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 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 $ algebraic numbers, and $\beta \in $ transcendental numbers. I know reciprocal of transcendental is transcendental. Question: Whether $\alpha\cdot \gamma \in$ transcendental?
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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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25 views

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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167 views

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 $\tau$ constants, are they transcendental, irrational, or rational numbers?

Below are three OEIS constant sequences and values. Are they transcendental, irrational, or rational numbers? Note: $\tau = 2*\pi$ and the last two values are in radians. A233700. Decimal ...
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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?
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2.71828. And then another 1828.

This may qualify as the silliest math.SE question ever, but am I really the first person ever to worry about this? The decimal expansion of $e$ has a 2. And then a 7. And then a 1828. And...well, then ...
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How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$

How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$? If $a$ is transcendental over ${\mathbb Q}[\alpha]$ then the ...
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What are some algorithms that can be used to test if a number is transcendental or not?

Well according to the definition of transcendental numbers I find that its any number that doesn't have any polynomial equation of any degree with integer coefficients summing up to 0. So ...
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The smallest class of numbers closed under addition, multiplication, and exponentiation

Let $\def\A{\mathfrak A}\A$ be the smallest subset of $\Bbb C$ that contains the algebraic numbers and also all numbers of the form $$\sum \alpha_i^{\beta_i}$$ where the $\alpha_i, \beta_i$ are ...
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Is $\log 2\pi$ rational?

Is it known whether $\log 2\pi$ is rational (where the base of the logarithm is $e$)? Or algebraic?
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Borel's result on transcendence measure

In "Sur la nature arithmétique du nombre e" (Comptes rendus de l'Académie des Sciences 128 (1899), 596-9) Borel presented his result on transcendence measure for e. This can be restated as follows: ...
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Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
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What kind of algebraic equations do trandescendal numbers not solve?

I know transcendental numbers cannot solve polynomials or rational functions (since they can always be written as a polynomial), but are they the solutions to equations containing a variable raised to ...
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Proof that cos(1) is transcendental?

So, I was playing around on Wolfram|Alpha (as we nerds like to do) and it said cos(1) was transcendental. Could someone provide me with the proof that cos(1) is transcendental?
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how fast does the proportion of associative operations on $S$ decrease with |$S$|?

as doubtless many have done before me, i recently fell into wondering how many of the binary operations on a finite set are associative. the stackexchange software fortunately pointed me to this ...
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Does the number $2.3\,5\,7\,11\,13\ldots$ exist and, if so, is it rational or irrational &/or transcendental? [duplicate]

Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$ if so, will it be rational or irrational &/or transcendental?
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Element in field of quotients is transcendental

Let $F\subseteq E$ be fields, and let $c\in E$. Let $F(c)$ be the field of quotients containing $F$ and $c$. Suppose $c$ is transcendental over $F$. Prove that every element in $F(c)$ but not in ...
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Transcendental Numbers (simple question)

Are all transcendental numbers irrational? I know that not all irrationals are transcendental (for example, $\sqrt{2}$); but I only know of a few transcendental numbers and they are all irrational.
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Uncountable sets of transcendental numbers

As a sort of follow up question to a previous question found here, besides the Liouville numbers, are there any other uncountable collections of transcendental numbers that are known? Clearly you ...
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“philosophical” question about the transcendence of $\pi$

I don't have any knowledge on transcendence proofs. I just heard that Lindemann proved that for any $\alpha \in \mathbb R^*$ algebraic, $e^\alpha$ is transcendental. Then, since $i$ is algebraic, and ...
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Transcendental numbers involving primes?

Is the prime zeta function value $$ P(2)=\sum_{p \in \mathrm{primes}} \frac{1}{p^2} = 0.452247420041065498506543364832247934173231343\ldots $$ a transcendental number ? What about the following sum ...
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Fast converging sums involving tetrations

Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check. Now my question : ...
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is the unique solution of $\cos t = t$ a transcendental number?

let $\alpha$ be the unique fixed point of $\cos:\mathbb{R} \rightarrow [-1,1]$ for any $t \in \mathbb{R} \setminus\{0\}$ if $t$ is algebraic then $\cos t$ is transcendental. thus if $\alpha$ were ...
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if $x\ne 0$, is at least one of $\{x, \cos\;x\}$ transcendental over $\mathbb{Q}$?

it seems at least superficially plausible that for real $x \ne 0$ then at least one of $\{x, \cos\;x\}$ is transcendental over $\mathbb{Q}$. has this assertion been proved to be true or false?
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Products of irrational numbers

Hello ladies and gentlemen! A friend of mine and I have been thinking about this particular issue: under what circumstances is the product of two irrational numbers rational? For example, multiplying ...
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Question about the proof that $K(u)$ is isomorphic to $K(x)$ if u is transcendental over $K$.

I have a few questions about a proof of the statement that if $u$ is transcendental over $K$ then $K(u)\cong K(x)$. My questions are marked as red and stated below the proof. The proof goes like ...