# Tagged Questions

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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### Do we know if all simple extensions of the field of rational numbers by transcendental numbers are not equal?

I understand that $\mathbb{Q}(x) \cong \mathbb{Q}(u)$ for all transcendental $u$, where $\mathbb{Q}(x)$ is the field of rational forms over $\mathbb{Q}$ and thus that all simple extensions of the ...
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### Is it known if $\pi + e$ is transcendental over the rational numbers?

I recall reading a comment on reddit that had stated that it is not known if $\pi + e$, (nor $\pi e$) is transcendental over $\mathbb{Q}$, nor even if it is irrational. Is this true? It strikes me as ...
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### Is there a direct proof that pi is not the root of an algebraic equation whose degree is a power of 2 [duplicate]

All known proofs that the circle cannot be squared are based on Lindemann's theorem that $\pi$ is not analgebraic number. But this seems to be a case of using an atomic bomb to kill a fly. What ...
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### Why is Chaitin's constant absolutely normal?

I have repeadetly seen claims that Chaitin's constant is normal in all bases (e.g. on Wikipedia), and I have also seen some proof sketches (e.g. here), but these only show the idea. For example, the ...
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### For which $x$ is $e^x$ rational? Transcendental?

Apart from the trivial cases, $x=\log a$ where $a\in\mathbb{Q}$, are all values of $e^x$ irrational? Are some transcendental?
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### Is a trigonometric function applied to a rational multiple of $\pi$ always algebraic?

Specifically, just to talk about cosine, is it true that $\cos(\frac{a\pi}{b})$ is algebraic for integers $a$ and $b$? Looking at this post and the link to trigonometric constants in the comments, it ...
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### Let$\ \lim_{n\to \infty} \frac{ \ln n}{f(n)}=1$. If$\ a,b,c$ are natural, can we have$\ a^{b+c \ln n}\sim a^{c f(n)}$?

I shall note that$\ n$ as well goes through the natural numbers and that$\ f(n)$ is rational for any$\ n$. Also, I'm obviously excluding$\ a=1$. I'm inclined to think my claim is not possible ...
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### What transcendental numbers are produced by $\sin{\alpha}$ when $\alpha$ is algebraic/constructible/rational (in radians)?

I know that by Lindemann–Weierstrass theorem(LW) sine and cosine of non-zero algebraic numbers (in radians) produce results that are transcendental. My question is what are the transcendentals ...
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### Are there classifications transcendental numbers that are similar to algebraic numbers for differential equations?

Considering that transcendental numbers are described as not a root of a non-zero polynomial equation with rational coefficients, are there classifications of transcendental numbers that are ...