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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 there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?

As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
ChemGuy's user avatar
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Is $\sqrt{2}$ an element of the set $\{k \bmod 2\pi \mid k \in \mathbb{N}\}$? [closed]

I'm exploring the properties of the set formed by taking the modulo $\pi$ of natural numbers, specifically $\{k \bmod 2\pi \mid k \in \mathbb{N}\}$. This set includes all values $k - 2n\pi$ where $0 \...
hans's user avatar
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The connection between $\pi$, $e$ and $20$ [closed]

It's well documented that $e^{\pi} \approx 20+\pi$. This can be explained using the following series: $$\sum\limits_{k=1}^{\infty}\frac{8\pi k^{2}-2}{e^{\pi k^{2}}} = 1$$ The series is quickly ...
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We know existence of Transcendental raised to Algebraic Irrational equals rational, but what about opposite?

Introduction: If we take $a=2^\sqrt[3]{2}$ which is transcendental by Gelfond-Schneider Theorem, and $b=\sqrt[3]{4}$ which is algebraic irrational because it is root of monic-irreducible polynomial ...
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If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then $b\neq p^k$

If $\zeta(5)=\frac{a}{b}\in\mathbb{Q}$ then prove that $b\neq p^k$ where $p$ is any prime and $k\in\mathbb{N}$ Take $a,b\in\mathbb{N}$ such that $(a,b)=1$. Now if $b=p^k$ then $$p^k\zeta(5)=a$$ So, by ...
Max's user avatar
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Is the imaginary unit $i$ contained in $\mathbb{Q}(e+i)$?

Is the imaginary unit $i$ contained in $\mathbb{Q}(e+i)$? $i \in \mathbb{Q}(e+i)$? Intuition tells me $i$ is not contained in $\mathbb{Q}(e+i)$ because it should somehow contradict the fact that $e$ ...
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Are the vast majority of irrational numbers, transcendental? [duplicate]

It is often stated that the vast majority of real numbers are irrational. Does it also follow that the vast majority of irrational numbers are transcendental?
Larry Freeman's user avatar
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Constructing number between zero and one by concatenating digits from square root of primes

Let $a_{n}=$ the first $p_{n}$th digits to the right of decimal point of the square root of the $n$th prime. Example: $\sqrt{2}=1.414213562...$ So, $a_{1}=41$ $\sqrt{3}=1.73205080...$ So, $a_{2}=732$ $...
Math Admiral's user avatar
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Is transcendental number to non zero algebraic power always a transcendental?

My motivation to this question is that we know $e^{a}$ is transcendental , where $a$ is Non-Zero-Algebraic, using Lindemann Theorem, but is it true for all transcendental numbers not only $e$?
Math Admiral's user avatar
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I want to ask for good references of linear algebra over rational numbers?

My current studies in algebraic number theory have led me to observe the frequent interplay between linear algebra concepts over the field of rational numbers. This connection becomes particularly ...
Math Admiral's user avatar
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A simple, concrete example of a transcendental element.

I am writing an article on transcendental numbers and I'm wondering if it is possible to construct a "simple" example of a transcendental element in a field extension. When thinking about ...
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Formula including divisors sum ($\sigma$), Euler Gamma ($\gamma$), $\pi$ and $\ln \pi$ [closed]

An interesting formula arose during the investigation of divisors sum efficient calculation. Actually the below series converges very slowly, as every series containing $\gamma$ :) $$\sum _{k=1}^{\...
Gevorg Hmayakyan's user avatar
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How do I check if this number is transcendental?

Two days ago, I tried to create an infinite series that might be able to generate a transcendental number, and when I checked the proper definition, it was mentioned that, it is a number that cannot ...
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Images of a vector under the Galois differential group span the solution set

I am reading the paper "A refined version of the Siegel-Shidlovskii theorem" by F. Beukers. In the proof of Theorem 1.5, he mentions the following results in Galois differential theory. Let ...
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"Nice" values of x such as sin(x) is transcendental

First time posting here so please forgive any lack of adherence to best practices. sin(x) is a transcendental function. However most common values for the angle x will yield an algebraic number result....
Gabriel Brown's user avatar
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Might there be an $n^{\text{th}}$ digit of $\pi$ where the sequence becomes palindromic?

