Hi explain why 0.999.... != 1 please.

[KoShiatar]

Does math make you frustrated? It should.

Yes it does!!!

[Raistlin]

well, the way i was taught, the limit is when you come close enough to a number/point but never to the number/point (or something along those lines). then again, that was only a calculus class and not a real analysis class.

No no, limits are for as x approaches, but never reaches, a number. Limits have definitive answers.

For instance, the limit as x approaches infinity of (1/x) = 0. x gets closer and closer to infinity, but obviously never gets there. However, the limit has a definite answer.

[Sefie1999AD]

It's like the fact that 1 - (1/10) - (1/100) - (1/1000).... carried on into infinity = 0.

Except 1 - (1/10) - (1/100) - ... isn't 0. It's 0.888... (= 8/9). Proof:

1 - (1/10) - (1/100) - ... = 1 - [ 1/10 + 1/100 + ... ] = 1 - [ 10^(-1) + 10^(-2) + ... ] = 1 - ∑(n=1...∞) 10^(-n) := 1 - S.

Here, the first term is 0.1 and q = 0.1, so you'll get: 1 - S = 1 - [ 0.1 / (1-0.1) ] = 1 - [ 0.1 / 0.9 ] = 1 - [ 1/9 ] = 8/9.

[Raistlin]

It's like the fact that 1 - (1/10) - (1/100) - (1/1000).... carried on into infinity = 0.

Except 1 - (1/10) - (1/100) - ... isn't 0. It's 0.888... (= 8/9). Proof:

1 - (1/10) - (1/100) - ... = 1 - [ 1/10 + 1/100 + ... ] = 1 - [ 10^(-1) + 10^(-2) + ... ] = 1 - ∑(n=1...∞) 10^(-n) := 1 - S.

Here, the first term is 0.1 and q = 0.1, so you'll get: 1 - S = 1 - [ 0.1 / (1-0.1) ] = 1 - [ 0.1 / 0.9 ] = 1 - [ 1/9 ] = 8/9.

Err... yeah. xD It's been a while.

I guess I was thinking about what I put in my last post: the limit (x -> infinity) (1/x) = 0.

[Avathar]

:kaolaugh:
http://mathpwned.ytmnd.com/

[Levian]

Just because there's no number in between doesn't make it equal to it. >:/ Then .9[insert 9s here]8 would be equal to .9[insert 9s here]9 who's apparantly equal to 1. So in the long run 1 would be equal to 9. And that just ain't right, girlfriend.

There's so many things wrong with that, I don't know where to begin.

First off, not only does any two distinct real numbers have to have a number between them, any two distinct real numbers have to have an <i>infinite</i> amount of numbers between them. 1 and 2 have an infinite amount of numbers between them. There exists not even ONE real number between .99999... and 1.

Also, .99999...8 isn't a number. That's saying "an infinite number of nines that never ends, but when they do end, tack an 8 on the end." It doesn't work like that. 0.9999... means "a decimal followed by an <i>infinite number</i> of nines.

There's an 8 at the end of the infinite nines. Now that wasn't so hard, now was it?

[Vincent, Thunder God]

Technically speaking, 0.9999 etc. does not equal 1. But if you round it off, it is 1 for all mathematical purposes.

There... an explantion so concise yet informative that even I am impressed. Even better... no long, drawn-out mathematical equations were used!

[Raistlin]

There's an 8 at the end of the infinite nines. Now that wasn't so hard, now was it?

Then the series of nines isn't infinite. "Infinite" means, obviously, that there is no end. Therefore, there can't be any other numbers in the series of infinite nines, nor can there be something on the end, because an end doesn't exist.

EDIT:

Technically speaking, 0.9999 etc. does not equal 1. But if you round it off, it is 1 for all mathematical purposes.

There is no rounding involved. 0.999... literally equals one.

[Vincent, Thunder God]

EDIT:

Technically speaking, 0.9999 etc. does not equal 1. But if you round it off, it is 1 for all mathematical purposes.

There is no rounding involved. 0.999... literally equals one.

