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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Proof of the transcendence of certain arcsin values

I'm aware of some transcendence proofs of certain special numbers like $\pi$ and $e$, and I'm aware that finding certain transcendence proofs can be quite difficult and involved. I also know that most ...
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Two transcendental elements and their shifting

This is not a homework assignment or question from an exam but it comes to my mind but I was not able to figure it out. Let $\{u,v\}$ be algebraically independent subset of $\mathbb R$ over $\mathbb ...
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About union of two algebraically independent sets

My question might be very simple for those that have a deep understanding for algebraically independent sets. Definition 1.1. Let $F$ be an extension field of $K$ and $S$ a subset of $F$. $S$ is ...
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Independent over rational number field

Is there some theorems which can make sure that $$1, \frac{\log 2}{\log 3}, \frac{\log 3}{\log 2}$$ are $\mathbb{Q}$-independent?
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Is this number transendental or irrational

This question was asked in a quiz on sets and my answer was wrong. So, I am asking for reasoning here. The number $\sqrt{2} e^m$ is: (i) a transcendental number (ii) an irrational number I wrote ...
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Integer rings and UFDs in transcendental field extensions of $\mathbb{Q}$

I recently began to study algebraic field extensions of $\mathbb{Q}$ aka number fields and especially the definition of algebraic integers in these fields. Some rings of algebraic integers are unique ...
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Infinity of the set of transcendental numbers.

I am looking for a proof of the infiniteness of the set of all (real) transcendental numbers. The first idea is to say that there exists, at least, one transcendental number (for example $e$). Now I ...
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What makes Hilbert's 7th problem important/relevant? [closed]

What was the motivation behind Hilbert's 7th problem? Looking into some of the history behind transcendental number theory, it seems that the field was almost non-existent in the late 1800's/early ...
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Improving the Dirichlet's approximation theorem.

Recall that Dirichlet showed the following: For every real number $x$ and every $Q>1$, there exists an integer vector $(p,q)\in \mathbb Z^2$ such that $|xq-p|<1/Q$ and $0<q<Q$. I wonder ...
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Fields Having Algebraic Numbers as Subfield

In the following text, we call a power series $\sum_{n=0}^{+\infty}c_nz^n$ rational if it satisfies $\forall n, c_n \in \Bbb{Q}(i)$ and finite if it is a polynomial. Let $\Bbb{A}$ be the set of ...
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Linearly independent over the rational field [closed]

Are $$ \frac{1}{\log 3}, \frac{1}{\log 5} , \frac{1}{\log 7} $$ $\mathbb Q$-linearly independent ?
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Reciprocal of a Liouville number is also a Liouville number

Prove that the reciprocal of a Liouville number is also a Liouville number I am using the definition of a Liouville number given in the book Transcendental Numbers by M. Ram Murty. A screenshot of ...
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Do you have an example of an algebraic number that is the sum of 2 transcendental numbers?

Do we know any non trivial algebraic number $a = b + c$ where $b$ and $c$ are transcendental? My question is related to the unsolved problem: we know that $\pi$ and $e$ are transcendental but nobody ...
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Degree of a field extension by a transcendental element

Let $F$ be a field, and let $F(x)$ be the field of fractions of the polynomial ring $F[x]$. I'm interested in the degree of the field extension $[F(x) : F]$. Obviously it is infinite, but what exactly ...
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Who all constitute algebraic numbers? [closed]

Algebraic numbers are numbers which are roots of polynomial with rational coefficients. What all constitutes these numbers. Transcendental numbers are those which can not be roots of above. And wiki ...
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Are there different types of transcendental numbers?

For example, algebraic integers are a special case of an algebraic number. Does something similar happens with transcendental numbers? Is possible to a trans. num. be different the other ...
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$e+1$ is transcendental on real [closed]

I have this: We know that $e$ is transcendent. Suppose $e + 1$ is algebraic, so $e + 1-1$ would also be algebraic, but it contradicts the fact that $e$ is transcendent, therefore $e + 1$ is ...
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Which irrationals become rational for some positive integer power?

