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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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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Connections of results in Harmonic analysis in the theory of Transcendental Numbers

Note :This question is proposed 2 years ago in MO , I see it appropriate for stackexhange math, i posted it here as it's unsolved problem and has a connection with Transcendental Numbers , mayeb we ...
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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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Nature of the range of $e^x$

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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Let$\ \lim_{n\to \infty} \frac{ \ln n}{f(n)}=1$. If$\ a,b,c$ are natural, can we have$\ a^{b+c \ln n}\sim a^{c f(n)}$?

I shall note that$\ n$ as well goes through the natural numbers and that$\ f(n)$ is rational for any$\ n$. Also, I'm obviously excluding$\ a=1$. I'm inclined to think my claim is not possible ...
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As$\ n \to \infty$, can a transcendental function$\ f\left(1+ \frac{1}{n}\right)$ to the power of$\ n$ tend to a rational power of$\ e$?

Let$\ f(n)$ be a transcendental function$\ \ne e^{g(n)}$, for any function$\ g(n)$. Does$$\ \lim_{n \to \infty} \left(f\left(1+ \frac{1}{n}\right)\right)^n =e^{ -k} = \lim_{n \to \infty} \left(1 - ...
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Quotient of two rational sequences and the nature of its limit

Suppose we have two sequences of rational numbers, $(p_i)_{i=1}^\infty$ and $(q_i)_{i=1}^\infty$, and suppose $$\lim_{i\to\infty}\frac{p_i}{q_i}=c<\infty,$$ where $c$ is known. Are there any ...
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Can a limit of form$\ \frac{0}{0}$ be rational if the numerator is the difference of transcendental functions, and the denominator a polynomial one?

Let$\ f_1(x)$ and$\ f_2(x)$ be transcendental functions such that$\ \lim_{x\to 0} f_1(x)-f_2(x)=0$, and$\ f_3(x) $ polynomial, such that$\ f_3(0)=0$. Can$\ \lim_{x\to 0} \frac{f_1(x)-f_2(x)}{f_3(x)}$ ...
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Changing digits of an irrational allowed?

Suppose you change every instance of a specific digit of π, e.g., suppose you make every "4" a "6" instead. I realize that this too would be irrational, but what I want to know is (1) on what basis is ...
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Does $6^x+3^x=10$ have an solution that is an algebraic number?

Does $6^x+3^x=10$ have a solution that is an algebraic number? I imagine not, but how would one go about proving such a thing?
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Is the root of $x=\cos(x)$ a transcendental number?

This question struck me when thinking about the fixed point of $x=\cos(x)$ being "obviously" not an algebraic number (unlike something like $\sqrt{2}$, see this question). If so, how would one prove ...
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Integers (strictly) between 0 and 1 form the basis of transcendental number theory?

In a MathOverflow comment on the question of "What is the most useful non-existing object of your field?", an answer is given A number which is less than 1 and greater than 1. Which elicited a ...
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Are all transcendental numbers theoretically accessible?

I apologize if the title (and the body) of this question is worded incorrectly, but I have no real experience in (transcendental) number theory, so it's probably the best I can do. I've been thinking ...
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Represent non-integer values on the factorial base

I want to compute the representation of the following values using the factorial number system: $\pi$ $e$ $\phi$ I know how to do it for integer values, but is it feasible for non-integer values? ...
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Is it correct: natural-logarithm maps algebraic numbers to transcendentals and vice-verse, over the domain it is defined?

Is it correct that the natural logarithm function maps algebraic numbers to transcendental and transcendental numbers to algebraic, other than 1? Of course, over the domain natural log is defined i.e. ...
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Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
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Is product of algebraic and transcendental number transcendental?

Let $\alpha \in \mathbb{A}$, and $\gamma \in \mathbb{T}$. I know that the reciprocal of a transcendental number is transcendental. Question: Is $\alpha\cdot \gamma \in \mathbb{T}$?
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Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
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Series of polynomials and uniformly convergence

It's part of the proof of a Lemma of an article I was reading (Algebraic values of transcendental functions at algebraic points). I couldn't understanding one thing: Let f be a complex function such ...
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Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

It's easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational. However, can it be shown whether it is algebraic or transcendental? My hunch is that it's transcendental but I don't know ...
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Exceptional Set and Schanuel's conjecture

I was reading an article about transcendental funtions (Algebraic values of transcendental functions at algebraic points, by Huang, J., Marques, D., Mereb, M.). The authors gave an example that says: ...
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Lebesgue measure of transcendental numbers in $[0,1]$.

What is the Lebesgue measure of the transcendental numbers in the $[0,1]$ interval? I was not able to find any information on this. (Does this question even make sense given what we currently know ...
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Is Champernowne's constant Liouville?

