1=.999999...?

[Souron]

Can infinity ever be used in this way? I don't think, atm, that it can.

In addition to what Atticus said, calculations are done with infinity very often actually: by computers. Computers use the IEEE floating point standard to represent decimal numbers. Virtually all calculations involving decimals use this kind of number. And it includes a representation for negative infinity and infinity, with rules for arithmetic on those numbers. Infinity is what really happens when you divide by 0 .

 

[Birdjaguar]

I am going to stop reading this thread; I swear it.

Oh Sauron is now posting.

 

[onejayhawk]

Why should it be justified? It's just a thing that happens, there's no reason to think it couldn't happen.

You can prove however that the limit has to be in the closure of the set where the sequence lies. In this case, [0,1].

(And btw., it's complement).

I often do not spell check. It's a bad habit, but deeply ingrained.

Of course you can prove that the limit is an element of the closure. That was conceded with the OP. What good does it do you?

If there was a consensus that 0.999... = 1, there would be no thread. But 0.9+0.09+0.009+... =/= lim(0.9+0.09+0.009+...), so exclusion of the limit is the point. Including it is not a given.

J

 

[Souron]

Oh Sauron is now posting.

Is this referring to my absence from CFC recently? Yes I'm back.

 

[Borachio]

In addition to what Atticus said, calculations are done with infinity very often actually: by computers. Computers use the IEEE floating point standard to represent decimal numbers. Virtually all calculations involving decimals use this kind of number. And it includes a representation for negative infinity and infinity, with rules for arithmetic on those numbers. Infinity is what really happens when you divide by 0 .

Well, I'll take your word for it, of course. But I was under the impression that computers can't represent infinity. If I use mine to divide 1 by zero it comes up with NaN, or Not a Number. Which always struck me as more than fair.

Mind you, I haven't done it in a while. And times do change.

Even that Maple package, which pretty routinely handled 30,000 digit numbers, didn't seem to be able to treat infinity as a number. As far as I recall. But again, I daresay I'm wrong.

Course, blah, blah, blah, something about graph sketching, blah, blah, I daresay.

 

[Souron]

Well, I'll take your word for it, of course. But I was under the impression that computers can't represent infinity. If I use mine to divide 1 by zero it comes up with NaN, or Not a Number. Which always struck me as more than fair.

Mind you, I haven't done it in a while. And times do change.

Even that Maple package, which pretty routinely handled 30,000 digit numbers, didn't seem to be able to treat infinity as a number. As far as I recall. But again, I daresay I'm wrong.

Course, blah, blah, blah, something about graph sketching, blah, blah, I daresay.

If you divide 0 by 0 you get NaN. If you divide a positive number by 0 you get inf. Also, integers don't have a representation for 0, so if the calculation is done with integers, you get an error instead. You can use Google calculator to see this, although there are probably better tools that allow you to enter Inf and Nan directly.

Maple should be the same, unless it's using arbitrary precision, in which case they are using their own libraries which may or may not represent infinity.

 

[onejayhawk]

If you divide 0 by 0 you get NaN. If you divide a positive number by 0 you get inf. Also, integers don't have a representation for 0, so if the calculation is done with integers, you get an error instead. You can use Google calculator to see this, although there are probably better tools that allow you to enter Inf and Nan directly.

Maple should be the same, unless it's using arbitrary precision, in which case they are using their own libraries which may or may not represent infinity.

It depends on your definition of 0. Anything is possible.

J

 

[Lohrenswald]

If there was a consensus that 0.999... = 1, there would be no thread. But 0.9+0.09+0.009+... =/= lim(0.9+0.09+0.009+...), so exclusion of the limit is the point. Including it is not a given.

J

That's not true though.

And Souron, just because a computer has to use like 10^-18 or whatever doesn't mean that dividing by zero is something you can't do (it's ill-defined, I think it's called)

 

[warpus]

There is consensus that 0.999... = 1, in the mathematical community. Which for the purposes of this thread is really the only one that matters.

If not then nothing at all will ever be in the "consensus is reached!" stage, since you'll always find someone who disagrees with something, even if it's something as widely accepted as... say... the theory of evolution.

 

[El_Machinae]

Or that you should switch doors!

 

[onejayhawk]

There is consensus that 0.999... = 1, in the mathematical community. Which for the purposes of this thread is really the only one that matters.

