Can somebody explain this to me?

1. Jan 14, 2007

Sivy78

2. Jan 14, 2007

cristo

Staff Emeritus
There are lots of threads on here discussing "0.99...=1." Try searching!

3. Jan 14, 2007

mathwonk

the point is there is a difference bewteen a real number and the numerals used to represent it.

just as 1/2 = 2/4, there are more decimals than there are reals, and1.000, and .99999 represent the same real.

one way to see this is to define the real number represented by an infinite decimal as the smallest number not smaller than any of the finite decimal approximations. then one eventually sees that the smallest number not smaller than.9, .99, .999, ..... is 1.00000.

if one does not eventually see this, one continues to post it here ad infinitum.

4. Jan 14, 2007

HallsofIvy

Staff Emeritus
The definition of the decimal representation 0.a1a2a3... is the limit of the sequence 0.a1, 0.a1a2, 0.a1a2a3, ...
In particular, 0.999... means the limit of the sequence 0.9, 0.99, 0.999, 0.9999, .... Since that is a geometric sequence, it is easy to show that its limit is [itex]\frac{0.1}{1- .9}= 1.