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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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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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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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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Is a complex number with transcendental imaginary part, transcendental?

A complex number that has transcendental real part is always transcendental? How about in the case of imaginary part?
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Is $6.12345678910111213141516171819202122\ldots$ transcendental?

My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer. After tiring of ...
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Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

I thought it was easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational, but found a gap in my proof. Simple finite approximations show the denominator cannot be small, though, ...
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90 views

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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65 views

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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Proving transcendental numbers

I'll warn now that this is probably a big question, but I am wondering if anyone can explain why it is so difficult to prove whether a number is transcendental or algebraic. For example, it is now ...
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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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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 ...
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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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Is $\pi/\sqrt{2}$ transcendental?

I believe that $\frac{\pi}{\sqrt{2}}$ is transcendental but I'm not sure about how to prove it. If $\frac{\pi}{\sqrt{2}}$ was algebraic, there would exist a polynomial $P \in \mathbb{Q}[X]$ such that ...
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Is a trigonometric function applied to a rational multiple of $\pi$ always algebraic?

Specifically, just to talk about cosine, is it true that $\cos(\frac{a\pi}{b})$ is algebraic for integers $a$ and $b$? Looking at this post and the link to trigonometric constants in the comments, it ...
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Is it known if $\pi + e$ is transcendental over the rational numbers?

I recall reading a comment on reddit that had stated that it is not known if $\pi + e$, (nor $\pi e$) is transcendental over $\mathbb{Q}$, nor even if it is irrational. Is this true? It strikes me as ...
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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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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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67 views

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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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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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 ...
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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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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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Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using Gelfond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental number ...
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Is there a number $x\neq0$ whose products with $\pi$ and with $e$ are both rational?

Does there exist a number $x\neq0$, such that $[x\cdot\pi\in\mathbb{Q}]\wedge[x\cdot{e}\in\mathbb{Q}]$? I thought this question would be easy to answer, but it turns out otherwise. Obviously ...
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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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Does there exist any positive integer $n$ such that $e^n$ is an integer (to show $\log 2$ is irrational)?

Does there exist any positive integer $n$ such that $e^n$ is an integer ? I was in particular trying to prove $\log 2$ is irrational; now if it is rational, then there are relatively prime ...
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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 $x^{x^{x^x}}$ 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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Is $ 0.112123123412345123456\dots $ algebraic or transcendental?

Since the decimal expansion $ 0.112123123412345123456\dots $ is non-terminating and non-repeating, clearly $ 0.112123123412345123456\dots $ is an irrational number. Can it be shown whether it is ...
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Equations involving factorial/Gamma function

Are there any known methods to formally solve equations like: 1)$x^3!+(2x^2)!-x!+3=0$ 2)$x!=e^x$ ($0$ is trivial but there must be another one) 3)$(2x!)^2+x!-1=0$ 4)$x!!+x!=7$ I don't need ...
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Do the Liouville Numbers form a field?

The Liouville numbers are those which are better-than-polynomially approximated by rationals. More precisely, we say $x\in\mathbb{R}$ is Liouville when for all $n\in\mathbb{N}$ there is a $\tfrac ...
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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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Why is Chaitin's constant absolutely normal?

I have repeadetly seen claims that Chaitin's constant is normal in all bases (e.g. on Wikipedia), and I have also seen some proof sketches (e.g. here), but these only show the idea. For example, the ...
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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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Calculating Closed form of Basel type problem.

I want to find the sum of $\displaystyle\sum_{n=1}^\infty \frac{1}{n^{\phi(n)}}$. where $\phi$ is Euler's totient. so for example 1/1^1 + 1/2^1 + 1/3^2 +1/4^2 + 1/5^4 + 1/6^2 etc... I also want to ...
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Irrational $x$ such that $\sin(\pi x)$ is algebraic

It's well-known that $\sin(\pi \frac pq)$ is always algebraic. In particular, as I understand, it can always be expressed in terms of radicals, because it can be connected to the abelian group of ...
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Is the sum of an algebraic and transcendental complex number transcendental?

I was wondering if the sum of an algebraic and transcendental complex number is transcendental. I was thinking if a is algebraic, and b is transcendental, then if a+b is algebraic, then a+b-a is also ...
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Prove that $E_0$ is transcendental

Consider the non-negative natural numbers: $0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19\dots$ Encode the primes as $1$, the rest as $0$. $E = 0,0,1,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1\dots$ ...
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Is $\ln n$ transcendental for all rational $n>1$?

I know that $\ln n$ is transcendental for all integer $n>1$. But does this still hold for non-integer rational values of $n>1$? For example, is $\ln 1.5$ transcendental? EDIT: Somehow managed ...
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When, how & who first gave this calculation of $\pi$

I came across this interesting method to calculate $\pi$. Why is it true and who first presented it? To calculate $\pi$, multiply by $4$, the product of fractions formed by using, as the numerators. ...
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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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When does the following construction generate a transcendental number?

Given $n\in[0,1]$ with base-b expansion $0.n_1n_2n_3\dots$, define $\Delta_b(n)$ to be the number with the following base-b expansion: $\huge{ 0.\underbrace{n_1}_{1^{st}\text{ ...