Questions tagged [transcendental-numbers]

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 $e$.

Let $\mathbb{A}\subset \mathbb{C}$ be the set of algebraic numbers: $$ \mathbb{A} = \{ z\in \mathbb{C}: p(z)=0, p(z)\in \mathbb{Z}[x] \} $$The set of transcendental numbers is $\mathbb{A}^c$. The first construction of a transcendental number, $\sum_{n=1}^{\infty} 10^{-n!}$, was due to Liouville in 1851.

If we fix a degree of polynomials in $\mathbb{Z}[x]$, there are only countably many such polynomials, each with finitely many roots. Then $\mathbb{A}$ is countable, whence $\mathbb{A}^c$ is uncountable. Put crudely, 'most' complex numbers are transcendental, though showing a particular number is transcendental is usually rather difficult.

All transcendental numbers have an irrationality measure of at least 2; see for more information.

748 questions
48
votes
1 answer

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 $x, y\in \mathbb{Q}$, does there exist $z \in \mathbb{Q}$…
39
votes
1 answer

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 process of some quite long calculations that are…
37
votes
1 answer

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 $\pi$ ? For example does there exists algebraic numbers…
jimjim
  • 9,517
  • 6
  • 34
  • 81
33
votes
0 answers

Smallest Subset of $\mathbb{R}_{>0}$ Closed under Typical Operations

Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$ Questions: 1. Do either $S$ or its elements have an…
32
votes
0 answers

Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

This has now been cross posted to MO. Let $F$ be a subset of $\mathbb{R}$ and let $S_F$ denote the set of values which satisfy some generalized polynomial whose exponents and coefficients are drawn from $F$. That is, we let $S_F$ denote $$\bigg…
Mason
  • 3,439
  • 1
  • 13
  • 39
30
votes
4 answers

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 independent over $\mathbb{Q}$ Any hints would be…
27
votes
1 answer

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)
Hooman
  • 3,517
  • 2
  • 21
  • 35
27
votes
2 answers

Do the Liouville Numbers form a field?

The Liouville numbers are those which are better-than-polynomially approximated by rationals. More precisely, we say $x\in\mathbb{R}$ is Liouville when for all $n\in\mathbb{N}$ there is a $\tfrac pq\in\mathbb{Q}$…
Oscar Cunningham
  • 15,389
  • 3
  • 43
  • 77
25
votes
2 answers

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 Dirichlet Eta function. Could it be proved that…
25
votes
5 answers

Every Real number is expressible in terms of differences of two transcendentals

Is it true that for every real number $x$ there exist transcendental numbers $\alpha$ and $\beta$ such that $x=\alpha-\beta$? (it is true if $x$ is an algebraic number).
M.H.Hooshmand
  • 2,163
  • 12
  • 24
24
votes
1 answer

Is $\sin(e)$ rational or irrational?

We know that $\pi$ and $e$ are transcendental numbers. Here $\sin(x)$ is a real trigonometric function. We know that $\sin(\pi)=0$ which is rational. Now I am wondering to know that whether $\sin(e)$ is rational or irrational. In addition, if it is…
24
votes
1 answer

Are all transcendental numbers a zero of a power series?

So I came across the concept of extending the notion of irrationality to higher degree polynomials. The base case of this is standard irrationality. That is, a number is irrational if it cannot be expressed as the ratio of two integers. This is…
22
votes
4 answers

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 numbers (given a transcendental number) closed under…
Adam
  • 1,948
  • 5
  • 20
  • 25
22
votes
2 answers

What would change in mathematics if we knew $\pi+e$ is rational?

It is well known that there's no conclusion now whether $\pi+e$ is rational or not. What would happen if we knew that $\pi+e$ is rational? Specifically, are there related open problems that would be settled?
zhangwfjh
  • 1,565
  • 1
  • 9
  • 24
22
votes
2 answers

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?
1
2
3
49 50