# 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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### Is this sum a transcendental number?

Question Is the sum $S$ a transcendental number? $$S = 1 + \frac{1}{12}+\frac{1}{123}+\frac{1}{1234}+\cdots$$ 123456789 is followed by 12345678910, 1234567891011 Some Search Results The denominator ...
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### positive(?) definition of a transcendental number - as opposed to negative def. not an algebraic number [closed]

Does this make sense? What's the formal definition of a transcendental number, but without saying it's not an algebraic number?
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### Elementary way of showing that $e^na-b=0$ is false for all $m,a,b\in\mathbb{N}$.

I want to show that $e$ (wich we will assume to be irrational) is no a root for the polynomial $p(x)=x^na-b$ with natural coeficients, in an elementary way (I'm aware that $e$ is transendental but a ...
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### What if $\pi e$ is algebraic? [duplicate]

Today I heard that $\pi$ and $e$ are transcendental. I don't think I have ever seen a proof of this, but I see this is available online. I also see that it is not know whether $\pi e$ is algebraic or ...
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### Proving that $\sum_{n = 1}^\infty 2^{-4^n}$ is transcendental

I want to prove that the following number is transcendental: $$\alpha = \sum_{n = 1}^\infty 2^{-4^n}$$ Liouville's theorem states that if $\alpha$ is algebraic with minimal polynomial of degree $d$, ...
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### How to prove that $k^{\pi}$ is not an integer for any integer $k\geq 2$?

I strongly suspect that $k^{\pi}$ is not an integer for any integer $k\geq 2$ (for otherwise this would be a famous result of which I am not aware). But how does one prove this? The answer to this ...
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### Is $e^a$ always algebraic for transcendental $a$?

From Lindemann–Weierstrass theorem, it is known that $e^a$ for non-zero algebraic $a$ is always transcendental. But if $a$ is transcendental, is the opposite ($e^a \in \mathbb A$) always true?
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### Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number?

Is the ratio of the perimeter of any shape with circular curves to its diameter result in an irrational number? I suppose it would depend one what would be defined as "circular curves", as ...
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### Disjoint algebraically independent sets

Let $A,B\subset\Bbb R$ algebraically independent sets over $\Bbb Q$ and $A\cap B=\emptyset.$ Let $T$ be a transcdental basis of $\Bbb R$ over $\Bbb Q$ such that $A,B\subset T.$ Now, $qA$ is ...
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### What is the simplest proof that $\sqrt{n}^\sqrt{n}$ is transcendental when $n$ is not a square? What is the most elegant proof?

Consider the following question -Are there two irrational numbers $a$ and $b$ such that $a^{b}$ is rational?- Well, suppose $\sqrt{2}^{\sqrt{2}}$ is rational, then we are done. If we suppose it is ...
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### Cardinality of transcendental basis over field generated by algebraic independent set

Recall that $T$ is a transcendental basis of $\Bbb R$ over $\Bbb Q$ if it is a maximal algebraically independent set. Also, For $B\subset\Bbb R$, the transcendental degree of $\Bbb R$ over $\Bbb Q(B)$...
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### Why Dottie$=2\sqrt{I^{-1}_\frac12(\frac 12,\frac 32)-I^{-1}_\frac12(\frac 12,\frac 32)^2} = \sin^{-1}\big(1-2I^{-1}_\frac12(\frac 12,\frac 32)\big)$?

Introduction: For some background information on the Dottie Number D, see the great posts at: What is the solution of $\cos(x)=x$? Some definitions: The “solution” to Kepler’s equation is Kepler E: ...
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### Does there exist a subfield $K\subsetneq \mathbb{C}$ such that $x$ is transcendental over $K$ for all $x\notin K$?

My conjecture is no. Proof: Suppose, to the contrary, that such a field $K\subsetneq \mathbb{C}$ exists. Then for all $x\notin K$, $p(x)\neq 0$ for all $p(t)\in K[t]\backslash\{0\}$. Screeching halt. ...
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### Show that $\sqrt[3]{\pi}$ is transcendental

A similar question about $\sqrt{\pi}$ was asked here and was beautifully explained. Now, how do we show that $\sqrt[3]{\pi}$ is also transcendental using similar techniques, without any advanced ...
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### Effective constants in Baker's bound on approximations of log(3)

Alan Baker gives an upper bound on $1/\log(3) - p/q$, providing a limit on how well $\log(3)$ can be approximated by $(p,q)$ of a given size. His bound is $3^p c / q^d$. The constants $c$ and $d$ ...
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### Question on a specific relation between rational and transcendental numbers

Heuristically, the below theorem/conjecture makes sense to me, but I do not have a formal mathematical proof to it; what I am actually looking for. May you perhaps, help with a formal proof or ...
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### Understanding a proof of transcendence of $\sum \frac{1}{2^{n!}}$

I am reading the proof of transcendence of $\sum \frac{1}{2^{n!}}$ from Milne's notes on Galois theory (p.20-21), in which I have some doubt(s). Let $\alpha=\sum \frac{1}{2^{n!}}$; it is irrational ...
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### Suppose that $c$ is transcendental over $\mathbb{Q}.$ Show that $\sqrt{c}$ and $c + \sqrt{c}$ are also transcendental.

