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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Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$ (Basel problem)

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler and he gave other proofs. I believe many of you…
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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 definitions are arguably rather cryptic to those…
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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 not be rational: Denote…
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Why is it hard to prove whether $\pi+e$ is an irrational number?

From this list I came to know that it is hard to conclude $\pi+e$ is an irrational? Can somebody discuss with reference "Why this is hard ?" Is it still an open problem ? If yes it will be helpful to any student what kind ideas already used but…
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Do numbers get worse than transcendental?

Mathematicians have come up with many ways of classifying how "exotic" some numbers are. For example, the most ordinary numbers are the natural "counting" number, and the next most exotic numbers are zero and the negative integers. Next are the…
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Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?

I thought it was easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational, but found a gap in my proof. Simple finite approximations show the denominator cannot be small, though, strongly suggesting irrationality. However, can it be…
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Is $ 0.112123123412345123456\dots $ algebraic or transcendental?

Let $$x=0.112123123412345123456\dots $$ Since the decimal expansion of $x$ is non-terminating and non-repeating, clearly $x$ is an irrational number. Can it be shown whether $x$ is algebraic or transcendental over $\mathbb{Q}$ ? I think $x$ is…
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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 $k$) came up, and I noticed that $f(1) = e-1$ was…
marty cohen
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Is $6.12345678910111213141516171819202122\ldots$ transcendental?

My son was busily memorizing digits of $\pi$ when he asked if any power of $\pi$ was an integer. I told him: $\pi$ is transcendental, so no non-zero integer power can be an integer. After tiring of memorizing $\pi$, he resolved to discover a new…
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Calculating pi manually

Hypothetically you are put in math jail and the jailer says he will let you out only if you can give him 707 digits of pi. You can have a ream of paper and a couple pens, no computer, books, previous pi memorization or outside help. What is the best…
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How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious what types of techniques are used and just how…
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Proof that a trigonometric function of a rational angle must be non-transcendental

Suppose I was working in radians and took $$\sin \left(\frac{a\pi}{b}\right)$$ where both $a$ and $b$ are integers. Is there a proof that the output of this function cannot be transcendental? Or, conversely, that $$\arcsin \left(t \right)$$ where…
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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? My choice of $\pi$ for this question isn't…
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What are examples of unexpected algebraic numbers of high degree occured in some math problems?

Recently I asked a question about a possible transcendence of the number $\Gamma\left(\frac{1}{5}\right)\Gamma\left(\frac{4}{15}\right)/\left(\Gamma\left(\frac{1}{3}\right)\Gamma\left(\frac{2}{15}\right)\right)$, which, to my big surprise, turned…
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Is $0.1010010001000010000010000001 \ldots$ transcendental?

Does anyone know if this number is algebraic or transcendental, and why? $$\sum\limits_{n = 1}^\infty {10}^{ - n(n + 1)/2} = 0.1010010001000010000010000001 \ldots $$
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