...
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If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then $a^b$ is transcedental.

For my math study, I have to prove the following: Let's denote the set of algebraic numbers with $\mathbb{A}$. Prove: If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then ...
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Is $e^{e^9}$ an integer?

I mean, of course $e^{e^9}$ is not an integer, but can we prove this? If you're thinking of asking Wolfram|Alpha, be warned: it gives different answers to the questions "is exp(exp(9)) an integer" (WA ...
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When $\cos x$ is transcendental?

About the transcendence of trigonometric functions I know that: 1) if $x$ is an algebraic number $\ne 0$ than $\cos x$ is transcendental. 2) if $p=\dfrac{m}{2^n}$ with $m,n \in \mathbb{Z}$ than $\...
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Algebraic subfield of transcendental extension

I was recently thinking about whether it is possible to generate an infinite dimensional algebraic extension over a base field using just finitely many transcendental elements. Specifically, given a ...
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How do I prove that $\forall \beta\in F(\alpha)\setminus F$ is transcendental?

Let $E/F$ be a field extension. Let $\alpha\in E$ be transcendental over $F$. Let $\beta\in F(\alpha)\setminus F$. Then, how do I prove that $\beta$ is transcendental over $F$? Here's how I tried: ...
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If $a, b$ are transcendental then $a+b$ is transcendental or $ab$ is transcendental [duplicate]

I have to prove the following: If $a, b$ are transcendental then $a+b$ is transcendental or $ab$ is transcendental, or both. I don't have any idea on how to solve this. I already proved this: ...
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Are all normal numbers transcendental?

Are all normal numbers transcendental? Just a question I've come up with.
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Is the product of a transcendental number by an integer transcendental?

I don't really know a lot about this subject but I was wondering if the product of a transcendental number by an integer is transcendental?
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Can transcendental to the power transcendental be rational?

Can a transcendental number to the power of a transcendental number be a rational number?
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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 ...