# 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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### Combinations of Transcendental Numbers are still transcendental numbers?

We know there are numbers like $\pi$, $e$, $\phi$ or also $\zeta(3)$ which are transcendental numbers. I was wondering if combinations of transcendental numbers are still transcendental numbers, like ...
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### Finding transcendental roots to an algebraic equation

So for equations with rational roots, there's a theorem that lists all the possible roots (Rational Root Theorem). If an equation has imaginary or irrational roots, their respective theorems say ...
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### A binary irrational with bits defined by primes

Define a number $q$ in binary notation whose $n$-th bit is $1$ for $n$ prime, and $0$ for $n$ composite. So its 2nd, 3rd, 5th, 7th, 11th, etc. bits are $1$, with all other bits $0$. Here is $q$ out to ...
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### Is $\cos\log a$ a transcendental for all nonzero algebraic $a$?

Is this known? (excluding a=1 as was corrected in comments)
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### If $a$ is a transcendental number, then is $a^n$ also a transcendental number? [closed]

If $a$ is a transcendental number (i.e., a number s.t. there does not exist a polynomial $P(x)$ s.t. $P(a) = 0$), is $a^n$ also transcendental? It would seem to me that it should be, but I can't ...
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### Algebraic values of the Gamma function using $\pi$

Is there any $x\in(0,1)\cap\mathbb Q$ different from $1/2$ such that $\Gamma(x)$ is algebraic over $\mathbb Q(\pi)$?
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### Is there a function whose limit approaches Pi? [closed]

I don't think my knowledge of Pi, irrationality, and transcendental numbers in general is complete. I've Googled for a day before posting this question. Intuitively, I understand why the ratio of ...
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### To prove that element $\frac{3}{n}+i\frac{4}{5}$ has an infinite order in $\mathbb{C}$ for any $n\in\mathbb{Z}\backslash \{0\}$

The problem is to prove that element $z=\frac{3}{n}+i\frac{4}{5}$ has an infinite order in the group $(\mathbb{C},\, \cdot\, )$ for any non-zero integer $n$. Let's consider the case $|n|\neq 5$. The ...
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### Determine a number is transcendental/algebraic

Determine: $(0.064)^{\frac{1}{3}}$ is transcendental or algebraic To show a number is transcendental/algebraic do I need to show there is a monic polynomial with integer coefficients such that the ...
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### For which values of $\sin(\theta)$ is the function algebraic?

Earlier today I stumbled upon a very long formula for the sine of 1 degree. (http://www.efnet-math.org/Meta/sine1.htm). When I reflected on this, it occurred to me that I could probably make a similar ...
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### Power of transcendental number.

There are some results that i found $1.$ if $a$ is an algebraic number other than $0$ and $1$ and $b$ is irrational algebraic then $a^{b}$ is transcendental like $2^{\sqrt{5}},3^{\sqrt{7}}$etc. $2.$ ...
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### Are there any known transcendental which measures something in the natural world except pi and e? [closed]

For pi, it measures the ratio of the circumference and diameter of a circle, etc. And e also means many special thing(mesuring growth, Prime Number Theorem, etc.).
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### Is the series: $\frac{\pi}{p_{1}!}+\frac{\pi}{p_{2}!}+…+\frac{\pi}{p_{n}!}$ convergent or divergent, where $p_n$ is the $n$-th odd prime?

Is the series: $$\frac{\pi}{p_{1}!}+\frac{\pi}{p_{2}!}+...+\frac{\pi}{p_{n}!}$$ convergent or divergent, where $p_n$ is the $n$th odd prime? And also why it is (the partial sums) transcendental? ...
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### Is $\pi^0$ actually rational? How about $\pi^i$? [duplicate]

Is there a rational argument that a transcendental or irrational number raised to zero should magically turn it into an integer, beyond obtuse convention? How about $\pi^i$? Is there a reasonable ...
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### Let $a$ and $n$ be integers, such that $a,n>1$ and $n$ is not a perfect square; show that: $a^{\sqrt{n}}$ is a transcendental number.

Although it is very hard to determine if a number is transcendental, I could appreciate any basic or simple insight or opinion concerning the statement, whether it is true or false. Regards
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### Transcendence of $\Gamma(1/3), \Gamma(1/4)$

Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a reference to that proof? Or maybe some details on the essential ideas ...
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### Wrong proof…But where is the mistake?

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\...
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### Do the second-last-digits of the primes $\ge 11$ form a transcendental number?

Suppose, the number $x$ is constructed from the second-last-digits from the primes $\ge 11$ The first $1996$ digits of $x\ =\ 0.11112...$ after the decimal point are : ...
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### Transcendental number

While reading on Wikipedia about transcendental numbers, i asked myself: Why is it so hard and difficult to prove that $e +\pi, \pi - e, \pi e, \frac{\pi}{e}$ etc. are transcendental numbers? ...