On Monday, October 2, 2017 at 12:46:00 AM UTC-6, John Gabriel wrote: > You think 0.333... is a shorter way of writing the well-defined number 1/3 ? > That's absurd. It's even more absurd when you consider that there is no way to > write 1/3 in base 10.
or _ 0.3
are other ways of writing 1/3, if the dot or the bar will line up properly over the 3.
0.333... has to be understood to imply that it is 3's that are to be assumed going on endlessly.
These are _other_ ways of writing 1/3. Whether they're shorter or not doesn't matter.
In other cases, though, using the convention that "S" really means "Lim S" does let things be written more compactly.
Thus, instead of saying "the limit as n goes to infinity of the sum from i = 1 to n of (expression)", we just say "the sum from i = 1 to infinity of (expression)", even though adding an infinite number of terms can't really be done.