Now, it was several years back, that I discovered a Logical error in current mathematics of representation of Reals. Several years ago I insisted that the Reals have a "suffix". So I was saying some years ago that 1/3 had to be .3333...33(+1/3).
Why did I come to that conclusion?
I came it because the truthful way of writing _______ 3| 1000 = 333+1/3
And any math teacher would find an answer as 333 as incorrect, for it fails to include the remainder of +1/3.
So, the logical reasoning is, why stop with whole numbers of inclusion of a remainder. Why not continue the practice of COMPLETING the DIVISION.
And I was happy to find out that Newton did the same way back in 1687 with his Compleat Quotient.
Unfortunately for mathematics after Newton, we seem to not have any one with enough logic in their mind, to demand rigor to the Reals.
And where we find fruitcakes saying 1/3 = .33333.... where they never Complete the Quotient, never fill in the remainder.
So, several years back, I do not remember exactly where I wrote that 1/3 = .3333....33(+1/3)
This way of writing Reals was sorely needed in the 1800s for if Reals had been written like this, then Cantor would not have been able to flood the world with his nonsense Diagonal argument, since there is no way of diagonaling .3333....33(+1/3) of its suffix.
And the silly ages old argument of .9999....=1 would never have gotten off the ground, because when you realize that 1 = .99999...99(+9/9) and never is it the case that 1 is the same as .9999....
So, here, now, today, we examine if the Greek ancient proof that sqrt2 is irrational, holds up to scrutiny once we include the suffix onto Reals.
Remember the Ancient Greeks never had the Reals, never had a decimal representation of numbers. They did however have whole numbers and fractions.
But let us see if their proof holds up, when rationals have a suffix.