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 $e$.

learn more… | top users | synonyms

6
votes
0answers
144 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?
0
votes
2answers
92 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 ...
1
vote
1answer
107 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.
2
votes
2answers
166 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 ...
1
vote
4answers
204 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 ...
3
votes
0answers
103 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 ...
1
vote
0answers
48 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
votes
0answers
123 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
votes
1answer
70 views

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?
4
votes
1answer
132 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 ...
2
votes
1answer
55 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 ...
5
votes
1answer
69 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 : ...
4
votes
1answer
90 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 ...
2
votes
2answers
136 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 ...
1
vote
0answers
77 views

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 ...
0
votes
1answer
98 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 $\{a_1,\dots,a_n,1\}$?
2
votes
1answer
118 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
votes
1answer
162 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 ...
0
votes
1answer
144 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 ...
22
votes
2answers
513 views

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, ...
4
votes
1answer
191 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 ...
8
votes
0answers
123 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 ...
2
votes
1answer
100 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 ...
5
votes
3answers
855 views

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 ...
20
votes
2answers
982 views

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
votes
1answer
123 views

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 ...
0
votes
1answer
72 views

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 ...
3
votes
2answers
151 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´╝Ühttp://en.wikipedia.org/wiki/Gamma_function I found this problem is ...
11
votes
1answer
374 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 ...
26
votes
2answers
531 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 ...
1
vote
2answers
51 views

Are elements of the range of a transcendental function themselves transcendental, excepting a “few” special cases?

Let $f(x)$ be a transcendental function with $x\in\mathbb{C}$. Then are the values $f(x)$ themselves transcendental, except perhaps for a "few" exceptions? For example, it is known that $f(x)=e^x$ is ...
1
vote
1answer
203 views

Is $a^b$ transcendental when $a$ and $b$ are?

I am being asked this question as an exercise in Garling's "A course in Galois theory". But isn't this an open question in math?
7
votes
1answer
195 views

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Transcendental?

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Clearly this series is convergent (compare to geometric series with ratio 1/2). I'm sure it's irrational since a rational number written in base 2 ...
4
votes
0answers
166 views

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.
1
vote
0answers
108 views

Gelfond-Schneider Constant $2^{\sqrt{2}}$

Someone knows a proof (books , articles) that $2^{\sqrt{2}}$ is irrational ? Without using that $2^{\sqrt{2}}$ is transcendent. Any hints would be appreciated.
41
votes
11answers
2k views

What if $\pi$ was an algebraic number? (significance of algebraic numbers)

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours? ...
2
votes
5answers
338 views

The origin of $\pi$

How was $\pi$ originally found? Was it originally found using the ratio of the circumference to diameter of a circle of was it found using trigonometric functions? I am trying to find a way to find ...
1
vote
3answers
188 views

Is there a proof that says that an operation that can take a transcendental number and make it an integer cannot exist?

Motivation: To get an integer to become a different integer, you have to add or subtract another integer, e.g. $1+2=3$ To get a rational number to become an integer, you have to multiply by the ...
3
votes
0answers
70 views

Please help me understand this proof that $e$ is transcendental

This started with my question "Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?". Kunnysan suggested that I model a proof on the standard proof that $e$ is transcendental. I ...
16
votes
0answers
248 views

Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?

This question is inspired by my answer to the question "How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?". The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer ...
2
votes
1answer
389 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
votes
2answers
92 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 ...
0
votes
1answer
74 views

number of the form $\frac{a_0+a_1\pi +\text{…}+a_n\pi ^n}{b_0+b_1\pi +\text{…}+b_m\pi ^m}$ [closed]

all numbers of the form $$\begin{align*}\frac{a_0+a_1\pi +\text{...}+a_n\pi ^n}{b_0+b_1\pi +\text{...}+b_m\pi ^m}\end{align*}$$ form an number field. $n,m$ are arbitrary non-negative itegers, ...
3
votes
1answer
366 views

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using a theorem.

Prove that ${\sqrt2}^{\sqrt2}$ is an irrational number without using Gelfond-Schneider's theorem. I'm interested in this problem because I knew that ${\sqrt2}^{\sqrt2}$ is a transcendental number ...
18
votes
1answer
661 views

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 $\pi$ ...
1
vote
1answer
137 views

$\pi$ and $\ln4$ relations. Even and Odd alternating sums.

Tonight, playing around on WolframAlpha, I discovered that the alternating sum of the odd numbers is $\frac\pi4$ and the alternating sum of the even numbers is $\frac{\ln4}4$ Are there any known ...
0
votes
1answer
82 views

Transcendental proofs vs. Irrational proofs

Why are proofs of the transcendence of certain numbers usually harder than irrationality proofs of those same numbers (for example, Lindemann's proof of the transcendence of pi vs. Niven's proof of ...
97
votes
1answer
20k views

Can $x^{x^{x^x}}$ be a rational number?

If $x$ is a positive rational number, but not an integer, then can $x^{x^{x^x}}$ be a rational number ? We can prove that if $x$ is a positive rational number but not an integer, then $x^x$ can ...
12
votes
7answers
472 views

Numbers with no finite representation on paper

It occurred to me that there must be a lot of numbers without any form of finite representation on paper. Is there a name for these numbers? For example... Integers and rationals have a very simple ...