EP, this is just a communication problem, I know you're no fool.
My opening premise was not "if x = 0.999" but "if x = 0.999 . . .", note the ellipsis which denotes a recurring decimal.
Thanks for the compliment, I know you aren't either. However, having said that....
I know what the ellipses denote, but you are talking about two different things. First, the infinity part is talking about math, the relationships between numbers, theories, proofs, sets of numbers and how they interoperate. The second is simple arithmetic.
In the arithmetic portion, the example starts with the idea of infinity with "x = 0.999..." but then switch to a finite set to do the actual arithmetic. In the example, they arbitrarily add an extra order or magnitude into the 10x portion but do NOT in the x portion. It's not valid to switch orders of magnitude. To remain consistent (a hallmark of good arithmetic) it would look more like this:
x = 0.999..... < consistent to 3 decimal places
10x-x = y
10*0.999... - 0.999... = y < consistent to 3 decimal places
9.99.... - 0.999 = y < consistent to 3 decimal places
The problem in the example is that they are NOT consistent to the same decimal places. If you say decimal places do not matter, then it's just as valid to do the arithmetic like this:
x = 0.999
10x - x = y
10*0.99999999 - 0.9 = y
Having said that, due to the real number system that we use that allows for infinities of numbers. It is absolutely true that it is a interestiing curiosity of our number system, but in practical terms or in calculable terms of physics or arithmetics, they are NOT equal.
And that's way more math than I thought I would talk about today.
