On Monday, March 6, 2017 at 8:02:46 PM UTC+2, bassam king karzeddin wrote:
> Any clever student can so easily now refute most of the well established illegal numbers in mathematics being considered as real numbers, even without any study to any higher branches of the so called advanced mathematical tools based on real or complex analysis
>
> Consider the infinite decimal representation of numbers that includes the following sets of illegal numbers
>
> 1) The transcendental numbers as for (pi), or (e)
> 2) The algebraic numbers as for (2^ {1/3}), or more generally (q^ {1/p}), where (p) is odd prime, and (q) is prime number
> 3) The infinite decimal representation for a constructible numbers as for (6/7) or sqrt(2)
>
> All those described above are unreal numbers (for sure)
>
> The easiest proof:
>
> Since all those described numbers above numbers are defined with endless digits (not terminating with infinite sequence of zero digits, then you may notify them with this simple notation in any number system as (N.M), where (N) is finite integer, and (M) is integer with infinite sequence of digits in any base number system, where the decimal notation is the dot (.)
>
> Then consider ONLY the integer (M), after the decimal notation,
>
> With its infinite sequence of digits in any base number system you adopt as denoted by the infinite sequence of digits as
>
> 0.M = 0. (d_1)(d_2)(d_3) … (d_n), where (n) tends to infinity
>
> And for more simplicity for students, consider the infinite representation of the arithmetical cube root of two denoted as (\sqrt[3]{2}), or (2^{1/3}), and in 10base number system for clarity
>
> Of course everybody knows that (2^{1/3} = 1.2599210498…), where actually this is forever as absolute rational approximations that never ends, which implies the non existence of such number (proved rigorously in my recent posts), thus (2^(1/3), and its infinite decimal representation are both nonexistent numbers (fiction numbers in mathematics), they have no proof of being existent but only intuitively or foolishly were concluded, or devilishly established as real numbers for a narrow purposes of showing unnecessary talents of few famous Jugglers from old centuries
>
> But it should be understood that if a carpenter was asked to make a cubic tank that contains two meter cube, then no harm to use any approximation for practical purpose, it is so easy task for a carpenter by trial or error calculation which is the same as the alleged modern mathematics of approximations
>
> Once you analyze those approximations step by step then you must understand the whole concept, and no harm if you go and convince your teacher in mathematics about it, before you become as a victim of those wrong concepts about the real numbers
>
> So, let us do it together, by writing the decimal or fractional representation as a rational number only
>
> First approximation of 2^(1/3) after the decimal notation is (1.2 = 12/10 = 6/5)
> Second approximation of 2^(1/3) after the decimal notation is (1.25 = 125/100 = 5/4)
> Third approximation of 2^(1/3) after the decimal notation is (1.259 = 1259/1000)
> Forth approximation of 2^(1/3) after the decimal notation is (1.2599 = 12599/10000)
> Fifth approximation of 2^(1/3) after the decimal notation is (1.25992 = 125992/10^5)
> Sixth approximation of 2^(1/3) after the decimal notation is (1.259921 = 1259921/10^6)
> Seventh approximation of 2^(1/3) after the decimal notation is (1.2599210/10^7) = Sixth approximation!*
>
> The tenth approximation of 2^(1/3) after the decimal notation is (1.2599210498/10^10)
>
> Then what is the (n) digit approximation, it is simply denoted by a rational number (M/10^n), where (M) is integer with (n + 1) number of sequent digits in 10base number system
>
> But our whole number is at infinity as by the definition, then you require both (n) and (M) to be with infinite sequence of digits, where this is not acceptable nor defined in the holy grail principles of mathematics, and also IMPOSSIBLE to attain (for sure), there for our alleged number is a fool number, that exists in the paradise of the fools and so Big Stupid’s minds (so unfortunately),
>
> Thus our alleged number is fake, nonexistent number (for sure)
>
> But, it is no harm if we rename it as (unreal number) for practical purposes only
>
> Same simple logic applies for any number with infinite sequence of digits as described above
>
> But be careful about the surd form of sqrt(2), that does indeed exist independently from it is fake decimal representation which is good for comparison only
>
> And be careful about (pi) too, which exist in two dimensions only as a closed line surrounding a maximum area, but never on a straight line which is only the number
>
> I posted this also in math.forum, but since they take long time to post it, then I post it here again for the protection of all school students minds
>
> Regards
> Bassam King Karzeddin
> 06th, March, 2017