Then think of this. There is a glass of water. 10/10ths of it would make it full. 0.99/10ths of it would make it almost completely full, but 0.01 of it would not be full.

That 0.01 can apply to any number of 0.99s. By this I mean that 0.9999 has 0.0001 empty, etc., so on and so forth.

[Raistlin]

Then think of this. There is a glass of water. 10/10ths of it would make it full. 0.99/10ths of it would make it almost completely full, but 0.01 of it would not be full.

That 0.01 can apply to any number of 0.99s. By this I mean that 0.9999 has 0.0001 empty, etc., so on and so forth.

That changes when you make an infinite series of nines. There is no such thing as an infinite series of zeros with a 1 at the end.

[Vincent, Thunder God]

Then think of this. There is a glass of water. 10/10ths of it would make it full. 0.99/10ths of it would make it almost completely full, but 0.01 of it would not be full.

That 0.01 can apply to any number of 0.99s. By this I mean that 0.9999 has 0.0001 empty, etc., so on and so forth.

That changes when you make an infinite series of nines. There is no such thing as an infinite series of zeros with a 1 at the end.

No offense, but clearly you don't understand decimals then. It exists. 0.01, 0.001, etc. They are all real decimals. It represents a very small number.

Anyone who knows math care to back me up on this one?

[Madame Adequate]

Then think of this. There is a glass of water. 10/10ths of it would make it full. 0.99/10ths of it would make it almost completely full, but 0.01 of it would not be full.

That 0.01 can apply to any number of 0.99s. By this I mean that 0.9999 has 0.0001 empty, etc., so on and so forth.

That changes when you make an infinite series of nines. There is no such thing as an infinite series of zeros with a 1 at the end.

No offense, but clearly you don't understand decimals then. It exists. 0.01, 0.001, etc. They are all real decimals. It represents a very small number.

Anyone who knows math care to back me up on this one?

No, actually, Raistlin is right. If it's recurring, it is not 0.000001, or 0.000000000001, or anything of the like. It is infinite, and therefore there is nowhere to place the last 1. Infinite numbers are a bitch to grasp, but.

So. 0.999... must equal 1 for the following reason:

1 - 0.999... = 0.000... Now because there IS no end to 0.000 recurring, there can BE no 1 to make it up to a total of 1. And if 0.000... + 0.999... = 1, then 0.999... must equal 1. There's no other way for it. If there is a 1 in 0.000... it's not recurring, and therefore the matter is settled in another way (eq the numbers are finite.)

[Vincent, Thunder God]

Then think of this. There is a glass of water. 10/10ths of it would make it full. 0.99/10ths of it would make it almost completely full, but 0.01 of it would not be full.

That 0.01 can apply to any number of 0.99s. By this I mean that 0.9999 has 0.0001 empty, etc., so on and so forth.

That changes when you make an infinite series of nines. There is no such thing as an infinite series of zeros with a 1 at the end.

No offense, but clearly you don't understand decimals then. It exists. 0.01, 0.001, etc. They are all real decimals. It represents a very small number.

Anyone who knows math care to back me up on this one?

No, actually, Raistlin is right. If it's recurring, it is not 0.000001, or 0.000000000001, or anything of the like. It is infinite, and therefore there is nowhere to place the last 1. Infinite numbers are a bitch to grasp, but.

So. 0.999... must equal 1 for the following reason:

1 - 0.999... = 0.000... Now because there IS no end to 0.000 recurring, there can BE no 1 to make it up to a total of 1. And if 0.000... + 0.999... = 1, then 0.999... must equal 1. There's no other way for it. If there is a 1 in 0.000... it's not recurring, and therefore the matter is settled in another way (eq the numbers are finite.)

Yes, if it is infinite, you are definetly right. Thanks for pointing that out.

[Raistlin]

Yes, if it is infinite, you are definetly right. Thanks for pointing that out.

That changes when you make an infinite series of nines. There is no such thing as an infinite series of zeros with a 1 at the end.

*bows*

[Madame Adequate]

Yes, if it is infinite, you are definetly right. Thanks for pointing that out.

Y'very welcome. Took me long enough to figure it out, might as well help others along