My solution (a bit rough): (a) A polynomial that $\sqrt{c}$ satisfies is $x^2 - c$. However, since $c$ is not algebraic over $\mathbb{Q},$ $c$ is not in $\mathbb{Q}$ otherwise, $x-c$ would be a ...
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### Is the real solution of $\ln(x)=-e^x$ transcendental?

The function $$f(x)=\ln(x)+ e^x$$ defined on $\mathbb R_+$ has a unique real root $u$. It satisfies $$\ln(u)=-e^u$$ and $$\frac{1}{u}=e^{e^u}$$ The numerical value is $$0.269874137573449223877\cdots$$...
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### A relative description of algebraic and transcendental numbers

An algebraic number is a solution to a polynomial with rational coeficients over a field $K.$ A transcendental number is a number that is not algebraic. Has anyone proposed a relative description of ...
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Suppose that $\overline{\mathbb{Q}}$ is the algebraic closure of $\mathbb{Q}$, and let $F \leq \overline{\mathbb{Q}}(x)$ be a subfield. Then I want to show that either $F/\mathbb{Q}$ or $\overline{\... • 71 0 votes 1 answer 61 views ### Proof that$\mathbb{Q}(x) \not \subseteq \mathbb{Q}(x+i)$Let$x\in \mathbb{R}$be transcendental over$\mathbb{Q}$. I want to show that$\mathbb{Q}(x) \not \subseteq \mathbb{Q}(x+i)$. So I assume that$\mathbb{Q}(x) \subseteq \mathbb{Q}(x+i)\Rightarrow $... • 1,406 4 votes 2 answers 83 views ### Showing there exists a countable subfield$F\subseteq\Bbb{R}$such that if$x\in F$,$\sin(x)\in F$I am asked to show that there exists a countable subfield$F\subseteq\Bbb{R}$such that if$x\in F$,$\sin(x)\in F$. I don't see a classic set-theory approach to solving this. I started out by looking ... • 2,835 3 votes 0 answers 30 views ### Is there any$p \in \mathbb{Q}[x,y]$with x and y degree both at least 1 such that$p(\pi,e)$is known to be irrational? The algebraic independence of$\pi$and$e$is a well known open problem, as is the specific case of rationality of$\pi + e$. My question is if there is any polynomial with rational coefficients (... -1 votes 1 answer 77 views ### Where does the linear equation in the proof of solvability in elementary numbers in [Chow 1999] come from? How do one get the equation below in the proof in [Chow 1999]? Chow writes on page 444 - 445: "If$A=(\alpha_1,\alpha_2,...,\alpha_n)$is a finite sequence of complex numbers, then for brevity we ... • 4,047 3 votes 0 answers 45 views ### Can transcendent degree of transcendent number be algebraic? (meaning can both$\alpha^{\beta}$and$\beta^{\alpha}$be algebraic) Here is my question. Are there$\alpha$and$\beta$transcendental numbers such that both$\alpha^{\beta}$and$\beta^{\alpha}$are algebraic? There isn't anything specific about roots of the question.... • 664 1 vote 1 answer 98 views ### Can every transcendental number be expressed as the infinite sum of a quotient of two polynomials? Is it possible to express all transcendental numbers (and more generally all real numbers$\in \mathbb R$) as the sum of an infnite series of the form $$\sum_{n=0}^{∞} \frac{p(n)}{q(n)}$$ where$p(n)$... • 2,783 1 vote 1 answer 50 views ### Rectangles with side lengths proportional to transcendental numbers BESIDES$\varphi$A rectangle with side lengths proportional to$\varphi$is a "golden rectangle". If we perform a procedure of subtracting the shorter side length from the longer side length, a golden ... 2 votes 0 answers 44 views ### Transcendence of Liouville-Like Numbers Liouville numbers such as $$\sum_{k=1}^\infty\frac1{10^{k!}}$$ are known to be transcendental, essentially from Diophantine approximation type arguments. Using stronger results than what Louiville had ... 