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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 $e$ transcendental when working with hyperreal numbers?

When working with strictly real numbers, there are a number of proofs that $e$ is transcendental. However, when dealing with non-standard analysis, one can express $e$ as $(1 + \frac{1}{H})^H$ for any ...
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Algebraic independence and $\overline{\mathbb{Q}}-$ linear independence

This article on Wikipedia about the Lindemann-Weierstrass theorem mentions two equivalent formulations, and says that they are equivalent by an argument about symmetric polynomials. But I cannot see ...
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Is there any “supertranscendental” number that doesn't satisfy a “polynomial” equation with algebraic coefficients and exponents?

Any number $x$ is called algebraic if there exist integer coefficients $a_0, a_1, ...,a_{n-1},a_n$ and integer exponents $b_0, b_1,...,b_{n-1},b_n$ where $b_0 = 0, b_1 = 1$ etc. such that $a_nx^{b_n}+...
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How to construct a transcendental number

I am doing a project about irrational and transcendental numbers and I was wondering how could I construct a "new" transcendental number. I know that all Liouville numbers are transcendental so this ...
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Sums and Products of Algebraic and Transcendental numbers

I am doing a project about irrational and transcendental numbers and part of this project involves looking at sums and products of various combinations of rational, irrational, algebraic and ...
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Interpolating data.

Consider the following formula $$_{n}q_x=1-exp[-n\times _{n}m_x-.008 \times n^3 \times _{n}m_x^2]\ldots(1).$$ Page 867 of this book shows values of $_{5}q_x$ associated with $_{5}m_x$ by the above ...
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Is it possible to solve polynomials with degree larger equal 5 using formulas with transcendental numbers?

Algebra tells us that it is not possible to solve a polynomial with degree larger equal 5 using formulas containing roots, multiplication etc. But the use of transcendental numbers ist not allowed in ...
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Is $\Bbb Q[\alpha]$ a field ? where, $\alpha \in \Bbb C$ such that $\alpha + \pi = {\pi}^2 \alpha $

Is $\Bbb Q[\alpha]$ a field ? Here $\alpha \in \Bbb C$ is such that $\alpha + \pi = {\pi}^2 \alpha $ . I tried to argue by contradiction. Tried to come up with tricks commonly used for concluding ...
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Is there any known transcendental $b$ such that $b^b$ is also transcendental?

Numbers such as $e$ and $π$ are known to be transcendental, however, $e^e$ or $π^π$ are not even known to be irrational, let alone transcendental. There are infinitely many transcendental numbers $a$...
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Is “indeterminate” a synonym for “variable” or for “transcendent”?

The following is from Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, Edited by H. Behnke, F. Bachmann, K. Fladt, W. Süss and H. Kunle. Bold text ...
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Why differentiate between irrational and transcedental numbers?

As the title says, I am wondering why transcedental numbers ever were introduced. Other common subsets of $\mathbb R$, such as natural numbers, integers rational and irrationals are obvious. For ...
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Major misunderstanding about field extensions and transcendence degree

So presumably this question is very basic, but I'm having some trouble with apparent contradictions in my reasoning. Let $k$ be a field and $k \subseteq K$ a field extension. We say that $K$ is a ...
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Every transcendental number satisfies a power series

I was reading this expository paper by Yves André in which he states a nice result: every transcendental number is the root of a power series over $\mathbb Q$. He accredits this theorem to Hurwitz in ...
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Is $\Gamma{(\frac{1}{5})}$ transcendental?

Wolfram Mathworld lists several transcendental numbers such as $$\Gamma{\left(\frac{1}{3}\right)},\Gamma{\left(\frac{1}{4}\right)},\Gamma{\left(\frac{1}{6}\right)}$$ I don't see the reason why Wolfram ...
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$e$ in Roth's Theorem

THEOREM 1.8 of the book Making Transcendence Transparent by Burger says: then it says: But $e$ is not algebraic how it satisfies Roth's Theorem ?
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On Liouville numbers and the Continuum Hypothesis [duplicate]

Collecting some theorems from the book Making Transcendence Transparent by its authors, there is some inconsistency, I think... : i. $L= \sum_{n=1}^{\infty} 10^{-n!}$ is transcendental. ii. ...
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Proof that $(n+1)m!>(m+1)!−1$ does not hold for $m>n$

I am currently doing a project that involves some work on Liouville's theorem for transcendental numbers and Liouville's constant. I have found a proof that Liouville's constant is transcendental ...
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Let $K$ be a field. Prove that every element in $K(x)\backslash K$ is transcendental

Let $K$ be a field. Prove that every element in $K(x)\backslash K$ is transcendental over $K$. Is proof of the question above similar to that of the question below?
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Can an algebraic function contain transcendental constants?