Related to Irrationals becoming rationals after being raised to some power. Let $r \in \mathbb{R} \setminus \mathbb{Q}$. True or false: there exists an $n \in \mathbb{N}$ (positive integers) such that ...
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Infinite product of transcendental numbers approaches 1

I am seeking infinite formulas connect transcendentals and rationals. We know $$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$...
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Translate the continuous fraction $[2;2^{1!}, 2^{2!}, 2^{3!}, \cdots]$ to a sum in order to prove that it's transcendental

Translate the continuous fraction $[2;2^{1!}, 2^{2!}, 2^{3!}, \cdots]$ to a sum in order to prove that it's transcendental The task is to prove that the number $[2;2^{1!}, 2^{2!}, 2^{3!}, \cdots]$ is ...
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A simple formula similar to the Koide Formula

In the question Something similar to the bizarre Koide formula? it was requested for constants that fit the Koide formula to make something similar to it, which the reason for was explained in the ...
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Is $e-1/e$ rational?

My intuition tells me that it's not rational, and even not algebraic (i.e., it's transcendental). But I'm having a hard time showing it. Taking it slightly further, $e-\frac1e=\frac{e^2-1}{e}=\frac{(e+...
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Expansion of analytic function on disk

In a proof of a lemma used for the Gelfond Schneider theorem, it is claimed that given functions $f_1(z), \dots, f_L(z)$ analytic on $D = \{z : \vert z \vert < R\}$ and continuous on $\overline{D} =...
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Way of expressing transcendental number in terms of known mathematical constants?

During my research I found a constant which is a transcendental number and I am able to calculate it to high precision (like 300 and more decimal places). Do you know of a computer/online program, ...
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Elementary proof that $\pi$ is transcendental

A popular (and maybe the only) approach to showing that $\pi$ is transcendental is to first prove that for every non-zero algebraic number $a$, the number $e^a$ is transcendental. That requires ...
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I'm looking for material to understand a bit of transcence of numbers and study whether they are rational or not

I graduated at engineering and I know a fair amount of real analysis, calculus, etc. I read Apostol's book on analytic number theory. I'm looking for some book to see a bit more about the theory on ...
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Power towers, sequences, fixed points, and a mysterious constant

Today I was thinking about the function $f(t)=t^{t^{t^{t...}}}$ Where $t>1$. We can define this notion more formally with a sequence, $a_{n+1}=t^{a_n}$ given $a_0=t$. If the sequence converges, say ...
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Numbers that are solutions to polynomials with coefficients that are rational numbers. Do $e$ and $\pi$ belong to this family? [duplicate]

We know $e$ and $\pi$ are transcendental. Meaning that they aren't the solutions to any polynomial equation with coefficients that are integers. The next obvious question to ask - what about ...
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51 views

Conversion of definite integral to continued fraction:

Consider integral of the form : $$\int_a^b f(x)dx$$ $f(x)$ is analytic and real valued for real domain. Now fix $a$ and $b$ ( most likely $[0,1]$ and $[0,\infty]$ ) . Can we construct a continued ...
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Can $\pi$ be defined using Dedekind cuts?

I have read that Dedekind cuts allow you to define the real numbers from the rationals. For example, $\sqrt{2}$ can be defined in the following way: Partition the rational numbers into two sets $A$ ...
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Proof verification on Field Extension

Let $K$ and $E$ be two fields, and let $u$ be transcendental over $K$. If $K\subset E\subseteq K(u)$, then $u$ is algebraic over $E.$ Proof. Since the degree of the transcendental extension is ...
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A Question about rationality , irrationality or Transcendence of definite integral

( this is my first question on the site so please forgive any possible mistake ) Consider integral of the form : $$\int_0^\infty f(x)dx$$ Can we have a set of conditions of $f(x)$ such that we can ...
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Does there exist integer $k >1$ for which $|e^{k\pi}-k\pi-2n|\leq 10^{-6}$

It is well known that $e^{k\pi}-k\pi$ is almost even integer for $k=1$, now ,are there others $k $ ? Assume if there is some finitly $k$ then what about periodicity of $e$ because we would have ...
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Is it possible to have $\text{transcendental}^\text{transcendental} =\text{algebraic}$?

