Articles of transcendental numbers

Three other transcendental(?) numbers

I have $3$ numbers which seem to be transcendental, but probably very difficult to be proven transcendental. PARI/GP gives no indication that one of the numbers is algebraic. The first one $$\int_0^1 \ln(x^2+1) dx=\ln(2)+\frac{\pi}{2}-2$$ This number has the same numerical status as $\ln(2)+\frac{\pi}{2}$, which is the sum of two transcendental numbers. Of course, this does […]

Real numbers of the form: “difference of two linearly independent transcendental numbers”

Regaridng to the question Real numbers which are writable as a differences of two transcendental numbers now, it is important to know answer of the following question: Is it true that for every real number $x\neq 0$ there exist linearly independent transcendental numbers $\alpha$ and $\beta$ (i.e., $r\alpha+s\beta=0$ implies $r=s=0$, for every rational numbers $r,s$) […]

if $x\ne 0$, is at least one of $\{x, \cos\;x\}$ transcendental over $\mathbb{Q}$?

it seems at least superficially plausible that for real $x \ne 0$ then at least one of $\{x, \cos\;x\}$ is transcendental over $\mathbb{Q}$. has this assertion been proved to be true or false?

Is a complex number with transcendental imaginary part, transcendental?

A complex number that has transcendental real part is always transcendental? How about in the case of imaginary part?

Please help me check my proof that the transcendental numbers are dense in $\mathbb{R}$

I need to prove that the set of all transcendental numbers is dense in $\mathbb{R}$, and to that end, I have written the following proof: Let $\mathbb{T}$ denote the set of transcendental numbers in $\mathbb{R}$. First, notice that $\mathbb{T} \subset \mathbb{R}\setminus \mathbb{A}$, where $\mathbb{A}$ denotes the algebraic numbers in $\mathbb{R}$. Next, since $\mathbb{A}$ is a […]

Complex transcendentals not known in component form?

Are there any transcendentals whose real or imaginary components have not been found in exact form?

Do transcendental numbers contain any string of digits?

It is often said that $\pi$ contains any string of digits. But does the property “transcendental” imply “contains any string of digits?

Producing infinite family of transcendental numbers

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 independent. I would like to know if the result holds for infinitely many numbers. Explicitely, if $\{a_1, a_2, \cdots \}$ is an infinite family of real numbers such that every finite […]

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?

Adding or Multiplying Transcendentals

Is it possible to add or multiply (no subtraction) only positive transcendental numbers and yield a solution that is algebraic? Exponential manipulation is excluded from this question, as $e^{\ln2} = 2$ EDIT: Also, excluding reciprocals of other transcendental numbers and subtraction?