0.999... = 1

how come 0.999.... exists and 0.999...8 doesn't?

 

Because there is an infinite amount of 9's. The 9's are never-ending, so how can they have an 8 after them?

 

head goes boom

 

wow
we're talking about infinity here but numbers that don't exist is just plain weird

 

Well, it's not that it doesn't exist, rather that you can't get numbers like that in the first place. Do you understand what I'm trying to say or does it all sound impossible?

 

Personally I would say numbers like .999...96458 exist but the numbers on the end are meaningless because there is always more 9s to deal with so the number ends up the same as .999... anyways.

 

I want to say this again: infinity numbers are crap

 

But are they equal to 1 or not?

 

I guess it depends on how exact you want to be, assuming this is in some sort of equation. I voted yes though, just because I hate decimals

 

Since .999...96458 is just a way of writing 0.999... with extra meaningless digits at the end, and since 0.999... = 1, then 0.999...96458 does equal 1. The 96458 is being listed as "after infinity" which means they have no effect on the number. Take infinity and add 7, you get infinity. As a matter of fact, take infinity and add infinity and you still get infinity. I am not beating around the bush with limits or any of that kind of stuff, I am saying actual infinity has no end by definition, so adding anything to the "end" means throwing it away.

The "infinity" we are dealing with, if I am not mistaken, is known as a "countable" infinity, aleph-null, which is not the infinity of calculus or the algebraic number line. Aleph-null represents the cardinality of the set of all natural numbers (1 and up). Some properties of this aleph number are for all real X:
aleph-null + X = aleph-null
aleph-null * X = aleph-null
The number of 9s in 0.999... is equal to aleph-null, which means that adding anything "after" the 9s will not change the cardinality of decimals at all, resulting in no change in value. Thus, 0.999... = 0.999...96458 = 1. It is not good notation to put the extra decimals at the end, but that is all there is to it.

0.999... is exactly equivalent to 1. There is absolutely no difference. There are no real numbers between 0.999... and 1.

 

I've already given proof in my previous post. Again: redundant zeros make 1.00' = 1. If 0.99' = 1, then 0.99' = 1.00'. Those two things are expressed in exactly the same way, with the huge difference that they are different numbers. If you say that's not true then you're claiming that 0.99 = 1.00, which, I'm hoping, you agree is nonsense.

 

Kuvasz, what you are saying about different representations of decimals is not true and can be disproved by using decimal long division.
1
9)9

or

0.9999 etc.
9)9.0000 etc.
-8.1
90
-81
90
-81
90
-81
9 etc.

The same mathematically legal procedure can generate two identical numbers based on the choice of the mathematician, and I stress that it is a mathematically legal procedure.

I am sure everyone agrees that 0.99 =/= 1.00. What he is saying is that 0.999... and 1 represent the same constant, and therefore 1 = 0.999.... Different representations of one value appear all the time in mathematics, it just isn't usually in decimal notation.

 

I don't know how much do you trust Wikipedia, but I agree with this
http://en.wikipedia.org/wiki/0.999...

There are mathematical proofs, which contain no cheating. No dividing by 0 etc..

 

I found some other websites that says the same...
DONT TRUST WIKI

 

Actually, some arcticles are good. You just have to sort which information you belive, and which you don't.
I think this one is acceptable.

 

@11-Sodium: You might have guessed it, but I'm not overly familiar with maths, and your post doesn't mean much to me
I'm using common sense when I say infinity does not justify marking the same number with two completely different set of digits (i.e. 0.99' and 1.00').

 

Oh if noone mentioned yet, thisícan be true cause there isn't a number which has infinite 9's at the end. You can always put there an other. It is only logically true, there is no use of this in the real life.

 

it definitely is equal

steps for fractioning repeating decimals
1.) x = 0.9999...
2.) 10x = 9.9999....
3.) 10x - x = 9.9999... - 0.9999...
4.) 9x = 9
5.) x = 1


plus whats 1/3 as a decimal 0.333333...
1/3 time 3 is 1 and 0.333333... times 3 is 0.999999999...

 

@ kuvasz. You're being stubborn. There have been several (how many exactly now?) proofs all showing that 0.999... is exactly equal to one. You're not using logic, you're just saying that they aren't the same because they contain different numbers. If you actually read the maths, even if you have a bit of trouble with understanding a few of the proofs, you'll find that it's undeniable. Numbers expressed in the same way can still look completely different, and there is no proof that they can't.
To other's who say that it's 'close enough' or that the 'bit at the end is insignificant' you're wrong. 0.999... is exactly equal to one. There is no difference between 0.999..., 1/1, etc.

 

^ Agree, 0,99...=1
0,99... is just an other way to show it.
Gah345 showed an excellent example