We know that it possible to have $\text{irrational}^\text{irrational}=\text{rational}$. To verify the possibility of this, we consider the expression $\sqrt{2}^\sqrt{2}$, if it is rational, then this ...
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Transcendence of $e(T)$

It is well known that Euler have proved that $e=[2, 1, 4, 1, 1, 6, \ldots, 2n, 1,1, \ldots]$ and that $e$ is a transcendental number by Hermite's evidence. Let us consider the function $e(T)$ ...
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Proof of irrationality of $\arcsin(\frac{1}{4})$

I was working to find a different approach to Niven's theorem from the one in my textbook taking a route via Chebyshev polynomials. It all comes to proving the irrationality of $\arcsin(\frac{1}{4})$ ...
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Would this be generating a transcendental number?

Based on my understanding, a transcendental number is a number that is not computable, or cannot be generated by an algorithm. Let's say that (theoretically) I randomly generate a number that is ...
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Zero equivalence in computer algebra systems

Testing whether an expression (say, on the real or complex numbers), with or without variables, is known to be difficult. There is an semi-algorithm by Daniel Richardson to solve it for the exp–log ...
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Are all (computable) transcendental numbers zeros of infinite polynomials?

Since $\pi$ is a zero of the Taylor series of $\sin{x}$, I wonder if all transcendental numbers (computable ones) are zeros of polynomials if we allow for infinite polynomials. Does anyone know about ...
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Estimating $f(x)=\frac{\sin(x)+\sec(x)+\tan(x)}{\cos(x)\csc(x)\cot(x)}$ where $x \in[-\frac{\pi}{3},\frac{\pi}{4}]$

Consider the function $$f(x)=\frac{\sin(x)+\sec(x)+\tan(x)}{\cos(x)\csc(x)\cot(x)}$$ in the interval $x \in[-\frac{\pi}{3},\frac{\pi}{4}]$. Find a combination of algebraic (not transcendental) ...
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Is this proof of $e^{W(\pi)}$ being transcendental correct

I have been experimenting with the lambert W function ($W(x)$) and transcendental numbers, and have attempted to prove this result, however I am unable to find any confirmation that $e^{W(\pi)}$ is ...
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What about of the irrationality and transcendence of expressions involving the omega constant, and/or $\pi$ and $e$?

I know that there are some open problems concerning the irrationality and trancescende of certain combinations (suitable expressions as sums/differences, products/quotients and exponentiations/...
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sine of non-zero algebraic number is necessarily transcendental

How can we prove or disprove that: $$\forall x \in \mathbb{A}\setminus\{0\} \implies \sin{x} \in \mathbb{R}\setminus\mathbb{A}$$ Is that holds for other trig functions?
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Looking for good book on transcendental number theory

I'm looking for advanced text book and more friendly text, especially in the advanced ones. One thing in particular that I'm looking for is a geometric approach to the theory, since I was unable to ...
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How to find the Gaussian function given probability and boundary?

I intend to find the Gaussian function, where $p$ is the probability, area under its curve, within the boundary $r_{min}=- r_0$ and $r_{max} = +r_0$. Consider the following Gaussian function: $$ f(r)...
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What is the solution for a general case of $ax^m = e^{b/x^n}$?

What is the solution for a general case of $ax^m = e^{b/x^n}$? I am a bit new to non elementary function, but it seems Lambert W function is a probable solution. Upon checking, it seems, it requires ...
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What it means to say $\pi$ and $e^\pi$ are algebraically independent $?$

$\pi$ and $e^\pi$ are algebraically independent. But what does that mean? Is that if $a_0, a_1, \cdots, a_n$ and $b_0, b_1, \cdots, b_n$ are algebraic numbers then $$a_0\pi + a_1\pi + \cdots + a_n\...
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Is $\cos(\ln(x))$ transcendental?

I am aware that according to Lindemann–Weierstrass theorem: 1) $\sin(a),\;\cos(a),\;\tan(a)$, and their multiplicative inverses $\csc(a),\;\sec(a),$ and $\cot(a)$, for any nonzero algebraic number $a$...
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Is this proof that $\tan a$ is transcendental correct?

First, a proof that $\sin a$ is transcendental, where $a$ is algebraic not zero. Given that, $$e^{ia}=\cos a + i\sin a$$ if $\sin a$ were algebraic, then $\cos a = \pm \sqrt{1-\sin^2a}\:$ is also ...
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Is $\tan^{-1}\tan^{-1}1$ irrational?

Here, it is proven that $\arctan(2)$ is irrational. Here, it is proven that $\arctan(x)$ is irrational for natural $x$. By a proof similar to that from the last linked post, it can easily be shown ...

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