Sigh, I miss PF for a couple days, and my punishment is to have to clean up one of these messes.
donaldcat said:
but we know that 0.9* = 0.999999999....... is close to 1 and not equal to 1
but how can we prove 0.9* is not equal to 1?
What you "know" is wrong. The decimal number system is defined so that 0.9* = 1, and calculations like the one you performed are the reasons such a definition was chosen.
Werg22 said:
The value 0.9* does not have any real meaning.
...
0.9* can be interpreted as the limit ... The limit is in fact 1.
Unless your arguing that 1 does not have any real meaning either, this contradiction sums up the primary objection I have to your posts.
0.9* is the notation for a decimal number. That decimal number has no more and no less meaning than any other decimal number. 1 just happens to be another notation for that same decimal number.
The secondary objection is that mathematics is a
much, much broader subject than the study of the arithmetic of terminating decimal expansions. There is absolutely no trouble with working with numbers like
pi or
e in their exact form, even to a staunch constructivist.
Mathematics has roots in the study of nature.
A tertiary objection is that nature is full of irrational numbers. So, they better have some meaning, even if we can only approximate them. (As an aside, note that most
rational numbers don't even have terminating decimal expansions)
Furthermore, approximating things with terminating decimals is a
convenient course of action for scientific pursuits... not a
necessary one. There is absolutely no reason at all why we couldn't approximate measured quantities with irrational values.
0.9* is simply another notation for:
C = 9(1/10) + 9(1/10)^2... 9(1/10)^n
Where C is always closer to reality as n grows.
And as a quaternary objection (I love that word), this is IMHO one of the most common fallacies that prevent a person from understanding decimal notation. 0.9* is a notation for
one particular number... it is not some ambiguous number with an unspecified, but finite number of 9's. "0.9...9 (n 9's)" is a correct alternative notation for your C. 0.9* is not.
And, to be precise, you should write something like C(n), to express the fact that what you've written is a function of n.