mjc0961 said:
ManInRed said:
That .9999...forever is less than 1. When they are in fact equal.
Ooh, the misconception that those two are equal when they aren't is one I hate. They are so close that it doesn't matter if you round up, but mathematically they are not the same.
Mathematically they are the same, two different ways to write the same real number, there is no rounding. It only seems like 0.999... would be smaller than 1. There are a few proofs out there demonstrating they are the same number.
For example:
1/9 = 0.111?
9 x 1/9 = 9 x 0.111?
1 = 0.999?
Or this one:
y = 0.999?
10y = 9.999?
10y ? y = 9.999? - 0.999?
9y = 9
y = 1
If 0.999... did not go on forever but stop at some very large point, then it would be less than 1. The last proof of shifting the digits to the left by one obviously is only possible because the 9's go on forever. It seems weird to people, because it is unusual to deal with infinitely long strings of numbers. It's not the only example of being able to write 1 real number in multiple ways, just a very strange looking one.