# 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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### Is $0.1010010001000010000010000001 \ldots$ transcendental?

Does anyone know if this number is algebraic or transcendental, and why? $$\sum\limits_{n = 1}^\infty {10}^{ - n(n + 1)/2} = 0.1010010001000010000010000001 \ldots$$
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### Are $\pi$ and $e$ algebraically independent?

Update Edit : Title of this question formerly was "Is there a polynomial relation between $e$ and $\pi$?" Is there a polynomial relation (with algebraic numbers as coefficients) between $e$ or $\pi$ ?...
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### Is $0.23571113171923293137\dots$ transcendental?

Is the following number transcendental? $$0.23571113171923293137\dots$$(Obtained by writing prime numbers consecutively from left to right, in the decimal expansion)
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### Different types of transcendental numbers based on continued-fraction representation

I've been reading Wikipedia's article on continued fractions. A few examples are given for the continued-fraction representation of irrational numbers: $\sqrt{19}=[4;2,1,3,1,2,8,2,1,3,1,2,8,\dots]$ ...
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### Is the solution of $e^x \log(x)=1$ transcendental?

Let $u$ be the solution of the equation $$e^x \log(x)=1$$ Is $u$ rational, irrational algebraic or transcendental? $u$ seems to be transcendental, but I cannot prove it. Perhaps, someone has an ...
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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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### 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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### Algebraic subfield of transcendental extension

I was recently thinking about whether it is possible to generate an infinite dimensional algebraic extension over a base field using just finitely many transcendental elements. Specifically, given a ...
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### Prove of trancendence of $\ln(2)$.

Where can I find some proofs for another transcendental numbers, like Hermite/Lindemann theorem proofs for $e/\pi$? For instance, prove that $\zeta(3)/\ln(2)$ is a transcendental number.
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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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### Numbers which are “Provably Difficult to Compute”?

We recall that a computable number $\alpha \in \mathbb{R}$ satisfies the following: there exists a computable function $f$ such that, given any positive rational error bound, $f$ outputs a rational ...
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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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### Do replacing distinct digits from distinct places of an algebraic irrational

Do replacing distinct digits from distinct places of an algebraic irrational number necessarily make it a trancsendendal number? Since my question isn't worded well, therefore I would clarify it by ...
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### Can permutating the digits of an irrational/transcendental number give any other such number?

Let $x_n$ be the infinite sequence of decimal digits of a fixed irrational/trascendental number. Can I obtain any other irrational/trascendental number's sequence of decimal digits through a ...
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### Other Algebraically Independent Transcendentals

I was thinking about a incomplete answer I gave earlier today to a interesting question made by the user lurker. The question was about wheter or not the sum or the product of two transcendental ...
So I've just watched this wonderful Numberphile video about transcendental numbers. In the video, the guy shows that $$e=\sum_{n=0}^\infty\frac{1}{n!}=1+\frac{1}{1}+\frac{1}{1\cdot2}+\frac{1}{1\... 3answers 642 views ### Can every transcendental number be expressed as an infinite continued fraction? Every infinite continued fraction is irrational. But can every number, in particular those that are not the root of a polynomial with rational coefficients, be expressed as a continued fraction? 1answer 191 views ### Irrationality/Transcendentality of values of e^{e^x} 1) Is e^{e^x} irrational for all rational x? It is known that e^x is transcendental for every nonzero algebraic x. But this dos not help here because for transcedental x, e^x can be ... 2answers 111 views ### For which x is e^x rational? Transcendental? Apart from the trivial cases, x=\log a where a\in\mathbb{Q}, are all values of e^x irrational? Are some transcendental? 1answer 55 views ### Changing digits of an irrational allowed? Suppose you change every instance of a specific digit of π, e.g., suppose you make every "4" a "6" instead. I realize that this too would be irrational, but what I want to know is (1) on what basis is ... 1answer 96 views ### Prove \log(x) is transcendental What is a proof that \ln(\alpha) is transcendental for \alpha. I believe I heard somewhere that the natural logarithm of any rational number is transcendental. Do you guys have any proofs of that ... 3answers 197 views ### Is the product of a transcendental number by an integer transcendental? I don't really know a lot about this subject but I was wondering if the product of a transcendental number by an integer is transcendental? 3answers 521 views ### Is there a general way to solve transcendental equations? To make things definite, let's narrow them and call transcendental equation of the form$$f(x) = 0$$where f is a real elementary function in the usual sense. For example$$\cos(\pi x) + x^2 = 0...
Weierstrass proved the result [Lindemann-Weierstrass theorem] that if $a_1, \cdots, a_n$ are reals linearly independent over the rationals, then $e^{a_1}, \cdots, e^{a_n}$ are algebraically ...