Articles of transcendental numbers

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 rational. 2) Is $e^{e^x}$ transcendental for all algebraic $x$? This would imply 1).

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 permutation of $x_n$?

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 means of a simplest example: $x=\sqrt{2}=1.4142\color{blue}{1}3\color{green}{5}6237..\text{which is algebraic}$ Lets replace the $5^{th}$ digit by $2$ and $6^{th}$ digit by $7$ $y=1.4142\color{blue}23\color{green}76237$ $\large \text{Question}:$ Is $y$ […]

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?

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 “Steven’s Number” :-)) as binary: $S = \sum_{n=1}^{\infty}(2^{-n!})$, since each digit can only be “0” or “1”: the corresponding power of 2 (instead of 10) included or not. Since […]

How to show $e^{2 \pi i \theta}$ is not algebraic.

I was wondering if someone could possibly help me figure out how to show $e^{2 \pi i \theta}$ is not algebraic when $\theta$ is irrational. Thanks!

Is product of algebraic and transcendental number transcendental?

Let $\alpha \in \mathbb{A}$, and $\gamma \in \mathbb{T}$. I know that the reciprocal of a transcendental number is transcendental. Question: Is $\alpha\cdot \gamma \in \mathbb{T}$?

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 something which is very easy to prove, in my head I could figure out the rough […]

What is the probability that $250$ random digits contain $7777$ , $8888$ and $9999$?

First, how I arrived at a number having the property that the first $250$ digits after the decimal point contain $7777$ , $8888$ and $9999$ I wanted to construct a number which can be shown to be transcendental using the irrationality measure Consider the sequence $$a_1=0\ ,\ a_2=1\ ,\ a_n=a_{n-1}^2+a_{n-2}\ for \ n>2$$ Then, […]

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 numbers in $\mathbb{C}/\mathbb{Q}$ could result on a algebraic number in the same set. A infinite amount of trivial examples can […]