Assuming $n>1$, would it be reasonable to think there is an $n^{\text{th}}$ digit of $\pi$ where stopping there would yield a palindromic number $(3.14159...951413)$? Would it be more likely that ...
Pickelhaube808's user avatar
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Lesser known forms of Euler's identity

https://twitter.com/martinmbauer/status/1763622278128947464?t=YObGhK4ZqjAXrwPaHxB8gw&s=19 Many know Euler's identity, but did you also know that $|e^{i\pi}| = |\pi^{ie}| = |i^{\pi e}| = 1$ While ...
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Is the real-imaginary Schanuel's conjecture equivalent to the full Schanuel's conjecture?

Schanuel's conjecture says the following about the transcendence of numbers related by the complex exponential function: Given any $n$ complex numbers $z_1, ... z_n$ that are linearly independent ...
Arvid Samuelsson's user avatar
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Does Schanuel's conjecture imply that $\pi^e$ is transcendental?

My understanding (and correct me if I'm wrong) is that it is unknown whether $\pi^e$ is algebraic or transcendental. I've also been led to believe that most open questions of this type would be solved ...
Akiva Weinberger's user avatar
15 votes
1 answer
499 views

Proof $\pi$ is transcendental without symmetric function theory

Recently for a bonus homework assignment in my algebra class, I was asked to review the literature and write up a proof that $\pi$ is transcendental. Essentially every source I found ("The ...
Alex Pawelko's user avatar
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203 views

Can $x\sin(x)$ be algebraic when it is not $0$?

It's easy to show (using the Lindemann-Weierstrass theorem) that, for $x\ne 0$, at least one of $x$ and $\sin(x)$ must be transcendental. But what about $x\sin(x)$? After all, the product of two ...
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How to prove that for all digits there exists a transcendental number which contains it infinite number of times in its representation?

Soon I was told a statement that for all digits a transcendental number can be found, containing the digit infinitely many times. It seems obvious, but cannot find good enough argument for it. Can ...
Az Sym's user avatar
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Deduction from Schanuel's conjecture

I am deducing from the Schanuel's conjecture the following statement: Let $\alpha, \beta$ be positive real algebraic numbers with $\alpha,\beta \neq 1$ and $\frac{\log \beta}{\log \alpha} \notin \...
Mystery girl's user avatar
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Understanding the proof of a transcendental criterion regarding palindromic continued fraction.

I am trying to understand the proof of theorem 2.1 from the following paper: https://aif.centre-mersenne.org/item/10.5802/aif.2306.pdf. Basically the authors prove that given $\alpha=[0;a_1,a_2,\dots,...
WiggedFern936's user avatar
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Cases where transcendental numbers can add up to a rational number? [closed]

Other than sums like $π + (1 - π)$, obviously. Can two transcendental numbers add up to a rational number? Or how about an infinite series of them?
Alexandra's user avatar
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Unsure about a step in proving $e$ is transcendental

I'm currently trying to understand the proof of e being transcendental in Julian Havil The Irrationals, but not sure how the following is concluded. A polynomial of degree $mp+p-1$ is defined on the ...
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Is $\sin n/\sin m$ irrational for nonzero integers $n\neq m$?

As shown in Is sin(x) necessarily irrational where x is rational?, the sine of any rational except for zero is transcendental. But what about the ratio of two such numbers, at least for integers? I ...
Tobias Kienzler's user avatar
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Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers.

This is one of the exercises in my abstract algebra book (Nicholson) and it's just the title: Show that $\pi$ is not algebraic over the field $\mathbb A$ of algebraic numbers. All I know what to do ...
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Can the occurrence of certain digits be proven/disproven, for any arbitrary irrational number?

$\pi$ is perhaps the most famous irrational number. We know it contains all decimal digits from 0-9, just by virtue that all digits occur, at least once, within 32 decimal places: $\pi = 3....
Alexander Kalian's user avatar
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Sum of reciprocals of $10$ to the power of factorials

$$\sum_{i=1}^{\infty} \frac{1}{10^{i!}}$$ I don't remember the name. Also, if someone would be kind enough to send the proof of its transcendence. Thanks, envy.
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Is the imaginary number i really both algebraic and and irrational?

Wikipedia states that an immediate corollary of the Gelfond Schneider theorem is that $i^i$ is a transcendental number. To me it not so obvious, because it first has to be shown that i is both ...
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A lemma used in Lindemann-Weierstrauss Theorem

Below is a question given in exercises of book of Galois theory by Patrick Morandi which is used in the proof Lindemann-Weierstrauss theorem. Let $$ \sum_{i=1}^r a_i x^{\alpha_i} \text { and } \sum_{i=...
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Transcendence of $\ln(a)/\ln(b)$ where $a$ and $b$ are rational numbers.

