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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Spigot algorithms for transcendental numbers

I'm trying to write a program that will compute digits of transcendental numbers using a spigot algorithm. While researching, I found the BBP Formula, and a Compendium of BBP-Type Formulas, alas, I ...
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41 views

Algebraic values of sine function

Are there algebraic inputs to the sine function that produce algebraic outputs? Other than zero, that is? This is assuming the sine function in radians.
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Is there an explicit irrational number which is not known to be either algebraic or transcendental?

There are many numbers which are not able to be classified as being rational, algebraic irrational, or transcendental. Is there an explicit number which is known to be irrational but not known to be ...
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relation betwn ln and e

If $f(x) = ln(x)$ and $f^-1(x) = e^x$ then is $e^x = 1/ln(x)$??? because I see $e^9 = 8103$ but $1/ln9 = .455$ How are they reverse? I don't understand!
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Is this a well defined problem in terms of Euclidean Geometry?

I am trying to construct an example of a geometric problem, stated in terms of Euclidean Geometry, that is not Machine Provable (or in an equivalent definition Automatically Provable)-i.e no computer ...
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Why do we care about the Champernowne constant?

I was browsing code golf and I came across this challenge: http://codegolf.stackexchange.com/questions/68685/the-rien-number It caught my interest and I wanted to learn a bit more about the ...
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If $\theta$ is a rational number, is $e^{i\pi\theta}$ algebraic?

I want to know if $\theta$ is a rational number, is $e^{i\pi\theta}$ an algebraic number or not? For the first step I tried to write it $(e^{i\pi})^\theta$, that equals $(-1)^\theta$, but I think ...
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Where are the transcendental numbers?

This question is motivated from an exercise from Rudin. The exercise says that prove that set of all algebraic numbers is countable. Proof: We know that a number $z$ is called algebraic if it is the ...
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Values of Eisenstein Series

I'm trying to prove the algebraic independence of $\pi, e^\pi$ and $ \Gamma(1/4)$ while using Nesterenko's Theorem ($\{q, P(q), Q(q), R(q)\}$ contains at least three algebraically independent numbers ...
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1answer
38 views

Is there an easy enough way to show that between two algebraic numbers there is an infinite number of transcendental numbers?

We know that between two different rational numbers there is an infinite number of irrational numbers and that between two different irrational numbers there is an infinite number of rational numbers. ...
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60 views

Do transcendental numbers contain any string of digits?

It is often said that $\pi$ contains any string of digits. But does the property "transcendental" imply "contains any string of digits?
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In simple English, what does it mean to be transcendental?

From Wikipedia A transcendental number is a real or complex number that is not algebraic A transcendental function is an analytic function that does not satisfy a polynomial equation However these ...
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Finite powers of fixed transcendentals

I was thinking about the complex unit circle $S_1\subset\mathbb{C}$ as a group under multiplication and how, if possible, an element $z$ of infinite order could generate the circle itself. A friend ...
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114 views

Can I construct a line segment with the length $e$ or $\pi$? [closed]

What I really mean is that without restriction(only circle and ruler),can we construct it with geometric method or something else. If we can or not,how or why?I am just interested in this ...
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59 views

How to prove the following formula is a constant?

For any $\rho\in\mathbb{R}^+$, prove that the following formula equals a constant: $$\dfrac{1}{\rho^2}{\int_{-\rho}^\rho x^2 e^{\left(\tfrac{\rho^2}{x^2-\rho^2}\right)}dx}\left({\int_{-\rho}^\rho ...
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71 views

Why is Gelfond's constant transcendental?

I have seen a proof of $\pi$ being transcendental by conclude that transcendental number powered by algebraic number must be transcendental and algebraic number powered by algebraic number must be ...
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1answer
105 views

Liouville numbers and continued fractions

First, let me summarize continued fractions and Liouville numbers. Continued fractions. We can represent each irrational number as a (simple) continued fraction by $$[a_0;a_1,a_2,\cdots\ ...
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2answers
74 views

Convert PI to base 4. Does my unique human genome exist in the sequence of digits?

The human genome consists of sequences of BASE Pairs A G C T Convert the number PI to base 4. Does my unique human genome exist in the sequence of digits?
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Something similar to the bizarre Koide formula?

In 1981, Koide found the empirical relation, $$\frac{m_e+m_\mu+m_\tau}{\big(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau}\big)^2} = 0.666659\dots\approx \frac{2}{3}\tag1$$ where $m$ are the masses of the ...
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190 views

Is $0.248163264128…$ a transcendental number?

