I am not usually one to post on forums often, as I feel my writing makes me appear to be a pedantic jerk. My apologies to anyone who thinks I am "
correcting"
them. I make no claim to be a mathematician, I just enjoy the discussion no matter how it turns out. I believe this to be one of the few places on the internet where this conversation will not degrade into baseless name calling.
I think I have found what my problem with the "
quick"
form of the proof is. It begins with the assumption that 0.9999~ is equivalent to 1. Without that assumption, 10*0.9999~ would not equal 9.9999~. Mind you I am not saying this to contradict the 0.9999~=1 statement, only that we are using the conclusion to prove it.
When I said multiplying by 10 was putting a 0 at the "
end"
of the number, I was not using any mathematical basis. It is a short hand we use to multiply quickly, just like the "
move the decimal to the right."
Multiplication is defined as repeated addition. So multiplying 0.9999~ by 10 is the same as
0.9999~
0.9999~
0.9999~
0.9999~
0.9999~
0.9999~
0.9999~
0.9999~
0.9999~
+ 0.9999~
9.9999~If it were not assumed
before this step that 0.9999~ were equal to 1, and not some infinitesimal nonzero number
x less than 1, then 9.9999~ would be 10
x less than 10.
This short hand we use to quickly multiplying a number by 10 is the key to why this proof "
works"
on a gut level. This is also why 10 is used, and not something else, like 2 or 736. Both these are just as valid, assuming 0.9999~=1
x = 0.9999~
2x = 1.9999~
-x -x
x = 1
x = 0.9999~
736x = 735.9999~
-x -x
735x = 735
x = 1
Remember, I am not saying that 0.9999~ isn't equivalent to 1, just that those steps are only valid if it is. This is where the "
proof"
breaks down. It is a clever example using the equality, however.
If only my nick was "=nottrue" 
is a considerably more stylish way to express sarcasm than 
I only wish I could have been as succinct. 
