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 $e$.

Let $\mathbb{A}\subset \mathbb{C}$ be the set of algebraic numbers: $$ \mathbb{A} = \{ z\in \mathbb{C}: p(z)=0, p(z)\in \mathbb{Z}[x] \} $$The set of transcendental numbers is $\mathbb{A}^c$. The first construction of a transcendental number, $\sum_{n=1}^{\infty} 10^{-n!}$, was due to Liouville in 1851.

If we fix a degree of polynomials in $\mathbb{Z}[x]$, there are only countably many such polynomials, each with finitely many roots. Then $\mathbb{A}$ is countable, whence $\mathbb{A}^c$ is uncountable. Put crudely, 'most' complex numbers are transcendental, though showing a particular number is transcendental is usually rather difficult.

All transcendental numbers have an irrationality measure of at least 2; see for more information.

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$\sum_{n=1}^{\infty} a^{-n!}$ is transcendental ??

Is $\sum_{n=1}^\infty a^{-n !}$ transcendental for any positive integer a ? I know $\epsilon =\sum_{n=1}^{\infty} 10^{-n!}$ is transcendental, for Liouville´s Theorem, ($p_k=10^{k!} \sum_{n=1}^k 10^{-n!}$ and $q_k=10^{k!}$. Then $| \epsilon…
Angelo
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If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then $a^b$ is transcedental.

For my math study, I have to prove the following: Let's denote the set of algebraic numbers with $\mathbb{A}$. Prove: If $a \in \mathbb{A}$\{$\mathbb{0,1}$}, $b \in \mathbb{A}$\ $\mathbb{Q}$ then $a^b$ is transcedental. I'm trying to use Baker's…
Peter
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Calculating Closed form of Basel type problem.

I want to find the sum of $\displaystyle\sum_{n=1}^\infty \frac{1}{n^{\phi(n)}}$. where $\phi$ is Euler's totient. So for example $\frac{1}{1^1} + \frac{1}{2^1} + \frac{1}{3^2} +\frac{1}{4^2} + \frac{1}{5^4} + \frac{1}{6^2}$ etc... I also want to…
Neil
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Are there any better asymptotics than Liouville for how fast a series of rational terms needs to converge to guarantee the sum being transcendental?

So the title basically says it all: A Liouville number is a number $x$ such that for any $n$, there exist integers $p,q$ with $q > 1$ and $0 < |x - p/q| < 1/q^n$. This implies that the number $x$ can be arbitrarily well approximated by rationals, to…
user2566092
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Algebraic relations between trigonometric numbers

Given $n\in2\Bbb N$, what is precise algebraic relation between $cos\frac{\pi}{n-1}$,$cos\frac{\pi}{n+1}$? Both numbers are algebraic, which implies there should be an algebraic relation between them. What is this precise relation? Is this relation…
Turbo
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How could we prove that it is not a spanning set.

Consider the space $\mathbb{R}$ as a linear space over the field $\mathbb{Q}$ of rational numbers. For any transcendental number x the set {1, $x$, $x^2$, $x^3$,......} is linearly independent. How could we prove that it is not a spanning set.
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Proof for $-4\pi^2+48\ne A+B\pi+C\pi^2$ when $(A,B,C)\ne (48,0,-4)$

I have to prove $-4\pi^2+48\ne A+B\pi+C\pi^2$ when $(A,B,C)\ne (48,0,-4)$ $A,B,C \in \mathbb{Q}$ As a part of question I try to solve.
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Where does the linear equation in the proof of solvability in elementary numbers in [Chow 1999] come from?

How do one get the equation below in the proof in [Chow 1999]? Chow writes on page 444 - 445: "If $A=(\alpha_1,\alpha_2,...,\alpha_n)$ is a finite sequence of complex numbers, then for brevity we write $A_i$ for the field…
IV_
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How to prove $\pi^i$ is irrational?

How to prove $\pi^i$ is irrational? Is there a method to prove this question? Thanks for the answers!
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Logic question about a interpretation of a result about transcendental numbers.

Suppose we have $\pi$ and $\pi x$ where $x>0$ is an integer. I) We don't know if $\pi$ is algebraic or not. (Let's pretend) II) We are dealing with two set of numbers: algebraic and transcendentals. In the sum: $$\pi+\pi x$$ suppose we can prove…
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Are there transcendental numbers that cannot be reached?

This is a hard question to ask. But I've been contemplating transcendental numbers. I know that there are infinitely many; simply multiply a known transcendental (like pi) by every rational number. So are there any transcendental numbers that we…
Jordan
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Why is :$\displaystyle {e}^\sqrt{2}$ is known to be transcedental number but ${\sqrt{2}}^ {e}$ is not known?

I'm confused why $\displaystyle {e}^\sqrt{2}$ is known to be transcendental number but in the same time ${\sqrt{2}}^ {e}$ is not even known , why we can't deduce any thing from $\displaystyle {e}^\sqrt{2}$ to know more about irrationality of…
zeraoulia rafik
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Product of Transcendentals

I want to prove or disprove that the product of two transcendental is transcendental: (However, not using an inverse identity such as $\pi$ and $\frac1\pi$) My attempt: Proof using Hilbert's number: $2^\sqrt2$ The product of Hilbert's number I…
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Is my proof that $\gamma$ (the Euler-Mascheroni constant) is transcendental correct?

The Euler-Mascheroni constant $\gamma$ can be defined as $\lim\limits_{n\to \infty}(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}-\ln n)$. For every positive integer n (except for 1), the value of this sequence is transcendental. So from the definition,…
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Is it possible that a non repeating expansion in a variable base to be rational?

or the reverse, where a repeating expansion in a variable base to be rational? been trying some trial and error cases without success by variable base we mean each digit can be in a different base but any digit can occur as many times in the same…
jimjim
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