On Thursday, November 17, 2016 at 3:50:08 PM UTC+2, bassam king karzeddin wrote:
> Even this was explained quite many times but never mind to prove it again and again:
>
> First, the Greek construction of a number by unmarked straightedge and a compass with finite number of steps, with arbitrary shortest distance between two distinct visualized locations on the straight number line defined as a unity, was the base of construction of an existing real number, which implies its true existence, wither it can be measured by rational numbers or not, since the construction of the rational numbers is defined by those definitions also, and was based on the perfection of the ideas behind this elementary principle
>
> So if we deny or disprove the Greek famous principle, then yes no numbers can be defined on a number line except two numbers by convention as you mentioned earlier, where then the concept of a constructing or measuring a number becomes so meaningless
>
> So, depending on the validity of the Greek principle, Sqrt(2) is a real existing number, being abstractly and physically a diagonal of a square of defined unity side, thus it is exactly constructed with unique location on a number line that no rational number can occupy, and was originated from the unity “one”, despite being immeasurable by rational.
>
> Doubling the square in real non zero integers is impossible, (n^2 =/= 2m^2), therefore taking the sqrt of both sides of this eqn. would yield:
> Sqrt(2) =/= n/m, where (n/m) is rational number, NO matter if both (n & m) tends to be infinite integers
>
> So, the rational representation (or the best approximation) of Sqrt(2) is always rational number, no matter if you can fill out the galaxy size of some of its endless digits
> And since Sqrt(2) was proved to be irrational number which is impossible to be rational number, hence it is rational representation in any rational number system is fake number that is impossible to exist, trying always and forever to replace the unique location of Sqrt(2) but always unsuccessfully
>
> Let us see it clearly in our decimal 10base number system representation, as a sequence of endless approximation, for Sqrt(2)
>
> First Sqrt (2) = 1 = 1/(10)^0
> Second = 1.4 = 14/10^1
> Third = 1.41 = 141/10^2
>
> …………. = …………………
>
> Tenth = 1414213562/10^9
>
> …….. = ……………
>
> K’th = n/m, where (m = 10^(k - 1),
> and (n) is integer with (k) number of digits
>
> (k + 1) = n/m, where (m = 10^k), and
> (n) is integer with (k + 1) number of digits
>
> Hence, if we go after exact value, we find this task is impossible to achieve, because we require both of
> (m & n) to be infinite integers with infinite sequence of digits, where this is obviously impossible, and also not permissible in mathematics, (then we better admit the truth of real best approximation), or say this is the magnitude of Sqrt(2) in rational number (which is not an exact number)- as John Gabriel continuously describes for immeasurable numbers
>
> The proof is therefore completed
>
> Regards
> Bassam King Karzeddin
> 17Th,Nov., 2016