Transcendental Numbers

[Guest]

What are transcendental numbers exactly? I completely know nothing about this math term.

 

[galactus]

As per tkh's post, I suppose I should elaborate.

Transcendental numbers cannot be expressed as the root of any

algebraic equation with rational coefficients. These are equations

involving simple integers with powers of pi. The

numbers pi (and e) can be expressed as an endless continued fraction or

as the limit of an infinite series. The fraction 355/113

expresses pi accurately to six decimal places.

In 1882, German mathematician F. Lindemann proved that pi is

transcendental, finally putting an end to 2,500 years of speculation. In

effect, he proved that pi transcends the power of algebra to display it in

its totality. It can't be expressed in any finite series of arithmetical or

algebraic operations.

from "Wonders of Numbers" by Clifford Pickover, Oxford University Press.

 

[ChaoticLlama]

Additionally, it is a number that can be generated infinitely without repetition.
ie: \(\displaystyle \L\pi = 3.141592653....\) Will continue forever without the series of decimals repeating themselves.

 

[tkhunny]

\(\displaystyle {\pi}\) is not the root of any polynomial.

\(\displaystyle x - {\pi} = 0\)?

 

[skeeter]

\(\displaystyle {\pi}\) is not the root of any polynomial.

\(\displaystyle x - {\pi} = 0\)?

\(\displaystyle {\pi}\) is not the root of any polynomial with rational coefficients.

 

[stapel]

Additionally, it is a number that can be generated infinitely without repetition.

That's the irrational numbers, like \(\displaystyle \sqrt{2}\). Transcendentals are a subclass of the irrationals.

Eliz.