If not then nothing at all will ever be in the "consensus is reached!" stage, since you'll always find someone who disagrees with something, even if it's something as widely accepted as... say... the theory of evolution.

Not really. There is a consensus that lim (0.9+0.09+0.009+...)=1. Not exactly the same thing.

That's not true though.

What's not true? If there was a consensus, there would be no thread.

J

 

[UWHabs]

There's a consensus among everyone who follows the currently accepted axioms of mathematics. There are a few people who disagree, but need to bring in non-standard analysis to try to prove their points.

It's the same way there's a consensus about climate change and the earth being more than 4000 years old. Not everyone agrees, but there is still a consensus.

 

[Souron]

It depends on your definition of 0. Anything is possible.

J

My point was that there is use in treating infinity as a number.

And Souron, just because a computer has to use like 10^-18 or whatever doesn't mean that dividing by zero is something you can't do (it's ill-defined, I think it's called)

True.

There is consensus that 0.999... = 1, in the mathematical community. Which for the purposes of this thread is really the only one that matters.

If not then nothing at all will ever be in the "consensus is reached!" stage, since you'll always find someone who disagrees with something, even if it's something as widely accepted as... say... the theory of evolution.

Scientific consensus and agreement on which axioms to use in mathematics is a very different thing. Math does not need to be understood "this is the way it is because the prof says so". Rather it is what the professor says it is because that makes it useful. Furthermore, this seems like a basic grade school question, but it actually illustrates a finer point in calculus, so an interesting question for that reason.

 

[Birdjaguar]

Is this referring to my absence from CFC recently? Yes I'm back.

Yes!

 

[dutchfire]

There's a consensus among everyone who follows the currently accepted axioms of mathematics. There are a few people who disagree, but need to bring in non-standard analysis to try to prove their points.

It's the same way there's a consensus about climate change and the earth being more than 4000 years old. Not everyone agrees, but there is still a consensus.

What does that even mean?
Is the parallel postulate a "currently accepted axiom of mathematics"? Is the Axiom of Choice?

 

[Folket]

Not really. There is a consensus that lim (0.9+0.09+0.009+...)=1. Not exactly the same thing.



What's not true? If there was a consensus, there would be no thread.

J

0.99.. is just shorthand for a zero followed by an infinite number of nines.

if you agree that lim (0.9+0.09+0.009+...)=1, then it follows that 0.99.. is also 1.
Since every term in the sequence (0.9, 0.99, 0.999, ....) is smaller then 0.99.. we know that lim(0.9, 0.99, 0.999, ....) is less or equal to 0.99..
so we get 0.99 >= lim(0.9, 0.99, 0.999, ....) = 1. i.e. 0.99.. >= 1.

 

[warpus]

Not really. There is a consensus that lim (0.9+0.09+0.009+...)=1. Not exactly the same thing.

That is exactly what 0.999... means actually.

Is there documentation of this rift in the mathematical community about the true meaning of 0.999... ? Can you link me to some literature outlining the controversy?

 

[onejayhawk]

0.99.. is just shorthand for a zero followed by an infinite number of nines.

if you agree that lim (0.9+0.09+0.009+...)=1, then it follows that 0.99.. is also 1.
Since every term in the sequence (0.9, 0.99, 0.999, ....) is smaller then 0.99.. we know that lim(0.9, 0.99, 0.999, ....) is less or equal to 0.99..
so we get 0.99 >= lim(0.9, 0.99, 0.999, ....) = 1. i.e. 0.99.. >= 1.

That is the weirdest attempted proof yet. Don't say "just". If you dig in the thread there is some confusion on that point. One thing is clear, 0.999 < 1. Your statement, that 0.99... >= lim(0.9, 0.99, 0.999, ....) is always false. I am unsure how you would think it was true.

J

 

[Lohrenswald]

One thing is clear, 0.999 < 1.

Actually: no

 

[onejayhawk]

Actually: no

0.999 < 1, clearly. I assume that you miss typed.

Every element of the sequence 0.9, 0.99, 0.999, ... < 1. No element is = 1. Hence less than and not equal.

That is exactly what 0.999... means actually.

No, which is the entire point.

0.999... is an element of the sequence (0.9, 0.99, 0.999, ...). But lim(0.9, 0.99, 0.999, ... ) is not an element. It is the conventional substitution. This entire conversation is about conventional substitutions.

J