My question is in the title: Is $a=0.248163264128…$ a transcendental number? The number $a$ is defined by concatenating the powers of $2$ (in base $10$). It is possible to express $a$ as a ...
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Swapping the digits of an algebraic number (e.g. $\sqrt 2$)

Let an algebraic number, say $ a=\sqrt 2 = 1.41421356237309504880...$, and define $$b=f(a)=1.14243165323790058408...$$ by swapping the digits $a_{2i+1}$ and $a_{2i+2}$ for $i≥0$, corresponding to ...
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Proof of irrationality of $\zeta(2)$ without explicitly calculating it

Question is pretty much the title. It is pretty easy to show that $\zeta(2n)$ is irrational for all $n$ once you know that $\zeta(2n)$ is a rational multiple of $\pi^{2n}$ (and then also use the fact ...
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100 views

Strange Algebraic Number

We call a number algebraic if and only if it is the solution of a polynomial with integer coefficients. A number (complex or real) is transcendental if and only if it is not algebraic. A while back ...
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106 views

Is $\frac{1}{2^{2^{0}}}+\frac{1}{2^{2^{1}}}+\frac{1}{2^{2^{2}}}+\frac{1}{2^{2^{3}}}+…$ algebraic or transcendental?

Inspired by this question, the series $\dfrac{1}{2^{2^{0}}}+\dfrac{1}{2^{2^{1}}}+\dfrac{1}{2^{2^{2}}}+\dfrac{1}{2^{2^{3}}}+\dots$ is clearly irrational. But is it algebraic or transcendental? I ...
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75 views

Logarithm of a transcendental number

Can anything be said about the nature of the number $\log y $ where $y $ is a transcendental number not of the form $y=e^x $ or written trivially in that form using $x=\log w $ for some $w $ ...
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Is $(-2)^{\sqrt{2}}$ a real number?

Is $(-2)^{\sqrt{2}}$ a real number? Clarification: Is there some reason why $(-2)^{\sqrt{2}}$ is not a real number because it doesn't make sense why it shouldn't be a real number. Mathematically ...
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35 views

Integration/Fundamental Calculus/Transcendental Numbers

Given $\int\sqrt{1 + \frac{-x}{\sqrt{4-x^2}}} dx$, how would this be integrated? On a universally-calculus side of things, do all functions have integrals? And is it easier to evaluate definite ...
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1answer
51 views

Is the sum of transcendental and algebraic number transcendental number?

I know almost nothing about transcendental numbers, I know the definition of them and maybe few results about them and that is all. But the question in the title somehow naturally arises when ...
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162 views

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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29 views

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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28 views

Sum of algebraicly independent transcendentals is transcendental?

We say the transcendental numbers $\xi_1,...,\xi_n$ are algebraicly independent if an algebraic combination of them satisfies: $$ \sum_{1\leq i\leq n}\psi_i\xi_i= 0 \iff \psi_i=0, \forall i=1,...,n. ...
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How to prove $1/ \log a + 1/ \log b$ for rational $a$ and $b$ is a transcendental number?

I know how to prove $\log a$ for rational $a$ is transcendental, because if it were algebraic it would imply $e$ is algebraic as well (namely if $\log a = b, e = a^{1/b}$), and I can prove $\log a + ...
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Number with non-constructive algebraicity proof

Is there a computable number which is known to be algebraic but no explicit polynomial of which it is a root is known?
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43 views

Is every normal number transcendental? [duplicate]

This question is related to question http://math.stackexchange.com/q/197507 but not quite the same (or is it?): is every normal number transcendental? Can it be proved or is there a counterexample to ...
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33 views

Is this number a Liouville number?

Suppose I have a binary constant $q = 0.1010000000000000000000000000000000001001..._2$. In base 10 this number is $q $~$ .6250000000077325..$ and is defined as $$q = \sum_{\rho}^{\infty} ...
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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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118 views

How to prove $e^{1/e}$ is irrational?

How do we prove $e^{\frac{1}{e}}$ is irrational ? Also how do we show it is transcendental ? The number $\eta = \exp(\exp(-1))$ occurs naturally in the context of tetration and power towers. Let ...
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Other Algebraically Independent Transcendentals

I was thinking about a incomplete answer I gave earlier today to a interesting question made by the user lurker. The question was about wheter or not the sum or the product of two transcendental ...
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3answers
68 views

Adding or Multiplying Transcendentals

Is it possible to add or multiply (no subtraction) only positive transcendental numbers and yield a solution that is algebraic? Exponential manipulation is excluded from this question, as $e^{\ln2} = ...
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2answers
173 views

Is there a function whose limit approaches Pi?

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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2answers
85 views

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 $|n|\neq 5$. ...
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486 views

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 ...
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1answer
32 views

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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1answer
40 views

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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3answers
92 views

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) ...
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268 views

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 ...