Quote:

**Insentience**:

Me and a friend were having a debate whether or not 0.999... (for those who are confused the 9 is repeating.). I'm saying it does = 1, but he says it doesn't. Though I think I may be right, but there is always the benefit of doubt, and I am open to hearing what others have to say.

So if 0.999... does equal 1, then explain how it equals 1.

but if 0.999... doesn't equal 1, then explain how it doesn't equal 1.

This has been bugging me for awhile.

Like everything in mathematics, the truth of a statement is always a consequence of rules and definitions. Mathematical rules and definitions don't necessarily have to translate into something that makes

*physical* sense.

This seeming contradiction to common sense has its roots in the definition of "infinite." Infinity is not a very large number. It is infinity.

*Infinitely* small is not a very small number, it's 0, by definition.

1 minus an infinitely small number is the same thing as 1 minus 0, because an infinitely small number is 0

*by definition*. Therefore 0.999... is 1 by definition. Get it?