There is a subtle, but important, error in the proof you gave.
Where 1/3 = .999..., it must be assumed that there is still a fractional remainder. 1/3 doesn't exactly equal .999... any more than pi exactly equals 3.14159.
For example, I could use your proof to make some pretty wild conclusions:
no subject
Where 1/3 = .999..., it must be assumed that there is still a fractional remainder. 1/3 doesn't exactly equal .999... any more than pi exactly equals 3.14159.
For example, I could use your proof to make some pretty wild conclusions:
1/6 = .16666...; 6 * .16666 = .9996; thus, .9996 = 1.
1/36 = ..02777...; 36 * .02777 = .99972; thus, .99972 also equals one!
Hmm, but wait.
If .9996 = 1, and .99972 = 1, then .9996 = .99972. Ooo. Hey, this could get fun!