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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 $e$.

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Proof that $(n+1)m!>(m+1)!−1$ does not hold for $m>n$

I am currently doing a project that involves some work on Liouville's theorem for transcendental numbers and Liouville's constant. I have found a proof that Liouville's constant is transcendental ...
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Let $K$ be a field. Prove that every element in $K(x)\backslash K$ is transcendental

Let $K$ be a field. Prove that every element in $K(x)\backslash K$ is transcendental over $K$. Is proof of the question above similar to that of the question below?
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Can an algebraic function contain transcendental constants?

Most definitions I've seen [1] [2] are agnostic about what kind of polynomials the function must satisfy. PlanetMath, for example, says A function of one variable is said to be algebraic if it ...
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How do you turn an irrational, non-transcendental number, like 1.618… back to its form of (a + sqrt(b))/c.

Looking at irrational numbers, I had an idea, as to computing square roots. Take the golden ratio. Numerically, it's 1.618.... but I can also write it like this: $\frac{1+ \sqrt{5}}{2}$ I want to ...
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Constructing an infinite chain of subfields of 'hyper' algebraic numbers?

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 $...
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Representation of Transcendental number via continued fractions

My question is quite simple. As far as I know, rational numbers has finite continued fraction representation. And if for any number we have infinite continued fraction, then it necessarily is ...
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Real irrational algebraic numbers “never repeat”

An oft-used phrase describing irrational numbers is that their (decimal) expansions "never repeat". The sense of "never repeating" intended is, of course, that their expansions don't repeat forever. ...
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unknown sequences with sum of $e$ [closed]

Starting with an transcendental number like $e$, it is known that we can write it as a sum of infinitely many rational numbers of the form : $e= \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {...
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Proving that a number is transcendental

Let $(x_n)_{n> 0}$ a sequence of $\{ 0,1 \}$ and $$ x=\sum_{n=1}^{\infty}\frac{x_n}{10^n}. $$ Prove that if $x$ is irrational then $x$ is transcendental. I tried to first start by proving that $...
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Unique solution of $x = \cos\left(\frac{x}{2}\right)$

How does one show that there is a unique solution to this equation? $$x = \cos\left(\frac{x}{2}\right)$$ Furthermore how can we find it?
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Proving an identity for $\sum_{m,n\in\mathbb{Z}_{>0}}\frac{\gcd(m,n)^r}{m^sn^t}$.

My task is to prove the well-known identity Here all variables are positive integers I only know I should use mobius inversion formula, but how to proceed I am getting confusion, please any one ...
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Is the definition of a transcendental field extension necessarily impredicative?

Let $t$ be transcendental over $K$, i.e. there is no $p(x) \in K[x]$ s.t. $p(t)=0$. Can you define the smallest field containing $K$ and $t$ in a predicative way? If not, how would you prove that ...
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proving transcendental numbers are irrational

I don't understand how every transcendental number is irrational, is there a way to prove that? I know it just means it's a non-algebraic number, but how does that correlate to irrationality?
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Real valued function which is continuous only on transcendental numbers

First of all, I am sorry for asking this question. We know that $R$ is uncountable. And also the set of all transcendental numbers is uncountable. How can I construct a function $f(x)$ on $R$ which ...
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Proof that e is transcendental in Herstein's Topics in Algebra (1st ed)

In the proof of Theorem 5.F. page 177. page 176 page 177 From the constructed $F(x)$ from $f(x),$ how can we choose $$f(x)=\frac{1}{(p-1)!}x^{p-1}(1-x)^{p}(2-x)^{p}\cdots(n-x)^{p}$$ where $p>n$ ...
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How many values of $\theta$ give $cos(\theta)$ is algebraic

I recently saw Lindemann's proof that $\pi$ is transcendental by using the fact that $e^{i\pi} = -1$, and this made me realize that the Lindemann-Weierstrass theorem implies that the $\cos$ , $\sin$ ...
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Is $(\ln 2)^2$ transcendental?

