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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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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42 views

Let $K = $ algebraic numbers. Then is $\operatorname{Span}_K(\pi, \pi^2, \dots)$ a vector space of transcendentals?

$V = {\rm Span}_K(\pi, \pi^2, \dots)$ is clearly a $K$-vector space. If we let $K = \Bbb{Q}$ temporarily, then every element of $V$ is transcendental as it's a finite linear combination $Q(X), \ X = ...
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580 views

Irrationality of $\pi$ another proof

Proposition. Let $\alpha\in\mathbb{R}$. If there is a sequence of integers $a_n,b_n$ such that $0<|b_n\alpha-a_n|\longrightarrow 0^+$ as $n\longrightarrow \infty$, then $\alpha$ is irrational. ...
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53 views

Strange notation for a decimal expansion of a transcendental number

I am checking page proofs for one of my papers right now and an editor changed $\zeta(3)=1.202$$\ldots$ to: $\zeta(3) = 1.202,...,$ I find this latter notation very strange and think it ...
2
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245 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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819 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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1answer
54 views

Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$ (where $T$ = transcendental numbers)

Question: Prove that if $t \in T$ and $q \in Q$, but $q \neq 0$ then $qt \in T$. This is Exercise 2.7.13(a) from Mark E. Watkins, Jeffrey L. Meyer: Passage to Abstract Mathematics. I'm currently ...
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89 views

Question about $e^{e^{e^e}}$

Is there a proof that the power tower of length $4$ of $e$ is irrational? Is it known whether or not $$e^{e^{e^e}}$$ is transcendental?
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1answer
130 views

2.71828. And then another 1828.

This may qualify as the silliest math.SE question ever, but am I really the first person ever to worry about this? The decimal expansion of $e$ has a 2. And then a 7. And then a 1828. And...well, then ...
3
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21 views

How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$

How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$? If $a$ is transcendental over ${\mathbb Q}[\alpha]$ then the ...
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66 views

What are some algorithms that can be used to test if a number is transcendental or not?

Well according to the definition of transcendental numbers I find that its any number that doesn't have any polynomial equation of any degree with integer coefficients summing up to 0. So ...
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79 views

The smallest class of numbers closed under addition, multiplication, and exponentiation

Let $\def\A{\mathfrak A}\A$ be the smallest subset of $\Bbb C$ that contains the algebraic numbers and also all numbers of the form $$\sum \alpha_i^{\beta_i}$$ where the $\alpha_i, \beta_i$ are ...
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2answers
97 views

Let $t$ be a transcendental number. Prove that the set $\{(a+bt) \mid a,b \in \mathbb{Q}\}$ is not a number field.

Can I just pick a number in the set and then prove it's not constructible? Thx
6
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1answer
237 views

Is $\log 2\pi$ rational?

Is it known whether $\log 2\pi$ is rational (where the base of the logarithm is $e$)? Or algebraic?
2
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74 views

How to prove this: If $t$ is a transcendental number, then $5t^{4}+8t+3$ is also transcendental?

Can I prove as follows? If $5t^{4}+8t+3$ is not transcendental, then $5t^{4}+8t+3$ is a solution of a polynomial $p$. If you expand $p(5t^{4}+8t+3)=0$ and write it in the form of another polynomial ...
2
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28 views

Borel's result on transcendence measure

In "Sur la nature arithmétique du nombre e" (Comptes rendus de l'Académie des Sciences 128 (1899), 596-9) Borel presented his result on transcendence measure for e. This can be restated as follows: ...
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137 views

Rational and trascendental numbers: $\pi$, $e$ and $\pi+e$ [duplicate]

The numbers $\pi,e$ are trascendentals, but if consider: $\pi+e$ then is rational, trascendental? Thanks
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1answer
47 views

What kind of algebraic equations do trandescendal numbers not solve?

I know transcendental numbers cannot solve polynomials or rational functions (since they can always be written as a polynomial), but are they the solutions to equations containing a variable raised to ...
10
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2answers
213 views

Proof that cos(1) is transcendental?

So, I was playing around on Wolfram|Alpha (as we nerds like to do) and it said cos(1) was transcendental. Could someone provide me with the proof that cos(1) is transcendental?
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32 views

how fast does the proportion of associative operations on $S$ decrease with |$S$|?

as doubtless many have done before me, i recently fell into wondering how many of the binary operations on a finite set are associative. the stackexchange software fortunately pointed me to this ...
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158 views

Does the number $2.3\,5\,7\,11\,13\ldots$ exist and, if so, is it rational or irrational &/or transcendental? [duplicate]

Does there exist a number which contains in its digits all of the prime numbers listed in order: $$2.3\,5\,7\,11\,13\ldots\ldots$$ if so, will it be rational or irrational &/or transcendental?
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109 views

Element in field of quotients is transcendental

Let $F\subseteq E$ be fields, and let $c\in E$. Let $F(c)$ be the field of quotients containing $F$ and $c$. Suppose $c$ is transcendental over $F$. Prove that every element in $F(c)$ but not in ...
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1answer
113 views

Transcendental Numbers (simple question)

Are all transcendental numbers irrational? I know that not all irrationals are transcendental (for example, $\sqrt{2}$); but I only know of a few transcendental numbers and they are all irrational.
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182 views

Uncountable sets of transcendental numbers

As a sort of follow up question to a previous question found here, besides the Liouville numbers, are there any other uncountable collections of transcendental numbers that are known? Clearly you ...
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214 views

“philosophical” question about the transcendence of $\pi$

I don't have any knowledge on transcendence proofs. I just heard that Lindemann proved that for any $\alpha \in \mathbb R^*$ algebraic, $e^\alpha$ is transcendental. Then, since $i$ is algebraic, and ...
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128 views

Transcendental numbers involving primes?

