obsrvr524 wrote:I explained why that red clause is not true in the partial post you just quoted. If you have some rebuttal to that explanation - give it.

I thought I already did.

That might be true except that you specified "NO FRACTIONAL DIGITS" - that means there is no "9.###" but merely "9."

And your "..." actually bumped your "999" upward from 0 to make it an infinite count.

The ellipsis is to the right side of the digits which indicates that the digits are expanding to the right, which is to say, in the direction of the least significant digit.

Let \(x\) be the index of the first (the leftmost) \(9\) in \(99\dot9\). With that in mind, the index of the second is \(x - 1\), the index of the third is \(x - 2\), the index of the fourth is \(x - 3\) and so on. And since the ellipsis indicates that this process continues without an end, there is no last digit, which means, there is no \(9\) with the lowest index ("the least significant \(9\)").

In the case of YOUR number, which is \(\sum_{i=0}^{i->\infty} 9\times10^i\), the least significant \(9\) exists.

That alone tells you there is a difference between the two numbers.

If the "..." is allowed to go below zero then your number "999..." is undefined and can't be rationally discussed.

If it's allowed to go below zero, the concept becomes contradictory, which means, it cannot stand for anything real, but that does not mean we can't talk about it and/or use it in other ways.

And note that I didn't say that \(999\dotso\)

necessarily represents that concept. It can be used to represent that concept but it can also be used to represent other concepts. It all depends on the index of the first digit. If the index of the first digit is \(infA^2\), and the number of \(9\)s is \(infA\), then "..." does not extend beyond the decimal point.

Normally, decimal numerals have "the rightmost digit before the decimal point". \(150\) is an example. The rightmost digit before the decimal point is \(0\). In such a case, we have a rule for determining the index of any digit within the numeral. The index of the rightmost digit, for example, is \(0\). The index of any other digit is based on how far away it is from the rightmost digit and whether it is to the right or to the left of it. For example, the index of \(5\) is \(1\) and the index of \(1\) is \(2\).

But \(999\dotso\) does not have "the rightmost digit", so we cannot use this rule. And since no other rule exists, there's no way to deduce the index of any digit.