Wolfram says $(\ln 2)^2$ is transcendental. I think it says numbers of the form $(\ln a)^b$ are all transcendental, at least for integer $a$ and $b$, I didn't check further. Maybe there is some ...
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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 $\sin(e)$ ...
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Transcendence of $x^x = p \in \mathbb{P}$

I'm going straight for the question, so: let $x^x = p \in \mathbb{P}$, then $x$ is irrational. The proof is obvious, $p \neq a^a$ for some integer $a$, so $x^x = (\frac{a}{b})^{\frac{a}{b}}$, suppose ...
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Solving $e^{t/\!\ln(x)}=x$ for $x$

How do you solve the equation: $e^{t/\!\ln(x)}=x$ for $x$? Here $t=1,2,3,\dotsc$ This is what I did: $\ln(e^{t/\!\ln(x)})=\ln(x), $ $t/\!\ln(x)=\ln(x),$ $t=(\ln(x))^2,$ $x=e^\sqrt{t},e^{-\sqrt{t}}...
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Importance of the Transcendence of $\pi$ and $e$ [closed]

Why do people care that $\pi$ and $e$ are transcendental?
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Defining number set sequences based on the fundamental theorem of arithmetic

The Fundamental theorem of arithmetic gives a natural way to write natural numbers $\mathbb{N}$ uniquely in terms of products in powers of primes: $$N = 2^{n_2} 3^{n_3} 5^{n_5} \ldots $$ for all $N \...
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Prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancendental over $\Bbb{Q}$.

We want to prove that $\alpha:=\sqrt{\pi}+\sqrt 2 \in \Bbb{C} $ is trancedental over $\Bbb{Q}$. Attempt. We use proof by contradiction and so assume that $\alpha \in \Bbb{C}$ is algebraic over $\...
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Are random numbers transcendental?

Do I see it right that a randomly chosen number $a$ of the form $ a = 0.abc...xyz$ with random digits $a,b,c,x,y,z$ an approximation of a transcendental real number is? It is an approximation of a ...
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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 ...
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If the sum of a sequence of positive rational numbers is transcendental, is the sum of every subsequence that is irrational, also transcendental?

Assume we have a sequence of positive rational numbers $(a_n)$, and $\sum_{n=1}^\infty a_n = x$ and $x$ is transcendental. If we have a subsequence of $(a_n)$, $(b_n)$ and $\sum_{n=1}^\infty b_n = y$ ...
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Is uncountable transcendental-additions enough to make $\mathbb{Q}$ into $\mathbb{R}$?

Consider $\mathbb{Q}$ and then consider "adding" a transcendental $\zeta$ to it, while still retaining the field axioms (i.e. $\mathbb{Q}(\zeta)$). We could add another transcendental in an obvious ...
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Are there any theorem relating transcendency/irrationality of Euler's constant and the Riemann hypothesis?

Several formulations of the Riemann hypothesis can be given in terms of Euler’s constant. For example, Theorem (Nicolas 1981) The Riemann hypothesis holds if and only if all the primorial numbers $...
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How to obtain the degree of this field extension?

Let $K/F$ be a field extension, $t,w \in K$ such that $t$ is transcendental over $F$ and $w$ is transcendental over $F(t)$. Prove that for any positive integer numbers $n$ and $m$ the equality $[F(t,w)...
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Is there a distinguishable characteristic between the summation / continuous fraction method in algebraic and transcendental numbers?

I try to illustrate the formation and derivation of algebraic numbers and transcendental numbers. I found that both categories of numbers can be made by continuous summation or division/fraction ...
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Proof of rational + algebraic is algebraic and rational + transcendental is transcendental?

I've heard of and seen some proofs that the product and sum of two algebraic numbers is algebraic, however many of them are quite complex and require a variety of machinery (from matrix eigenvalues to ...
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Verification and rigorousness of my own proof of transcendence of Liouville's number.

Theorem: Liouville's number: $$L=\sum_{i=0}^\infty 10^{-i!}$$ is transcendental. Proof: Denote $$L_k=\sum_{i=0}^{k!} 10^{-i!}$$ We have: $$L_1=0.1$$ $$L_2=0.11$$ $$L_3=0.110001$$ $$L_4=0....
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What is the usefulness of classification into Transcedental and Algebraic numbers?