Is the prime zeta function value $$ P(2)=\sum_{p \in \mathrm{primes}} \frac{1}{p^2} = 0.452247420041065498506543364832247934173231343\ldots $$ a transcendental number ? What about the following sum ...
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49 views

Fast converging sums involving tetrations

Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check. Now my question : ...
6
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130 views

is the unique solution of $\cos t = t$ a transcendental number?

let $\alpha$ be the unique fixed point of $\cos:\mathbb{R} \rightarrow [-1,1]$ for any $t \in \mathbb{R} \setminus\{0\}$ if $t$ is algebraic then $\cos t$ is transcendental. thus if $\alpha$ were ...
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?
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136 views

Products of irrational numbers

Hello ladies and gentlemen! A friend of mine and I have been thinking about this particular issue: under what circumstances is the product of two irrational numbers rational? For example, multiplying ...
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59 views

Question about the proof that $K(u)$ is isomorphic to $K(x)$ if u is transcendental over $K$.

I have a few questions about a proof of the statement that if $u$ is transcendental over $K$ then $K(u)\cong K(x)$. My questions are marked as red and stated below the proof. The proof goes like ...
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1answer
70 views

Number made from the first digits of $2^n$

Consider the number c made from the first digits of $2^n$. To be more precise, the n-th decimal digit of c is the first digit of $2^n$. The first digits from c are : ...
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96 views

Transcendental Basis

Can you say that because $\pi$ is transcendental, that a basis of $\{\pi, \pi^2, \pi^3, \dots\}$ in the rational numbers $\mathbb{Q}$ spans the entire real numbers? It seems likely, although I can't ...
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153 views

Extend a rational number field $\mathbb{Q}$ by using a transcendental number?

Here denoting a set of real transcendental numbers $\mathbb{T}$, what can we then say about the structure $$ \mathbb{Q}(t) = \left\{\, \sum_{k=0}^{+ \infty} a_k t^k\mathrel{}\middle|\mathrel{} a_k \in ...
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Can you make some sort of structure out of transcendental numbers?

Let $T$ be the set of trancendental numbers over $\Bbb{Q}$. Then it is an easy proof that for all $a,b \in T$, either $a + b$ or $ab$ or both are transcendental. What if you defined the operation ...
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121 views

Algebraic independence

Let $a_1,\dots,a_n$ be transcendental numbers. If the set $\{a_1,\dots,a_n\}$ is algebraically independent over $\mathbb{Q}$, then so is the set $\{a_1,\dots,a_n,1\}$?
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131 views

Does the Thue-Morse sequence form a Sturmian Word?

Does the Thue-Morse sequence form a Sturmian Word? The Thue-Morse sequence 011010011001001..., formed by appending the negation of the existing string, yields the ...
2
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1answer
178 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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155 views

algebraic dependence over Q

Are numbers $\sqrt{2}$ and $e$ algebraically dependent over $\mathbb{Q}$? If yes, they belong to the same Mahler class. However, $\sqrt{2}$ is A-number, while $e$ is S-number. On the other hand, if ...
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Is $\{\tan(x) : x\in \mathbb{Q}\}$ a group under addition?

A student asked me the following today : Is $S:= \{\tan(x) : x\in \mathbb{Q}\}$ a group under addition? I am quite perplexed by it. Clearly, the only non-trivial part is to check For any $x, ...
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216 views

Proof that ${\pi}$ can(not) be expressed as a root or as a root in combination with a fraction

I was doing some math for a programming project of myself and ran into decimal numbers and how to define them without losing precision while calculating an expression, so I tried writing them down as ...
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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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119 views

Where is the least upper bound property used in transcendence proofs?

The second-order theory of real numbers is what you get when you take the axioms for ordered fields and add one more axiom, the least upper bond property, also known as Dedekind completeness: that ...
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Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: $$c_{\small d}x^{\small ...
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Proving that $\frac{\pi}{4}$$=1-\frac{\eta(1)}{2}+\frac{\eta(2)}{4}-\frac{\eta(3)}{8}+\cdots$

After some calculations with WolframAlfa, it seems that $$ \frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}} $$ Where $$ \eta(n)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k^{n}} $$ is the ...
9
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1answer
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On Bailey and Crandall's sum for $\sum_{n=0}^\infty \frac{1}{5^{5n}}\left(\frac{5}{5n+2}+\frac{1}{5n+3}\right)$

On page 20 of "On the Random Character of Fundamental Constant Expansions", Bailey and Crandall gave the rather unusual sum, $$u_2 = \sum_{n=0}^\infty ...
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Given $\log(p(x)) = q(x)$ are $p$ and $q$ algebraically independent?

Since $e^x$ and $\log y$ are transcendental functions, does $$\log p(x) = q(x)$$ mean that polynomials $p$ and $q$ (of finite degree $n$ and $m$ respectively) are algebraically independent? What ...
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154 views

How find $\Gamma{\left(\frac{8}{9}\right)}=\frac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$

show that $$\Gamma{\left(\dfrac{8}{9}\right)}=\dfrac{9-\sqrt{14}+\sqrt{75-32\sqrt{3}}}{33}\cdot\sqrt[4]{182}$$ where Gamma function:http://en.wikipedia.org/wiki/Gamma_function I found this problem is ...
11
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384 views

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

Does someone know a proof that $\{1,\pi,{\pi}^2\}$ is linearly independent over $\mathbb{Q}$ ? The proof should not use that $\pi$ is transcendental. $\{1,e,e^2,e^3\}$ is linearly independent over ...
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628 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 ...