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Topic: Why do we need those real non-constructible numbers?
Replies: 68   Last Post: Dec 11, 2017 1:42 AM

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 bassam king karzeddin Posts: 2,182 Registered: 8/22/16
Re: Why do we need those real non-constructible numbers?
Posted: Nov 9, 2017 9:42 AM

On Thursday, November 9, 2017 at 12:01:55 PM UTC+3, WM wrote:
> Am Donnerstag, 9. November 2017 08:38:59 UTC+1 schrieb bassam king karzeddin:
> > whereas the absolute (Pi) exists exactly in the perfect circle that doesn't exist in any imaginable reality
>
> It does not exist in reality. But it exists in imagination. And its circumference can be approximated as closely as we like (in ideal mathematics - not in reality).

Yes and exactly as an endless approximation (generally in rational numbers) to something assumed existing from an old naive definition of the need for calculating the circle's area
>
> To accept it as a number or value or lenght is questionable. But a rather convincing argument is this: If it exists on the real line, then it must be a point, because if only adding any positive eps like 10^-1000000000000 gives a number that provably is not the limit of so many sequences (Vieta, Wallis, Gregory-Leibniz, Euler, Ramanujan, Borwein,... see for instance GU03.PPT in https://www.hs-augsburg.de/~mueckenh/GU*/)

But epsilon > 0, thus a distance, not a sizeless point, and 10^{n} or 10^{-n} both existing rational numbers no matter if (n) is positive integer with arbitrary sequence of digits (say in 10base number) that can fill say only seven galaxy size, where every trillion of sequence digits can be stored say only in one (mm) cube and regardless what are those many mentioned big names say or describe it

so, if the Lim(1/n) = 0, when n tends to Infinity, implies strictly that

(1/n) > 0, when n tends to infinity

And the limit of a series or a function generally doesn't exist since its absolute convergence is actually a ratio of two diverging integers that are impossible to exist in any imaginable reality but making epsilon relatively small is actually for engineering problem solving and never perfect mathematics
> >
> > Had the Greek recognized this simplest fact, then never they would have raised their three famous impossible constructions for sure

>
> These constructions are not impossible. They are only impossible with the tools allowed by Plato. (See for instance the Quadratrix, loc cit p. 40)
>
> Regards

They are indeed impossible constructions by all means and all the alleged known construction methods such as (Origami, Dedekind cuts, paper folding, Newton's approximation, intermediate theorems, ... etc) are actually well-exposed and cheating methods that are even far less accurate than eye marking or using direct protector or far less than any numerical approximations for the many critiques they have, however they may succeed in constructing a constructible number or constructible angle only which isn't any new additions

Famous cheating construction: the modern maths had claimed to construct exactly a regular pentagon by using non-existing real and imaginary numbers as (pi, e, i, -1, 0), whereas Euclide had constructed it EXACTLY thousands of years back by real physical geometry and without using all that cheating nonsense fictional real NUMBERS

The Wolfram-Alpha shamelessly claim (up to date) to construct EXACTLY any regular polygon by solving a polynomial (x^n + 1 = 0), whereas it is impossible geometrical constructions for many n, where (n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, ...), since those angles must correspond to non-constructible numbers, hence non-existing and fictional angles too

However, I wrote few topics with so simple proofs in my posts about those unreal numbers or unreal angles too

Regards
Bassam King Karzeddin
Nov. 9th, 2017

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