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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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 $\pi$ ...
23
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801 views

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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Closed form for a pair of continued fractions

What is $1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cdots}}}$ ? What is $1+\cfrac{2}{1+\cfrac{3}{1+\cdots}}$ ? It does bear some resemblance to the continued fraction for $e$, which is ...
66
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Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

I thought it was easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational, but found a gap in my proof. Simple finite approximations show the denominator cannot be small, though, ...
24
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1answer
757 views

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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Prove that $\pi$ is a transcendental number

Does anyone has a link to a site that confirms that $\pi$ is a transcendental number? Or, can anyone show how to prove that $\pi$ is a transcendental number? Thank you in anticipation!
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4answers
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Non-existence of irrational numbers?

I realize the title of my question will probably cause the raising of some eyebrows, so let me explain. Not sure whether to file this under "math" or "philosophy". This also might be able to be ...
2
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1answer
229 views

Sum and product of two transcendental numbers can't be both algebraic

Suppose $a$ and $b$ are complex numbers and both transcendental over $\mathbb Q$. I am wondering why $ab$ and $a+b$ can not both be algebraic. Thanks for any help.
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Proving that $\frac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\frac{(-1)^{(p_{k}-1)/2}}{p_{k}} \right )^{-1}$ an identity of Euler's.

This is another identity of Euler's relating $\pi$ to the prime numbers, available here \begin{align*} \dfrac{\pi}{2}=\prod_{k=2}^{\infty}\left(1+\dfrac{(-1)^{\dfrac{p_{{k}}-1}{2}}}{p_{k}} \right ...
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2answers
540 views

Linear independence of the numbers $\{1,e,e^2,e^3\}$

Does someone know a proof that $\{1,e,e^2,e^3\}$ is linearly independent over $\mathbb{Q}$? The proof should not use that $e$ is transcendental. $e:$ Euler's number. $\{1,e,e^2\}$ is linearly ...
2
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1answer
411 views

Is $\large \frac {\pi}{e}$ rational, irrational, or trandescendal?

Is there an argument for why $\large \frac {\pi}{e}$ is rational, irrational, or trandescendal? Can the quotient of any two transcendental numbers (which are not rational multiples of each other) be ...
5
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95 views

For integer $k > 1$, is $\sum_{i=0}^{\infty} 1/k^{2^i}$ transcendental or algebraic, or unknown?

Title says it all, I have an itch about series like this that seem to fall in the gray area where classical proofs that rational partial sums that converge too quickly must converge to transcendental ...
6
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3answers
847 views

How do we prove the existence of uncountably many transcendental numbers?

I know how to prove the countability of sets using equivalence relations to other sets, but I'm not sure how to go about proving the uncountability of the transcendental numbers (i.e., numbers that ...
3
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2answers
389 views

Sum and Product of two transcendental numbers cannot be simultaneously algebraic

If $\alpha$ and $\beta$ are real number and $\alpha$ and $\beta$ are transcendental over $\mathbb Q$, show that $\alpha \beta$ or $\alpha +\beta$ is also transcendental over $\mathbb Q$ Attempt: ...
31
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730 views

What is the role of mathematical intuition and common sense in questions of irrationality or transcendence of values of special functions?

I got the number $$\frac{\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)}{\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)}=0.824326275998351470388591998726842...$$ in the ...
19
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1answer
422 views

Is the positive root of the equation $x^{x^x}=2$, $x=1.47668433…$ a transcendental number?

I can prove using the Gelfond–Schneider theorem that the positive root of the equation $x^{x^x}=2$, $x=1.47668433...$ is an irrational number. Is it possible to prove it is transcendental?
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Is $ 0.112123123412345123456\dots $ algebraic or transcendental?

Since the decimal expansion $ 0.112123123412345123456\dots $ is non-terminating and non-repeating, clearly $ 0.112123123412345123456\dots $ is an irrational number. Can it be shown whether it is ...
34
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2answers
1k views

Is this number transcendental?

My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer. After tiring of ...
20
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4answers
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Uncountable set of irrational numbers closed under addition and multiplication?

Is such a thing even possible? There's not much to say really. Obviously if there was a set it would be full of transcendental numbers. This led me to think of a function generating transcendental ...
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Calculating pi manually

Hypothetically you are put in math jail and the jailer says he will let you out only if you can give him 707 digits of pi. You can have a ream of paper and a couple pens, no computer, books, previous ...
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Does this show that the Apery Constant is transcendental?

Last August I posted this on mathoverflow: http://mathoverflow.net/questions/71856/a-serendipitous-riemann-identity. I show the (slightly revised) equation below: $$\zeta (3)=\frac{2\pi^4}{315} ...
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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? ...
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Proof that at most one of $e\pi$ and $e+\pi$ can be rational

$e$ and $\pi$ are rather peculiar numbers. It turns out that, in addition to being irrational numbers, they are also transcendental numbers. Basically, a number is transcendental if there are no ...
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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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222 views

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 ...
8
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130 views

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 ...
5
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294 views

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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Proof of $\pi$ not being a quadratic irrational number.

Does someone know a proof (books , articles) that $\pi$ is not a quadratic irrational? The proof should not use that $\pi$ is transcendental. Any hints would be appreciated.
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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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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 = ...
5
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1answer
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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 $\alpha\cdot \gamma \in \mathbb{T}$?
3
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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 ...
3
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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?
3
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372 views

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 ...
2
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1answer
63 views

Extending the set of algebraic numbers

I have been trying to extend the countable set of algebraic numbers, by adding a countable amount of transcendental numbers (so that the resulting set is also countable). Now, of course I could ...
2
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170 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 ...
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386 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?
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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 ...
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Is $\ln n$ transcendental for all rational $n>1$?

I know that $\ln n$ is transcendental for all integer $n>1$. But does this still hold for non-integer rational values of $n>1$? For example, is $\ln 1.5$ transcendental? EDIT: Somehow managed ...