Most definitions I've seen [1] [2] are agnostic about what kind of polynomials the function must satisfy. PlanetMath, for example, says A function of one variable is said to be algebraic if it ...
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How do you turn an irrational, non-transcendental number, like 1.618… back to its form of (a + sqrt(b))/c.

Looking at irrational numbers, I had an idea, as to computing square roots. Take the golden ratio. Numerically, it's 1.618.... but I can also write it like this: $\frac{1+ \sqrt{5}}{2}$ I want to ...
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Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn ...
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Representation of Transcendental number via continued fractions

My question is quite simple. As far as I know, rational numbers has finite continued fraction representation. And if for any number we have infinite continued fraction, then it necessarily is ...
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Real irrational algebraic numbers “never repeat”

An oft-used phrase describing irrational numbers is that their (decimal) expansions "never repeat". The sense of "never repeating" intended is, of course, that their expansions don't repeat forever. ...
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unknown sequences with sum of $e$ [closed]

Starting with an transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form : $e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {...
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Proving that a number is transcendental

Let $(x_n)_{n> 0}$ a sequence of $\{ 0,1 \}$ and $$ x=\sum_{n=1}^{\infty}\frac{x_n}{10^n}. $$ Prove that if $x$ is irrational then $x$ is transcendental. I tried to first start by proving that $...
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Unique solution of $x = \cos\left(\frac{x}{2}\right)$

How does one show that there is a unique solution to this equation? $$x = \cos\left(\frac{x}{2}\right)$$ Furthermore how can we find it?
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Proving an identity for $\sum_{m,n\in\mathbb{Z}_{>0}}\frac{\gcd(m,n)^r}{m^sn^t}$.

My task is to prove the well-known identity Here all variables are positive integers I only know I should use mobius inversion formula, but how to proceed I am getting confusion, please any one ...
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Is the definition of a transcendental field extension necessarily impredicative?

Let $t$ be transcendental over $K$, i.e. there is no $p(x) \in K[x]$ s.t. $p(t)=0$. Can you define the smallest field containing $K$ and $t$ in a predicative way? If not, how would you prove that ...
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proving transcendental numbers are irrational

I don't understand how every transcendental number is irrational, is there a way to prove that? I know it just means it's a non-algebraic number, but how does that correlate to irrationality?
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Real valued function which is continuous only on transcendental numbers

First of all, I am sorry for asking this question. We know that $R$ is uncountable. And also the set of all transcendental numbers is uncountable. How can I construct a function $f(x)$ on $R$ which ...
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Proof that e is transcendental in Herstein's Topics in Algebra (1st ed)

In the proof of Theorem 5.F. page 177. page 176 page 177 From the constructed $F(x)$ from $f(x),$ how can we choose $$f(x)=\frac{1}{(p-1)!}x^{p-1}(1-x)^{p}(2-x)^{p}\cdots(n-x)^{p}$$ where $p>n$ ...
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How many values of $\theta$ give $cos(\theta)$ is algebraic

I recently saw Lindemann's proof that $\pi$ is transcendental by using the fact that $e^{i\pi} = -1$, and this made me realize that the Lindemann-Weierstrass theorem implies that the $\cos$ , $\sin$ ...
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Is $(\ln 2)^2$ transcendental?

Wolfram says $(\ln 2)^2$ is transcendental. I think it says numbers of the form $(\ln a)^b$ are all transcendental, at least for integer $a$ and $b$, I didn't check further. Maybe there is some ...
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Is $\sin(e)$ rational or irrational?

