# Tagged Questions

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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### Cardinality of the set of algebraically independent transcendental numbers

It is not difficult to see that there must be an uncountable infinity of transcendental numbers – but has anybody yet proved that there is an uncountable infinity of algebraically independent ...
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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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### 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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### What is the name of this irrational math constant and is there a compact way to write it? 0.10110111011110…

I think this number is a transcendental number and I've tried looking online to see who first made it, I'm not sure if it's a Liouville Number or if there is a more common or better name for it. Does ...
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### How to show $\tan \alpha$ is transcendental?

I need to show that $\tan \alpha$ is transcendental for each non-zero algebraic number $\alpha$. Can any one give me an idea ? Thanks.
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### Can permutating the digits of an irrational/transcendental number give any other such number?

Let $x_n$ be the infinite sequence of decimal digits of a fixed irrational/trascendental number. Can I obtain any other irrational/trascendental number's sequence of decimal digits through a ...
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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 integers ...
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### Algebraic relations between trigonometric numbers

Given $n\in2\Bbb N$, what is precise algebraic relation between $cos\frac{\pi}{n-1}$,$cos\frac{\pi}{n+1}$? Both numbers are algebraic, which implies there should be an algebraic relation between ...
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### Why do transcendental numbers exist?

(This is a revision of the below question, which was not clear. If I have used incorrect terminology, please offer corrections.) Given the sets $A$ and $B$, $B$ contains transcendental elements ...
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### How could we prove that it is not a spanning set.

Consider the space $\mathbb{R}$ as a linear space over the field $\mathbb{Q}$ of rational numbers. For any transcendental number x the set {1, $x$, $x^2$, $x^3$,......} is linearly independent. How ...
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### Is $\frac{\pi}{e}$ an algebraic integer?

From what I know, it is still an open question whether or not $\frac{\pi}{e}$ is irrational, but is there a proof that $\frac{\pi}{e}$ is not an algebraic integer?
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### Prove that $e^n$ is irrational for any natural number $n$ [closed]

I have a question and it will be appreciated that you tell me some more details. Here is the question. For an arbitrary natural integer n, prove for any $n$, $e^n$ is irrational.
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### How to find a transcendental number where no two adjacent decimal digits are equal?

By using WolframAlpha, I couldn't find any transcendental number without equal adjacent digits among the numbes $\tan(n)$, $\sin(n)$, $\cos(n)$, $\sec(n)$, $\cot(n)$, $\csc(n)$, $e^n$, and $\log(n)$...
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### Proof that at most one of $e\pi$ and $e+\pi$ can be rational

$e$ and $\pi$ are rather peculiar numbers. It turns out that, in addition to being irrational numbers, they are also transcendental numbers. Basically, a number is transcendental if there are no ...
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### Is $\pi$ a rational multiple of e? [duplicate]

Does $\pi = re$ for some rational $r$? I assume the answer is no but cannot prove so.
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### Transcendental a infinitely close to rationals?

Apologies that this question is rather vague, but I do not know how to state it more precisely. Is, say pi, infinitely "close" to some rational number? More importantly, are all transcendental numbers ...
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### Extending the set of algebraic numbers

I have been trying to extend the countable set of algebraic numbers, by adding a countable amount of transcendental numbers (so that the resulting set is also countable). Now, of course I could ...
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### Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ ...
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### Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
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### Proof for $-4\pi^2+48\ne A+B\pi+C\pi^2$ when $(A,B,C)\ne (48,0,-4)$

I have to prove $-4\pi^2+48\ne A+B\pi+C\pi^2$ when $(A,B,C)\ne (48,0,-4)$ $A,B,C \in \mathbb{Q}$ As a part of question I try to solve.
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### If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$.

Let $a$ and $b$ be elements in extension field $F$. Is it true that: If $a$ is algebraic over $F$ and $b$ is transcendental over $F$, then $a+b$ is transcendental over $F$? I just did the same ...
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### Why are numeric methods the only technique available to solving $\ln(x) = \sin(x)$? Is this $x$ transcendental?

I just read this question about finding the solution to the equation $\ln(x) = \sin(x)$. All the answers focus on using a numerical method to approximate the solution. This is interesting in its own ...
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### What is the probability that a number chosen at random in $[0,1]$ is transcendental?

Consider the interval $[0,1]$. What is the probability that a number chosen at random in $[0,1]$ is transcendental? Please give me some points on how to start this problem.
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### Existence of a $\varphi \in \mathbb{R}$ such that $\cos(\varphi)$ is transcendental

Does anybody know an elementary proof that shows that there is a $\varphi \in \mathbb{R}$ such that $\cos(\varphi)$ is a transcendental number? I have read about the Lindemann-Weierstrass Theorem but ...
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### Do we know if all simple extensions of the field of rational numbers by transcendental numbers are not equal?

I understand that $\mathbb{Q}(x) \cong \mathbb{Q}(u)$ for all transcendental $u$, where $\mathbb{Q}(x)$ is the field of rational forms over $\mathbb{Q}$ and thus that all simple extensions of the ...
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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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### Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2 [duplicate]

All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly. What ...
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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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### For which $x$ is $e^x$ rational? Transcendental?

Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?
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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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