# Questions tagged [transcendental-numbers]

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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### Criteria for irrationality of Euler's constant

Define for $n\in\mathbb{N}$, $$I_n=\int_0^1\int_0^1 -\frac{(x(1-x)y(1-y))^n}{(1-xy)\log xy}dx dy$$ In this article it is proved that $$I_n=\binom{2n}{n}\gamma+L_n-A_n$$ where $L_n=d^{-1}_{2n}\log S_n$,...
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### Are angles of Pythagorean triples, e.g. $\tan^{-1}\frac34$, transcendental multiples of $\pi$?

Simple question, but I don't know the answer and can't easily find good resources. Sometimes we can give quite surprising exact forms to the circular functions at peculiar arguments through quite ...
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### Does reinterpreting the base of a number leave transcendence unchanged?

Define the 'reinterpret $x$ from base $a$ to base $b$' function $R_{a,b}(x)$ as $$R_{a,b}(x)=\sum_{i\in\mathbb{Z}}b^iD_{a,i}(x)$$ where $D_{m,i}(x)$ is the $i$-th term in the $m$-ary expansion of $x$...
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### Is $\left\{\left.\sum_{k\ge1}10^{-p^{k!}}\,\right|\,p\text{ prime}\right\}$ algebraically independent?

I'm trying to come up with a countable set of real numbers that are transcendental and algebraically independent that's concrete and easy to work with. The idea I had was to come up with very sparse ...
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### Minimal polynomial of $x$ over $K(x^n)$ where $x$ is transcendental

I am struggling with the following problem: Let $L/K$ be a field extension and $x \in L$ be transcendental. Show that the field extension $K(x^n) \subset K(x)$ is algebraic of degree $n$. Determine ...
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### Is $\sum_{k=1}^\infty \frac{1}{p_{p_k}}$, where $p_k$ is the $k$-th prime, irrational? transcendental?

I was reading about the reason why the reciprocals of the primes have a divergent sum. So I was thinking of changing the index to the $k$th prime. We get: $$\sum_{k=1}^\infty \frac{1}{p_{p_k}}=S$$ ...
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### Is the number satisfying $\eta=\sin(\cos(\eta))$ transcendental?

I was graphing the function $\sin(\cos(\sin(\cos(\sin(\cos...$ when I realized it started to flatten out. This meant that this approaches a constant. Since the sine and cosine repeat, we can make a ...
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### Is my claim that this number is irrational correct?

Define a number $c$ in the following way: $$c=\ln \left(\prod^\infty_{k=1}\frac{e^{1/k}}{1+\frac{1}{k}}\right)$$ (I can assure you that this converges). Isn't this number transcendental since the ...
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### Classification of real numbers $\ x > 1\$ such that $\ \{\ \{x^n\}: n\in\mathbb{N}\}\$ is dense in $[0,1].$

I am looking for real numbers $\ x > 1\$ such that $\ \{\ \{x^n\}: n\in\mathbb{N}\}\$ is dense in $[0,1].\$ Here, $\ \{y\}\$ means the fractional part of $\ y\in\mathbb{R}.$ What I know: I ...
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### What is the reason that equations such as $\tan x = 2x$ can only be solved with the help of algorithms?

This is my first StackExchange question: What is the reason that equations such as $\tan x = 2x$, $\cos x = x$, $\sin(x) = x^2$ and other questions that involve the same variable within a ... 97 views

### Solution to transcendental equation

I'm looking for solutions to the equation $$x^2+2^x+x^x = 12$$ Which is satisfied obviously by $x=2$ and somewhat less obviously by $x\approx-3.4512$. By plotting $|z^2 + 2^z + z^z - 12|$ on the ...
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### At least one of these numbers are irrational: $a$ or $\ln a$.

I've found a proof that: $(1)$ if $a\neq1$ is a positive real number than at least one of these numbers are irrational: $a$ or $\ln a$. I was told that this result is a corollary of Lindemann theorem, ...
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### A constant written in terms of non-elementary functions

I found out that: $$\frac{\int \Pi(x)H(x)dx-\Pi(x)}{\int \Pi(x)dx}=\gamma$$ Where $H(x)$ is the $x$th harmonic number, $\Pi(x)$ is the analytic continuation of $z!$, and $\gamma$ is the Euler-...
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### About the algebraic character of the solution of $\cos x=x$ [duplicate]

The equation $x=\cos x$ is well-known because some facts. For example, with an old calculator, you can find approximations of the solution by typing any number and pressing the $\cos$ button ...
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### Fractional Part of Algebraic Number

I want to know whether a fractional part of algebraic number is still algebraic (moreover with the same degree). Is the statement true? I was trying to find the minimal polynomial explicitly from the ...
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Two complex numbers $\alpha,\beta$ are called algebraically independent if there is no polynomial $p(x,y)\in\mathbb{Q}[x,y]$ such that $p(\alpha,\beta)=0$. For a complex number $x$, let $A(x)=\{z\in\... 1 vote 0 answers 37 views ### Difficulty understanding a proof of "Baker's lemma" on bounding complex functions, involved in the proof of Baker's theorem?$\newcommand{\d}{\mathrm{d}}$I'm reading a proof of a weak version of Baker's theorem, presented here. I've managed to get almost all of the way through, but I'm struggling with section$4$, page$6$, ... 4 votes 0 answers 69 views ### Can pi be written as algebraic+log(algebraic)? The Universal parabolic constant is "the ratio between the arc length of the parabolic segment formed by the latus rectum to the focal parameter". It can be seen as '$\pi$but for parabola'. ... 5 votes 1 answer 100 views ### Herstein's proof of transcendence of$e$This question concerns Herstein's proof (in Topics in Algebra) that$e$is transcendental. If you have the book, it's Theorem 5.2.1, page 218, or there's a transcription here: https://sites.math.... 1 vote 1 answer 167 views ### Show that:$e^{\pi}-\pi^{e} >\frac12$without calculator Show that: $$e^{\pi}-\pi^{e} >\frac12$$ I am not sure, if anything can be done using elementary methods. Because$e$and$\pi$are not algebraic numbers. Therefore, I find it impossible to prove ... 0 votes 1 answer 87 views ### Knowing that$\pi \in \mathbb R$is a transcendent element over$\mathbb Q$.... Knowing that$\pi \in \mathbb{R}$is a transcendent element over$\mathbb{Q}$. Find a subfield$\mathbb{F}$of$\mathbb{R}$such that$\pi$is algebraic over$\mathbb{F}$, with$\deg(\pi, \mathbb{F}) =...
Definition $1.$ An irrational number $x$ is called a Liouville number if for any positive integer $n$ there exists a pair of integers $(p,q)$ such that $q\gt 1$ and $|x-p/q|\lt 1/q^n$. Definition $2.$ ... ### What is the problem in proving $\pi$ is transcendental over $\mathbb{Q}$ in purely algebraic method
Why can’t we prove for the field extension $\mathbb{R}$ over $\mathbb{Q}$ ,$\pi$ is transcendental over $\mathbb{Q}$ in a purely algebraic method? Or is there any prove that proves $\pi$ is ...