We know that $\pi$ and $e$ are transcendental numbers. Here $\sin(x)$ is a real trigonometric function. We know that $\sin(\pi)=0$ which is rational. Now I am wondering to know that whether $\sin(e)$ ...
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Transcendence of $x^x = p \in \mathbb{P}$

I'm going straight for the question, so: let $x^x = p \in \mathbb{P}$, then $x$ is irrational. The proof is obvious, $p \neq a^a$ for some integer $a$, so $x^x = (\frac{a}{b})^{\frac{a}{b}}$, suppose ...
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Solving $e^{t/\!\ln(x)}=x$ for $x$

How do you solve the equation: $e^{t/\!\ln(x)}=x$ for $x$? Here $t=1,2,3,\dotsc$ This is what I did: $\ln(e^{t/\!\ln(x)})=\ln(x), $ $t/\!\ln(x)=\ln(x),$ $t=(\ln(x))^2,$ $x=e^\sqrt{t},e^{-\sqrt{t}}...
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Importance of the Transcendence of $\pi$ and $e$ [closed]

Why do people care that $\pi$ and $e$ are transcendental?
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Defining number set sequences based on the fundamental theorem of arithmetic

The Fundamental theorem of arithmetic gives a natural way to write natural numbers $\mathbb{N}$ uniquely in terms of products in powers of primes: $$N = 2^{n_2} 3^{n_3} 5^{n_5} \ldots $$ for all $N \...
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Prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancendental over $\Bbb{Q}$.

We want to prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancedental over $\Bbb{Q}$. Attempt. We use proof by contradiction and so assume that $\alpha \in \Bbb{C}$ is algebraic over $\...
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Are random numbers transcendental?

Do I see it right that a randomly chosen number $a$ of the form $ a = 0.abc...xyz$ with random digits $a,b,c,x,y,z$ an approximation of a transcendental real number is? It is an approximation of a ...
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Are all transcendental numbers a zero of a power series?

So I came across the concept of extending the notion of irrationality to higher degree polynomials. The base case of this is standard irrationality. That is, a number is irrational if it cannot be ...
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If the sum of a sequence of positive rational numbers is transcendental, is the sum of every subsequence that is irrational, also transcendental?

Assume we have a sequence of positive rational numbers $(a_n)$, and $\sum_{n=1}^\infty a_n = x$ and $x$ is transcendental. If we have a subsequence of $(a_n)$, $(b_n)$ and $\sum_{n=1}^\infty b_n = y$ ...
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Is uncountable transcendental-additions enough to make $\mathbb{Q}$ into $\mathbb{R}$?

Consider $\mathbb{Q}$ and then consider "adding" a transcendental $\zeta$ to it, while still retaining the field axioms (i.e. $\mathbb{Q}(\zeta)$). We could add another transcendental in an obvious ...
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Are there any theorem relating transcendency/irrationality of Euler's constant and the Riemann hypothesis?

Several formulations of the Riemann hypothesis can be given in terms of Euler’s constant. For example, Theorem (Nicolas 1981) The Riemann hypothesis holds if and only if all the primorial numbers $...
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How to obtain the degree of this field extension?

Let $K/F$ be a field extension, $t,w \in K$ such that $t$ is transcendental over $F$ and $w$ is transcendental over $F(t)$. Prove that for any positive integer numbers $n$ and $m$ the equality $[F(t,w)...
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Is there a distinguishable characteristic between the summation / continuous fraction method in algebraic and transcendental numbers?

I try to illustrate the formation and derivation of algebraic numbers and transcendental numbers. I found that both categories of numbers can be made by continuous summation or division/fraction ...
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Proof of rational + algebraic is algebraic and rational + transcendental is transcendental?

I've heard of and seen some proofs that the product and sum of two algebraic numbers is algebraic, however many of them are quite complex and require a variety of machinery (from matrix eigenvalues to ...
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Verification and rigorousness of my own proof of transcendence of Liouville's number.

Theorem: Liouville's number: $$L=\sum_{i=0}^\infty 10^{-i!}$$ is transcendental. Proof: Denote $$L_k=\sum_{i=0}^{k!} 10^{-i!}$$ We have: $$L_1=0.1$$ $$L_2=0.11$$ $$L_3=0.110001$$ $$L_4=0....
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What is the usefulness of classification into Transcedental and Algebraic numbers?

Classification into rationals and irrationals makes complete sense because irrational numbers seem to be completely different from rational numbers, which are terminating or repeating. We know that ...
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Is this :$\int_{-1}^{1}\exp (-\tanh^{-1} (x))dx=\pi$ a well known integral representation for $\pi $?

I have tried to get interesting integral for interesting transcendental number , I have got the following :$\int_{-1}^{1}\exp (-\tanh^{-1} (x))dx=\pi$ , really I didn't accross that integral in the ...