Posted: Feb 25, 2010 6:20 pm
RPizzle wrote:Isn't it just infinitesimally close to 1, but not actually 1.
Nope. In the set of numbers known as the Reals, to which (regardless of whether or not they are distinct) both the numbers 0.9~ and 1 belong, there is no such thing as two numbers which are "infinitesimally close" to one another.
In other words, if a and b are both real numbers, then either a = b exactly, or the difference between them is some finite, non-zero, real number. Since we can name no finite, non-zero, real number corresponding to the difference between 0.9~ and 1, the only alternative is that they are equal.
(By the way, 0.9~ is shorthand for 0.9999(repeating forever).)