2 votes 1 answer 154 views ### Books for transcendental number theory I would like to start reading about transcendental numbers. I am familiar with the basics of field theory, number fields, and complex analysis. I have the least exposure to Galois theory. I am ... • 4,370 -3 votes 1 answer 63 views ### A countable set of transcendentals [closed] Let$S$be a countable subset of$\mathbb{R}$. Prove that there is a real number$c$, such that$s+c$is transcendental for all$s\in S$. Any hint? Edit: I was trying in vain to solve this by seeking ... • 61 1 vote 0 answers 67 views ### Transcendental numbers are decidable: a proof? My opinion is that the statement, for a real number$X$: "$X$is transcendental." is decidable. The sketch of the "proof" should be the following. If transcendency or algebraicity ... • 1,266 0 votes 0 answers 26 views ### Function and its range contained in$\Bbb Q\cdot S$where$S$is algebraically independent I have question about a good term to be used in such a situation. Let$f\colon\Bbb R\to \mathbb Q\cdot S$be afunction where$S\subset\Bbb R$is algebraically independent set over$\Bbb Q. $Clearly, ... • 2,153 0 votes 0 answers 24 views ### Lower bound on the run time for computing$n$digits of a transcendental number like$\pi$? For example, the Chudnovsky Algorithm for calculating$\pi$has run-time$O(n \log (n)^3)$for computing$n$digits. What are the best theoretical lower bounds we have for the run-time of computing$n$... • 1,612 2 votes 0 answers 30 views ### Explaining Algebraic Independence of$\delta$and$\gamma$I have been attempting to read this paper by Alexander Aptekarev. In it, he proves that$\delta$and$\gamma$cannot both be rational simultaneously. He also notes that this result follows from a ... • 3,726 1 vote 1 answer 54 views ### For$K=\mathbb Q(\alpha,\beta)$with$\beta^2=\alpha^3$if$\beta \in \mathbb Q(\alpha)$, then$|K:\mathbb Q|$is finite. Let$K/\mathbb Q$be a field extension and there exists$\alpha,\beta \in K$with$K=\mathbb Q(\alpha,\beta)$and$\beta^2=\alpha^3$. Then we have to show that: If$\beta \in \mathbb Q(\alpha)$, then ... • 2,241 6 votes 1 answer 161 views ### Is the infinite product$\prod_{i=0}^{\infty}(1+\frac{1}{2^{3^i}})$transcendental? Is the following number algebraic or transcendental? $$P:=\prod_{i=0}^{\infty}\left(1+\frac{1}{2^{3^i}}\right)$$ We could also define it as follows: let A be the set of natural numbers which contain ... • 486 1 vote 1 answer 88 views ### Find the splitting field of$x^2 - \pi^4$over$Q(\pi^4)$This is the first time I encountered a problem asking to find the splitting field of a polynomial with transcendental coefficients over$Q$adjoined to a transcendental number over$Q$. I have no idea ... 0 votes 1 answer 369 views ### What is the symbol of the transcendental numbers? [closed] the symbol of the natural numbers is$\mathbb{N}$, for integers it is$\mathbb{Z}$, for rationals it is$\mathbb{Q}$, for irrationals it is$\mathbb{I}$, and for transcendental numbers it is *not ... • 81 1 vote 0 answers 72 views ### How can these two arguments relate? I found the excerpt here that said the second involves the p-adic analog of the above. How can these two statement related? The transcendence of$2^{\sqrt2}$and$e^\pi$: Gelfand's proof. (Assuming ... 1 vote 1 answer 139 views ### Are$\mathbb{Q}(π)$and$\mathbb{Q}(π^2)$the same field? I've tried to check if$\mathbb{Q}(π)$and$\mathbb{Q}(π^2)$are equal fields. I know there exists a field isomorphism between$\mathbb{Q}(π)$and$\mathbb{Q}(π^2)$since$π$and$π^2$are ... 2 votes 0 answers 69 views ### Is$\sum_{p\text{ is a prime}}\frac{1}{p^{2}}$known to be irrational/transcendental? [duplicate] Is$\sum_{p\text{ is a prime}}\frac{1}{p^{2}}$known to be irrational/transcendental? The sum exists, because$\sum_{p\text{ is a prime}}\frac{1}{p^{2}}\leq\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^...
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(Please all of you be patient and give me your feedbacks whether the following ideas false or true ,whether they are useful or useless, also tell me in your opinions how can I improve these ideas if ...
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### Are there any known infinite series of rational terms that are just irrational (not transcendental)?

I have probably encountered hundreds of infinite series where each term is rational. In each case (as far as I can remember), the value of the infinite series was either rational or transcendental. ...
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### $t+1$ transcendent over $K$ if $t$ is transcendent
I've been solving some problems from my Galois Theory course and I want to check if the solution I came up with is correct. The question was: Given that an element $t$ is transcendent over a field $K$...