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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 $e$.

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.

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Do replacing distinct digits from distinct places of an algebraic irrational

Do replacing distinct digits from distinct places of an algebraic irrational number necessarily make it a trancsendendal number? Since my question isn't worded well, therefore I would clarify it by means of a simplest…
Hashir Omer
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Reciproc of the Lindemann theorem and the arc cosine of the golden ratio

Via the Lindemann theorem it is easy to prove that the cosine of any rational multiple of $\pi$ is an algebraic number; however its contrapositive only tells us that the arc cosine of an algebraic number is a transcendental number, not necessarily…
Mateus Araújo
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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?
David Holden
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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 $\sqrt{2}$ by any nonzero rational multiple of…
Steve Pap
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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 $x$ can't be the root of a second degree polynomial…
Tengen
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Applications of the property of transcendence

I was learning the proofs of the transcendence of $e$ and $\pi$, and I was wondering what would be the usefulness of these results. So my questions are: Are there any (relatively simple) application of the transcendence of $e$? More generally, why…
E. Joseph
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How to show $ \tan \alpha $ is transcendental?

I need to show that $ \tan \alpha $ is transcendental for each non-zero algebraic number $ \alpha $. Can any one give me an idea ? Thanks.
Chese Cat
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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.
math1234567
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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 searched for a proof and found this…
marty cohen
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Is this sum a transcendental number?

Question Is the sum $S$ a transcendental number? $$S = 1 + \frac{1}{12}+\frac{1}{123}+\frac{1}{1234}+\cdots$$ 123456789 is followed by 12345678910, 1234567891011 Some Search Results The denominator has a recursive relationship: $$a_{n}=a_{n-1}…
Aster
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Can transcendent degree of transcendent number be algebraic? (meaning can both $\alpha^{\beta}$ and $\beta^{\alpha}$ be algebraic)

Here is my question. Are there $\alpha$ and $\beta$ transcendental numbers such that both $\alpha^{\beta}$ and $\beta^{\alpha}$ are algebraic? There isn't anything specific about roots of the question. (I was just thinking over). Of course $e^{\ln2}…
kolobokish
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Why is $e$ transcendental, even though it can be written as a sum of algebraic numbers?

I am, of course, referring to the Maclaurin series for $e$ evaluated at $1$. Then we have $$e = \sum_{n = 0}^{\infty}\frac{1}{n!} = 1+\frac{1}{2}+\frac{1}{3!}+\frac{1}{4!}...$$Then $e$ is the sum of an infinite series of algebraic numbers, which…
Ty Jensen
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Is $e^{\pi \alpha}$ known to be transcendent for all real algebraic $\alpha$?

The MathWorld article Transcendental Number contains a reference to Yu. V. Nesterenko proof of transcendence of $e^{\pi \sqrt{2}}$. Is there a more general result about transcendence of $e^{\pi \alpha}$ for all real algebraic $\alpha$? I observed…
Vladimir Reshetnikov
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Applications of Gelfond's constant

Gelfond's constant is the transcendental number $e^\pi$ which is approximately equal to: $$ 23.140692632... $$ Are there any useful (real-world) applications of this constant besides its appearance in the Gelfond-Schneider Theorem? Is is "used" in…
Klangen
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Can I construct a line segment with the length $e$ or $\pi$?

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 question.Maybe your explanation will be beyond my ability.
Man Big
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