does this number exist?

[Itryatmath]

Okay so I was wondering, does the number 0.0000.....1 exist? I was talking to my teacher and he said that it doesnt exist because you can never reach the 1 due to the infinite amount of 0s in between. However I believe it does exist in a theorectical sense because 1 - .9999... = 0.000...1.

thanks

[sqrtneg1]

Well, it is generally agreed that [unparseable or potentially dangerous latex formula]. Here is one possible proof:

Let [unparseable or potentially dangerous latex formula] Note that

[unparseable or potentially dangerous latex formula]

The RHS is an infinte geometric series, with first term [unparseable or potentially dangerous latex formula] and common ratio [unparseable or potentially dangerous latex formula] whose sum is equal to

[unparseable or potentially dangerous latex formula]

Hence, [unparseable or potentially dangerous latex formula]. Q.E.D

Therefore, I believe that the subtraction you mentioned in your post is simply [unparseable or potentially dangerous latex formula].

[AuSmith]

I don't think any of these answers get to the heart of the matter. You might expect a number like [unparseable or potentially dangerous latex formula] to exist... something like "imagine the smallest positive number possible." But, you need to ask about definitions. The decimal expansion [unparseable or potentially dangerous latex formula] is defined to be the limit of the sums

[unparseable or potentially dangerous latex formula]

That is, decimal expansions are not defined for the sort of number you are trying to describe. For, those sorts of numbers, you have to get into "nonstandard analysis," in which they are commonly referred to as "infinitesimals." Actually, to form a "hyperreal number," you can tack onto a real number an "infinitesimal part" much like you tack on an "imaginary part" for complex numbers.