Classification into rationals and irrationals makes complete sense because irrational numbers seem to be completely different from rational numbers, which are terminating or repeating. We know that ...
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Is this :$\int_{-1}^{1}\exp (-\tanh^{-1} (x))dx=\pi$ a well known integral representation for $\pi $?

I have tried to get interesting integral for interesting transcendental number , I have got the following :$\int_{-1}^{1}\exp (-\tanh^{-1} (x))dx=\pi$ , really I didn't accross that integral in the ...
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Is this class of series all demonstrably transcendental?

Question: For a vector with integer entries $[a_0, a_1, \dots, a_{k-1}]$ is it true that when $\sum_{n=1}^\infty{\frac{a_{n-1 \mod k}}{n}}$ is not divergent it limits to some transcendental number ...
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How to calculate $e^\pi$ by hand? [closed]

There are some ways to find the values of the form "rational power of rational number" what about nonrecurring numbers? Is there any way to calculate that?
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Why mathematicians attempt to find more digits for $\pi$ since it has no last digit?

Many works and many research about irrationality of $\pi$ appear every year for predicting more digits for the tanscendental number which is $\pi$, The question that made me confused is :Why ...
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Interleaving of digits of $e$ and $\pi$ [closed]

What happens if we form a number by taking every other digit of, say, $\pi$? Has this type of construction been studied? Also, what if we interleave the digits of $e$ and $\pi$?
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What is the solution to $x! = i$?

I wanted to know what $x$ is in this equation: $x! = i$. I tried using normal calculators which just gave a math error, tried searching google which was pretty useless, zero results, and last but not ...
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Transcendental continued fraction

Can anyone here show that the continued fractions $$[2^1,2^2,2^3,...]$$ and $$[1!,2!,3!,...]$$ are transcendental. I tried using Liouville's criterion plus some basic inequalities but got nowhere. Any ...
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Transcendental numbers

Can transcendental numbers be plotted? Also, can a computer recognize a transcendental number? I mean, for example, a computer, while computing, understand that the number it is computing is not ...
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Transcendental Equation. [duplicate]

I was able to solve $ \ln x = - x $ using Lambert's function. I was wondering how does one solve $ \ln x = x$ Does the solution even exist for this equation?
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Are there transcendental numbers that cannot be reached? [closed]

This is a hard question to ask. But I've been contemplating transcendental numbers. I know that there are infinitely many; simply multiply a known transcendental (like pi) by every rational number. So ...
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Irrational and transcendental numbers of the form $\sum_{n=1}^\infty\frac{\operatorname{rad}(n)^\alpha}{s_n}$: proof verification and examples

We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,\tag{1}$$ with the definition $\operatorname{rad}(1)=1$. You can see ...
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What about of $\sum_{n=1}^\infty 1/10^{N_n}$ as transcendental number?

In this post we denote the primorial of order $n$ as $N_n=\prod_{k=1}^n p_k,$ where thus $p_k$ denotes the $k$th prime number. This section of the Wikipedia's article dedicated to transcendental ...
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Finite formula for behaviour of infinite continued fractions

I was surprised to find out that the continued fraction expansion of $e$ is $[2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10,...]$. Even though $e$ is transcendental, this expansion is very nice. We ...
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Can We prove Transcendence of $e$ if we know that $e^{m}$ is Irrational?

So I just managed to prove that $e^{m}$ where $m$ is an integer is irrational. I was wondering If I can prove using this fact that $e$ is Transcendental. The only thing I came up with was that if we ...
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Rationality of circumference of an ellipsis with rational semi-axes

We all know that the ratio of circumference of a circle to the radius is a transcendental number, but how about ellipses? It is well known that the circumference of an ellipse with semi-axes lengths $...
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Show that $[k(t): k(t^4 + t) ] = 4$

Let $k$ be the field with $4$ elements, $t$ a transcendental over $k$, $F = k(t^4 + t)$ and $K = k(t).$ Show that $[K : F] = 4.$ I think I have to use the following theorem, but I'm not quite putting ...
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density of $\{\sin(x^n)|n\in\mathbb{N}\}$ for $x>1$

While reading other topics, e,g, Is $n \sin n$ dense on the real line? or Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $[-1,1]$ for every natural number $m$?, the following problem appeared in ...