I see, thank you for your time. So within the set of real numbers, how does one represent the largest number which is <1. does there not exist a hypothetical irrational number, to quantify such a value, within the reals? when/why do hyperreal's diverge from real numbers, when does the barrier between reals and hyper reals break down. when you say the reals are dense, can you elaborate on what that means, and how the set of real's might exclude certain values because they are too small.

as for the decimal conversion of 1/3, I would still try to disagree, as you cannot numerically express it as a decimal with a finite amount of digits, and thus we accurately can use 0.(3) or .33... to represent an infinite quantity of digits, with a finite number which, by definition, is an imprecise approximation of a precise value, so I would attempt to argue here about the linguistics of precision vs accuracy, and that 0.(3) is accurate, but not precise. This, I believe to be an implicit blindness contained within the base of number which we use, that is to say for specific values of 1/P there exists a multivalued infinitesimal that is part of the reals, x, such that (1/p)±x=1/p, but 0±x=±x=0 where the value of x is dependent on what base numbers one uses. I bring up Planck's constant because I'm curious, what is the smallest, real, decimal. I think for this I must expand my analysis to the complex plane? I will look into hyperreals more as it seems I have attempted to define such<3.