By looking at extreme spikes of Champernowne's constant and how well it's approximated by some rational numbers I think it's reasonable to think that this is a Liouville number. However, no source I ...
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Do replacing distinct digits from distinct places of an algebraic irrational

Do replacing distinct digits from distinct places of an algebraic irrational number necessarily make it a trancsendendal number? Since my question isn't worded well, therefore I would clarify it by ...
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Can $x^x$ be a natural number for non-integer $x$?

Does some real non-integral $x$ exist such that $x^x$ equals a natural number? Thanks, Tom
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A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
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What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental. My question is what are the transcendentals ...
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Are there classifications transcendental numbers that are similar to algebraic numbers for differential equations?

Considering that transcendental numbers are described as not a root of a non-zero polynomial equation with rational coefficients, are there classifications of transcendental numbers that are ...
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Real numbers that are not the roots of any polynomial equation with algebraic coefficients

An algebraic number is a number which is a root of some non-zero polynomial equation with rational coefficients. A transcendental number is a number which is not a root of any non-zero polynomial ...
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Is this function continuous on transcendental number

This question is motivated from Thomae's function continuity at irrationals together with the fact that transcendental numbers are dense in real numbers. Let $$f(x) = \begin{cases}1 &, \text{x ...
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Related to $\pi$ and $\tau$ - are these numbers transcendental, irrational, or rational?

Here are three numbers whose decimal expansions are listed in OEIS. Are they transcendental, irrational, or rational numbers? Note: $\tau = 2*\pi$ and the last two values are in radians. A233700. ...
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Transcendentals as the Roots of Infinite Polynomials

I have always been taught that the difference between an algebraic and a transcendental number is that the former is the root to a polynomial of ${\bf finite}$ degree with integer coefficients. I did ...
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The “trick” functions in the “$\pi$ is transcendental” proofs

I was reading this paper and I wondered how did Hermite decide to define a function $$f(x)=\frac{x^{p-1}(x-1)^p\cdots (x-m)^p}{(p-1)!}$$ Are these functions only tricks or there is a deeper meaning?
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Do transcendental numbers outnumber real numbers?

I am not a mathematics student, but just out of curiosity I was checking out a website which explains the basics of 'Chaos Theory' to the layman. In this site was the sentence : transcendental ...
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$n^n$ cannot be expressed as a recurrence with polynomial coefficents

We say that a sequence $a(n)$ is $P$-recursive if there exist polynomials $p_0(n),\ldots,p_k(n) \in \mathbb{Q}[n]$ such that $$p_k(n) a(n+k) + \cdots p_0(n) a(n) = 0.$$ I would like to show that the ...
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Prove of trancendence of $\ln(2)$.

Where can I find some proofs for another transcendental numbers, like Hermite/Lindemann theorem proofs for $e/\pi$? For instance, prove that $\zeta(3)/\ln(2)$ is a transcendental number.
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Let $K = $ algebraic numbers. Then is $\operatorname{Span}_K(\pi, \pi^2, \dots)$ a vector space of transcendentals?

$V = {\rm Span}_K(\pi, \pi^2, \dots)$ is clearly a $K$-vector space. If we let $K = \Bbb{Q}$ temporarily, then every element of $V$ is transcendental as it's a finite linear combination $Q(X), \ X = ...
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Irrationality of $\pi$ another proof

Proposition. Let $\alpha\in\mathbb{R}$. If there is a sequence of integers $a_n,b_n$ such that $0<|b_n\alpha-a_n|\longrightarrow 0^+$ as $n\longrightarrow \infty$, then $\alpha$ is irrational. ...
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Strange notation for a decimal expansion of a transcendental number

I am checking page proofs for one of my papers right now and an editor changed $\zeta(3)=1.202$$\ldots$ to: $\zeta(3) = 1.202,...,$ I find this latter notation very strange and think it ...
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Sum and product of two transcendental numbers can't be both algebraic

Suppose $a$ and $b$ are complex numbers and both transcendental over $\mathbb Q$. I am wondering why $ab$ and $a+b$ can not both be algebraic. Thanks for any help.
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Is $0.1010010001000010000010000001 \ldots$ transcendental?

Does anyone know if this number is algebraic or transcendental, and why? $$\sum\limits_{n = 1}^\infty {10}^{ - n(n + 1)/2} = 0.1010010001000010000010000001 \ldots $$
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Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$ (where $T$ = transcendental numbers)

Question: Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$. This is Exercise 2.7.13(a) from Mark E. Watkins, Jeffrey L. Meyer: Passage to Abstract Mathematics. I'm currently ...
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Question about $e^{e^{e^e}}$

Is there a proof that the power tower of length $4$ of $e$ is irrational? Is it known whether or not $$e^{e^{e^e}}$$ is transcendental?