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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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 ...
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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 ...
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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 ...
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Prove that $\pi=2i\ln\left(\dfrac{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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I am confused on the substitution method for the following integral. I've put it in symbolab and wolframalpha and idk how they get -1/2*e^(2/x^2)
Here is the integral:
$\int \frac{2e^{\frac{2}{x^2}}}{x^3}dx$ or integral of $[2e^{(2/x^2)}]/x^3\,dx$
When I tried solving for it with substitution, I simply used $u=2/x^2$ whilst symbolab seemingly ...
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Show that if $z\in \mathbb{C}$ is a non-rational solution to $\sqrt{3}(1+z)=\tan\left(z\pi/2\right)$, then $z$ is transcendental.
This is exercise 1 from chapter 3 of Pillars of Transcendental Number Theory (Natarajan, Thangadurai), which is on the Gelfond-Schneider Theorem. There are a few clear ways to use the Gelfond-...
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infinitely many algebraically independent numbers without cardinality argument
I was thinking about the field of computable numbers, and it's intuitively pretty obvious that they'll be isomorphic to the characteristic $0$ algebraically closed field, but I realised I don't know a ...
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Are $\text{exp}(β_1), . . . , \text{exp}(β_n)$ are algebraically independent over $Q$?
Lindemann-Weierstrass’s Theorem: Let
$β_1, . . . , β_n$ be algebraic numbers that are linearly independent over $\mathbb Q$. Then the $n$ numbers $\text{exp}(β_1), . . . , \text{exp}(β_n)$ are
...
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Power series for functions that have $e$ as a root
Define an 'algebraic sympathizer' as any transcendental number that is root of at least one function that when expressed as a power series, each of the coefficients are rational. That is, for all $n$ ...
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On the irrationality of $\zeta(\frac{3}{2})$
It is known that $$\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$$Where $\zeta$ is riemann's zeta function. Usually people make $s$ an integer. But I thought of non integer values of $s$ and started with $s=...
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elements of $E\setminus\mathbb{Q}$ are transcendental over $\mathbb{Q}$
Let $u$ be a real transcendental number over $\mathbb{Q}$ and $\alpha$ a root of $X^2 + u^2 + 1$ in $\mathbb{C}$. We note $K=\mathbb{Q}(u)$ and $E=K(\alpha)$.
I have to show that
elements of $E\...
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Questionable proof of Liouville's Theorem (number theory)
I saw a proof of Liouville's theorem on ProofWiki. It goes like this:
Let $x$ be a Liouville number. Also suppose that it is an algebraic number.
Liouville numbers are irrational, so there exists $c &...
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Real number known not to be a period
I am working a bit with problems in non-archimedean settings inspired by the famous periods conjecture by Kontsevich-Zagier. I was preparing a talk and wanted to give of background about the initial ...
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What's an example of an element in $\mathbb R \setminus \mathbb Q[\pi]$?
Since $\Bbb Q[\pi]$ consists of expressions of the form
$$a_0 + a_1\pi + \ldots + a_n\pi^n \quad\quad a_i \in \Bbb Q$$
for $n\in \Bbb N$, the following isomorphism of sets is immediate:
$$\Bbb Q[\pi] \...
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Evaluating Legendre $\int_a^b P_{2n}(\text{Ei}(x))dx,\int_a^b P_{2n}(\text{li(x)})dx$ to solve li$(\mu)=0$ with logarithmic/exponential integral
$\def\li{\operatorname{li}}\def\Ei{\operatorname{Ei}}\def\P{\operatorname P}\def\W{\operatorname W}$
The Soldner Ramanujan constant $\mu$ is possibly expressible via a Dirac $\delta(x)$ Legendre $\P_v(...
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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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How is Siegel's lemma applied in number theory?
In the Wikipedia page, Siegel's lemma is stated as follows:
Consider the system $$ \begin{cases} \sum_{i=1}^Na_{1i}X_i=0\\
\vdots\\ \sum_{i=1}^Na_{Mi}X_i=0 \end{cases}, $$ where the
coefficients $a_{...
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$\int_0^\pi t^2\sin(t)\sin(n(\sin(t)-t\cos(t)))dt$ for alternate Goat problem Fourier sine series solution
The goat problem has the following transcendental equation:
$$\sin(a)-a\cos(a)=\frac\pi2,a=1.905695729\dots$$
If $f^{-1}(x)$ is odd, its Fourier sine series of period $\left[-\frac L2,\frac L2\right]$ ...
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Does $\pi$ have countably or uncountably many decimal digits?
I think I know the answer - countably many, and intuitively it does make sense i.e. it wouldn't make sense that a number has uncountably many decimal digits (is that even possible).
However, I've been ...
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Disjunctive numbers are transcendental
I struggle to understand why disjunctive numbers are necessarily transcendental.
A rich number (or disjunctive number) is a real number whose expansion, in a given base $b$ is a disjunctive sequence ...
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Citation showing a number is transcendental
On the Wikipedia page about transcendental numbers
there is a claim that for $\beta > 1$, if $\alpha = \displaystyle \sum\limits_{k=0}^{\infty }10^{-\left\lfloor \beta ^{k}\right\rfloor }$ ($\beta \...
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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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Can we find a function which vanish at the decimals of the Liouville's constant?
Thinking to Liouville's constant I was hoping If we can find with the decimal of the famous constant a function wich vanish at these decimals .
I found currently a very simple function wich is :
$$f\...
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Can we find exact value of $x$ satisfying $x^{x^x}=\sqrt{2}^{\sqrt{2}}$ [duplicate]
Can we find exact value of $x$ satisfying $$x^{x^x}=\sqrt{2}^{\sqrt{2}}$$
My effort:
Let us start with the form $x=2^t$, we have
$$\Rightarrow\left(2^t\right)^{2^{t\left(2^t\right)}}=2^{\frac{1}{\...
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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 ...
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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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Schanuel's Conjecture $\implies $ no surprises on the integers. $2^t+3^t=1 \implies t\notin\overline{\mathbb{Q}}$
Simple Question:
Is the proof given below correct?
Let $t$ satisfy $2^t+3^t=1$. We have
$t \approx -0.787884911025869783628555917298434738269083137354182194199 \dots$ according to wolfram alpha.
Then $...
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Do smooth "almost-algebraic" transcendental curves exist?
Is it possible for a smooth, transcendental plane curve to be such that for every point $(a,b)$ on the curve, there is a polynomial with rational coefficients $p(x,y)\in\mathbb{Q}[x,y]$ where $p(a,b)=...
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Can a pair of complex numbers be "completely" algebraically independent?
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\...
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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 $6$, ...
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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'.
...
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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....
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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 ...
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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}) =...
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Equivalent definitions of a Liouville number
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 $2.$ ...
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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 ...
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