# 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 there an exact expression for the full width half maximum of a sech^2 curve convolved on itself?

As some simple math can show, a Gaussian convolved onto itself is also a Gaussian. Importantly, the FWHM of the original gaussian compared to that of its convolved counterpart is different by a factor ...
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### Is transcendental number to non zero algebraic power always a transcendental?

My motivation to this question is that we know $e^{a}$ is transcendental , where $a$ is Non-Zero-Algebraic, using Lindemann Theorem, but is it true for all transcendental numbers not only $e$?
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### I want to ask for good references of linear algebra over rational numbers?

My current studies in algebraic number theory have led me to observe the frequent interplay between linear algebra concepts over the field of rational numbers. This connection becomes particularly ...
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### A simple, concrete example of a transcendental element.

I am writing an article on transcendental numbers and I'm wondering if it is possible to construct a "simple" example of a transcendental element in a field extension. When thinking about ...
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### Transcendence of $\ln(a)/\ln(b)$ where $a$ and $b$ are rational numbers.

In this post I proved that ln(3)/ln(2) is transcendental and an immediate corollary is that ln(x)/ln(y) is transcendental where x and y are natural numbers $x,y \neq 0,1$ if x is odd and y is even or ...
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### Proof that $\ln(3) / \ln(2)$ is transcendental?

I was trying to figure out if $\ln(3)/\ln(2)$ is transcendental, when I found this post by b_jonas But there's a proof just as simple showing that $\log 3/\log 2$ is irrational. Suppose on contrary ...
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### Is the natural logarithm of $a$ irrational if $a \neq 0,1$ is algebraic?

Starting with the result of the Gelfond–Schneider theorem: $$a^b = c$$ where a and b are complex algebraic numbers with $a \neq 0,1$, and $b$ not rational and $c$ is transcendental. Substitute the ...
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### Does a transcendental number raised to the power of an irrational algebraic number, always result in an algebraic number?

If $b$ is transcendental number and $a$ is an irrational algebraic number, is it safe to conclude that $b^a$ must be an algebraic number? My reasoning is: Write the result of the Gelfond–Schneider ...
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### How to identify transcendental numbers?

Given the relationship: $$\pi = \left(\frac{15ab^2c^3}{2d^4}\right)^{\frac{1}{5}}$$ where a, b and c are non-zero positive constants, can it be assumed at least one of a, b or c must be a ...
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### House of the inverse

Let $\alpha,\beta\in\overline{\mathbb Q}$. Denote by $h(\alpha)$ the house $\alpha$, that is the maximum of $|\sigma(\alpha)|$ when $\sigma$ describes $\mathrm{Gal}(\overline{\mathbb Q}/\mathbb Q)$. ...
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### Why a root for an integer polynomial in definition of transcendental number?

I've read many definitions for a transcendental number and some of them say that a transcendental number is a number that is not the root of any integer polynomial, while other say is a number that is ...
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### transcendental numbers as solutions to hyperpowers [duplicate]

By the Gelfond-Schneider theorem, if $x^x=2$, $x$ must be transcendental. What can be said of $x$ if $x^{x^x}=2$, $x^{x^{x^x}}=2$ etc.? Must it be transcendental? Of course, $2$ can be replaced by any ...
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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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