Infinitely Approaching Equals Being There

[David Morris]

http://polymathematics.typepad.com/polymath/2006/06/no_im_sorry_it_.html

And now for something completely different.  A math puzzle.  Or
conundrum, if you will.

In the figure to the left [not shown here], the bar above the number
.9 indicates that it is to be repeated forever.  For the remainder of
this post, we will represent that concept by several nines with an
ellipsis (.999…).

Now, here is the conundrum.  .9 repeating is EQUAL TO ONE.   Not CLOSE
to one, mind you, but EQUAL to one.

Nonsense, you reply.  It is obviously less than one.  Not by much – by
an infinitely small amount, in fact.  But the simple fact (?) that it
is not one is enough to demonstrate that it can’t be equal to one.
It’s as close as you can get to one without being one.

Wrong.  It is in fact equal to one, and that fact can be demonstrated
mathematically in several ways.

The most easily understood is to revert to other familiar repeating
digits.  Everyone knows that 1/3 is 0.333… and that 2/3 is 0.666…  If
you add them together, you get 3/3, which is one.

But now note that the sum of the decimals on the right side of the
equation is 0.999…

Therefore, one is equal to (not close to) .999…