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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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Are there any irrational/transcendental numbers for which the distribution of decimal digits is not uniform?

I conjecture that for irrational numbers, there is generally no pattern in the appearance of digits when you write out the decimal expansion to an arbitrary number of terms. So, all digits must be ...
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One to one correspondence between transcendental and uncomputable numbers

I know that both sets are uncountable infinite but the transcendentals are not a subset of the uncomputables. I don’t know if there exist uncomputable numbers that are not transcendental. But my ...
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What reason is there to conjecture that every finite string is really in the decimal expansion of $\pi$?

One of my students asked me this, and it occurred to me that I had never really questioned it. Apparently, it is only conjectured but widely believed that the decimal expansion in base $10$ of $\pi$ ...
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numbers that cannot be expressed in closed form?

Irrational numbers can be divided into two categories: - Algebraic - Transcendental But there are some numbers that are roots of polynomial ie. are algebraic but cannot be expressed in closed ...
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Question about the Gamma function

My question is fairly simple: I was wondering if $\,\,\,\Gamma\left(\pi\right) = 2.2880377\ldots\,\,\,$ had any special meaning. Is it irrational ?. transcendental ? is it useless ? ...
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Is this number Liouville?

If $b\in\mathbb{Z}_{\geqslant2}$, is the number given by the following sum: $M=\displaystyle\sum_{n\in\mathbb{Z}_{\geqslant0}}b^{-10^{n}}$ a Liouville number? It has lots of zeros, for sure, and by ...
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Suppose $\ \pi+e\ $ is transcendental. What about $\ \pi-e\ $?

Suppose $\ \pi+e\ $ is a transcendental number. Can we conclude whether $\ \pi-e\ $ is rational, algebraic irrational or transcendental ? If I understood the consequences of Schanuel's conjecture ...
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Transcendental in the infinite limit, rational otherwise

I am confused at the very core my understanding of numbers for this particular subject, for some $n \in \mathbb N \backslash {\{1}\}$,the following appears to become an increasingly accurate ...
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Density of power of some transcendental number

Let's note that where $\{x\}$ means fractional part of $x$. I am trying to figure out if $\{e^n\}_{n \in \mathbb{N}}$ and $\{\pi^n\}_{n \in \mathbb{N}}$ are dense in $[0,1]$. In general do we know if ...
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Rainbow numbers: Can mapping digits to different bases produce different varieties of irrationality?

This is a follow-up to the question, "Irrationality of 0.123456789101112 … and similar numbers." There I took some decimal number, in one case Champernowne's constant, $$ n_{10} = 0....
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Irrationality of 0.123456789101112 … and similar numbers

Consider four numbers in $(0,1)$: $n_1$ in base $10$ is formed by listing the decimal digits $1,2,3,4,\ldots$; $b_1$ in binary is formed by $0$ and $1$ for each even and odd digit of $n_1$: $$ n_1 = 0....
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Algebraic number to Liouville number

If $a\in\mathbb{R}\setminus\left\{0,1\right\}$ is an algebraic number, can $\ln\left(a\right)$ ever be a Liouville number? This is not a homework question, nor do I know much about the innards of ...
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For $\epsilon > 0$, is there always $n,m \in \mathbb{N}$ so that $e^{n}$ is $\epsilon$-close to $m$? [closed]

I don't have much to go off of, so I can't demonstrate any attempts here. I just want to know if there has been any answer or partial answer to this question.
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What is the intuition behind the proof that e is transcendental (David Hilbert's proof)?

I have tried to look at David Hilbert's proof that e is transcendental (the one which uses integral, a polynomial and a very large prime number p and looks at the divisibility to (p-1)!), but I find ...
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Properties of complex exponentials.

I saw the following property in wikipedia \begin{align} (e^z)^n=e^{zn} \tag{1} \end{align} where $z \in \mathbb{C}$ and $n \in \mathbb{Z}$. It's possible to change the $e$ in $(1)$ to some other ...
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Is the Prouhet-Thue-Morse constant transcendental in any integer base $b>2$?

The Prouhet-Thue-Morse constant, defined as $$ \tau =\sum _{{i=0}}^{{\infty }}{\frac {t_{i}}{2^{{i+1}}}}=0.412454033640\ldots $$ where the $t_i$ are elements of the Thue-Morse sequence, is ...
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If $u\in F$ is Transcendental over $K$, $F$ an extension field of $K$, Show every element in $K(u)$ not in $K$ is transcendental over $K$.

Doing some problems out of Beachy’s Algebra text, I came across that problem, and I’m at a loss how to show it without a bit of hand waving. Do I make some statement about spaces, and prove by ...
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How does Siegel's Lemma and properties of auxiliary functions helps in transcendental/irrationality proofs?

I know this Lemma appears more in trascendental number theory, but I saw a proof of the irrationality of $e^\pi$ it's a proof by contradiction, finding a integer between $0$ and $1$, but to that it's ...
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Solution of equation in $k$, $\sin k$ and $\cos k$

Some physical problem reduces to some equation that looks pretty transcendental: $$ k \sin\mathopen{}\left(\sqrt{k^2+2\delta}\,\pi/2\right)\mathclose{}\cos(k\pi/2) + \sqrt{k^2+2\delta}\cos\mathopen{}\...
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How do I prove that $ex/\ln(x)$ is the function traced out by the successive functions?

Why is the purple function equal to $ex/\ln(x)?$ How do I prove this? After tracing out successive blue functions I wanted to find the function that passed through the minimum points of the ...
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If the Liouville Constant is transcendental, is its exponentiation also transcendental?

So we have Liouville's Constant: $L_b=\displaystyle\sum_{n\in\mathbb{Z}^+}b^{-n!}=\left(0.\textbf{1}\textbf{1}000\textbf{1}00000000000000000\textbf{1}00000000...\right)_b$ And let $M$ be the ...
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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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1answer
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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?