In this post I proved that ln(3)/ln(2) is transcendental and an immediate corollary is that ln(x)/ln(y) is transcendental where x and y are natural numbers $ x,y \neq 0,1$ if x is odd and y is even or ...
KDP's user avatar
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2 votes
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Proof that $\ln(3) / \ln(2)$ is transcendental?

I was trying to figure out if $\ln(3)/\ln(2)$ is transcendental, when I found this post by b_jonas But there's a proof just as simple showing that $\log 3/\log 2$ is irrational. Suppose on contrary ...
KDP's user avatar
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Is the natural logarithm of $a$ irrational if $a \neq 0,1$ is algebraic?

Starting with the result of the Gelfond–Schneider theorem: $$ a^b = c$$ where a and b are complex algebraic numbers with $a \neq 0,1$, and $b$ not rational and $c$ is transcendental. Substitute the ...
KDP's user avatar
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1 vote
1 answer
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Does a transcendental number raised to the power of an irrational algebraic number, always result in an algebraic number?

If $b$ is transcendental number and $a$ is an irrational algebraic number, is it safe to conclude that $b^a$ must be an algebraic number? My reasoning is: Write the result of the Gelfond–Schneider ...
KDP's user avatar
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600 views

How to identify transcendental numbers?

Given the relationship: $$\pi = \left(\frac{15ab^2c^3}{2d^4}\right)^{\frac{1}{5}}$$ where a, b and c are non-zero positive constants, can it be assumed at least one of a, b or c must be a ...
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House of the inverse

Let $\alpha,\beta\in\overline{\mathbb Q}$. Denote by $h(\alpha)$ the house $\alpha$, that is the maximum of $|\sigma(\alpha)|$ when $\sigma$ describes $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$. ...
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Why a root for an integer polynomial in definition of transcendental number?

I've read many definitions for a transcendental number and some of them say that a transcendental number is a number that is not the root of any integer polynomial, while other say is a number that is ...
æîōü's user avatar
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transcendental numbers as solutions to hyperpowers [duplicate]

By the Gelfond-Schneider theorem, if $x^x=2$, $x$ must be transcendental. What can be said of $x$ if $x^{x^x}=2$, $x^{x^{x^x}}=2$ etc.? Must it be transcendental? Of course, $2$ can be replaced by any ...
Harry Lande's user avatar
4 votes
2 answers
372 views

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 $L_n=d^{-1}_{2n}\log S_n$,...
Max's user avatar
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4 votes
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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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12 votes
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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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automorphism between transcendental elements

Problem: Let $\alpha$ and $\beta$ be transcendental elements over $\mathbb{Q}$, show that there exists automorphism $\sigma$ of $\mathbb{C}$, such that $\sigma(\alpha) = \beta$ My attempt: Because $\...
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An "easy" way to get $\ln 7$ from $\ln 2$?

I was trying to convince someone that there's no "easy formula" to get $\ln 7$ from $\ln 2$ (in the sense that it is easy to get $\ln 8$ by $3\ln 2$). I guess the formal way to state his ...
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Necessary and sufficient condition for a number to be algebraic

I need a necessary and sufficient condition for a number to be algebraic over integers. I got this on wikipedia see here If $\alpha$ is an irrational number which is the root of an irreducible ...
Max's user avatar
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2 votes
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Proof of $e$ is transcendental

On the top of page 405 (413 of the pdf) of lecture notes by Professor Elman https://www.math.ucla.edu/~rse/algebra_book.pdf, it was asserted that $$\sum_{j=0}^m a_je^jF(0)-\sum_{j=0}^m a_jF(j)=-\sum_{...
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Does an algebraic irrational number always have all digits from the base?

If you take $\sqrt{2}$ in base $10$ and remove all digits $2$-$9$ you will get something like $1.1100010010101$... which i believe to be transcendental, and it got me curious, could there be an ...
Wagner Martins's user avatar
1 vote
1 answer
105 views

If the algebraic expression of a transcendental number is algebraic, is the number algebraic?

For instance, if the ‘silver ratio’ of a transcendental number is algebraic, how would someone classify that number? …. perhaps a special subset of transcendental numbers? Also, perhaps related: are ...
newby's user avatar
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2 answers
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Prove that $\pi=2i\ln\left(\frac{1-i}{1+i}\right)$

So I was looking through the homepage of Youtube to see if there were any math equations that I thought that I might be able to solve when I came across this video by Cipher which proposed the ...
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