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Stability of I in tetration and scaled infinities/infinitesimals?
bo198214 Wrote:
Ivars Wrote:Numbers like e , pi , i , irrationals, periodic decimals etc are not really constants but symbols, as they are incomputable, like infinity and 0.
Infinity is not a number, 0 is a number. e, pi, i, irrationals and periodic decimals are numbers too. There are uncomputable irrational numbers, however e, pi, i and periodic decimals are computable (in the sense that there is an algorithm to determine the n-th digit after the point).
Dear Ivars,

Unfortunately (ops ... , sorry Bo Wink , I didn' mean that!) I think that Henryk is absolutely right. The entities that you mentioned are indeed constant. Where did you find the idea that they are only "symbols"? They are constant numbers, ... because they are not variable. We should not confuse "variability" with the technical impossibility of "fully knowing them". If a number is computable, that means that it is mathematically known, i.e. that "there is an algorithm to determine the n-th digit before and after the point".

The fact that, techically, we don't have an infinite computer memory for storing ... all ... its representative figures (say, by using the decimal fixed point notation) is an engineering and not a mathematical problem. Engineers say that Pi cannot be fully known (or only by a certain approximation), but mathematicians say that it CAN be known, if we take the appropriate (infinite) number of necessary steps. For a math-man, it is sufficient to demonstrate that the algorithm exists. Then, the number is known. There are also cases where we can demonstrate the existence of a number, but we cannot compute it. But, in all these cases, it is out of question to consider them as variables or, ... just symbols.

Believe me! The decimal figures of Pi are not random, they are deterministically determined, if you see what I mean. The fact that an American Corporation provided in the past "pseudo-random" sequences of figures based on the decimal figures of PI, for random testiong nuclear reactor cores (Monte Carlo method), is another story. We are only human beings and we don't know what we are doing. For the non-destructive testing it was sufficient, but the sequence was not random.

From another point of view, expression e^(i*Pi/2) = i is a well known equality (identity) and it does not mean at all that i has .. an internal structure. Actually, i = sqrt(-1), with plus/minus, if you wish.

Please finally accept the fact that |d(Pi)| = 0. Don't try to differentiate constants, you will loose time ... for nothing. Move higher, as I already friendly suggested you, ... there is a lot of space!.


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