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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Reciproc of the Lindemann theorem and the arc cosine of the golden ratio
Via the Lindemann theorem it is easy to prove that the cosine of any rational multiple of $\pi$ is an algebraic number; however its contrapositive only tells us that the arc cosine of an algebraic ...
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Can $\pi$ be a root of a polynomial under special cases?
What if we consider polynomials whose coefficients are either rational or $e$, that is, a polynomial in $\mathbb{Q} \cup \{e\}$ with $\pi$ as a root.
Can this happen? Does it matter if we change ...
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What is $\sum\limits_{n=0}^{\infty} r^{an^2 + bn + c}$ ? or: is $0.0100100010000100001…$ transcendental?
The idea is a more convenient form for $N = 0.01001000100001000001...$ in base $r$, hopefully to show whether it is transcendental.
Sorry for brevity.
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Extrapolating properties of rational numbers to irrational/transcendental numbers
I've had this idea in my head for a while, but I've never told anybody because... well, I really don't know. I just never thought that it might even be remotely correct, but here goes. Here is just an ...
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Is this number Transcendental?
I've been messing around with microtonal scales and I came up with this number and I was wondering if it is Transcendental.
U = 17.312340490667609 to 15 decimal places
U = ...
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Transcendentality of the $\log$ of the golden mean
We know that $\phi$, the golden ratio, is algebraic. Is it known whether $\log(\phi)$ is algebraic?
Thank you!
PS. I am not in number theory, so I apologize in advance if this is obvious.
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How do we prove the existence of uncountably many transcendental numbers?
I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that ...
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Prove that $\pi$ is a transcendental number
Does anyone has a link to a site that confirms that $\pi$ is a transcendental number?
Or, can anyone show how to prove that $\pi$ is a transcendental number?
Thank you in anticipation!
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Is this number transcendental?
My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer.
After tiring of ...
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Polynomials in irrational powers and their roots
When solving an equation with irrational (or algebraic in the case) powers, are the roots likely to be transcendental or algebraic, or does it vary?
As an example, I was trying to figure out if $(x + ...
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Sets of Constant Irrationality Measure
Let $\mu (r)>2$ be the irrationality measure of a transcendental number $r$, and consider the following set of points $P \in\mathbb{R}$:
$P=\{r\in \mathbb{R}: \mu(r)=Constant\}$
Is this set a ...
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Does the “no gaps”-property with transcendental numbers mean that there is only “one number”?
I admit, the question is a little provocative, but is asked in all earnestness.
It is inspired by this one: Does .99999... = 1?
Many of the arguments here point to the fact that 0.999... and 1 are ...
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Liouville's number revisited
Liouville's Number is defined as $L = \sum_{n=1}^{\infty}(10^{-n!})$. Does it have other applications than just constructing a transcendental number?
(Personally, I would have defined it (as ...
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Certain Liouville Numbers
A Liouville number is a number which can be approximated very closely be a sequence of rational numbers (here is the rigorous definition I am working off of: ...
