(November 15, 2011 at 12:34 am)Pendragon Wrote:(November 3, 2011 at 1:37 pm)Tiberius Wrote: True, you cannot use infinite values in algebra, but we aren't. 0.999... isn't infinite, it is infinitely long. There is a big difference. Technically, all values can be represented as infinitely long:
1 = 1.000...
3.64 = 3.64000...
Etc.
There are various proofs that 0.999... = 1, not all of them are algebraic.
Sometimes there are questions when "infinity" is used. it is a concept, some say a limit. But what about the "set" of infinite limits?
Can anyone demonstrate to us what the difference is between "infinite", and/or "infinitely long"?
I have already shown one equation in which 0.999... and 1.000... are not interchangeable. The problem is infinity and we just need a better way to handle infinities.
(Technically, all numbers are infinitely long, but we will stick with tradition and consider any infinitely long number that does not end in zero.)
Take for instance this 'infinite' set -> ( 0.1, 0.01, 0.001, ...) in which only the last number is 'infinitely long' (in the set of transcendentals, all numbers are 'infinitely' long. )
The "last' number would be 0.000...1, but our math at this time does not allow anything tagged to the end of an infinite string (and actually that is a judgement call 1.0 x 10E-infinity) which invalidates the 'last' number in this set, but because the set is infinite the 'last' number is discounted (or "called" zero). This makes the set short one number but again, because of infinity and the way we handle it, infinity-one still equals infinity and the set is still complete without the 'last' number. Without this 'last' number we have nothing to subtract from one to get 0.999...(again, judgement call 1-(1.0 x 10E-infinity)) so it is accepted as 1, but they still are not interchangeable.
0.000... = 0.000...1 = 0.000...pi all equal 0
An infinite string of zeros is not necessary because it does not change any values, however a string of any other number changes the value every time it is repeated.
By definition 0.1x10E-infinity = 0.1x10E-infinity x 10
But if you divide both sides by 0.1x10E-infinity then 1=10 and that is why one cannot use infinities for anything but limits. An infinitely long number is considered valid and when complications arise, just say 0.999...=1 and all is well.
A number can be infinitely long, but not infinite. A set can be either.
You make people miserable and there's nothing they can do about it, just like god.
-- Homer Simpson
God has no place within these walls, just as facts have no place within organized religion.
-- Superintendent Chalmers
Science is like a blabbermouth who ruins a movie by telling you how it ends. There are some things we don't want to know. Important things.
-- Ned Flanders
Once something's been approved by the government, it's no longer immoral.
-- The Rev Lovejoy
-- Homer Simpson
God has no place within these walls, just as facts have no place within organized religion.
-- Superintendent Chalmers
Science is like a blabbermouth who ruins a movie by telling you how it ends. There are some things we don't want to know. Important things.
-- Ned Flanders
Once something's been approved by the government, it's no longer immoral.
-- The Rev Lovejoy
