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0.999...

http://en.wikipedia.org/wiki/0.999

Argued with a couple people about this today. Basically the idea is that in a real number system 0.9999999 with 9 extending for infinity, is equal to 1. They are the same number.

I understand the mathematical concept with real numbers, but I prefer the infinitesimal number system when it comes to this. I feel it is far more logical to assume that 0.999... never reaches 1, it just infinitely gets closer and closer, but never becomes 1. This I think is far more representative of physical world realities.

Any math people have any opinion on this?

I like the geometric series example when it comes to this. Basically
0.999...= 9/10 + (9/10)^2 + (9/10)^3 + .... = 9(1/10) / (1-(1/10)) = 1.

So 0.999... converges to 1. For all intents and purposes, any use of the real number system would use this, and would not treat both numbers as different.

.333 + .666 = .999
1/3 + 2/3 = 3/3 = 1

you can't explain that.
science.

quote:

Originally posted by LAdazeNYnights .333 + .666 = .999 1/3 + 2/3 = 3/3 = 1 you can't explain that. science.

/thread.

so what is 0.999...+0.999...?

I might not have believed you a semester or so ago. The logicians who make up these rules are nuts

quote:

Originally posted by LAdazeNYnights .333 + .666 = .999 1/3 + 2/3 = 3/3 = 1 you can't explain that. science.

0.333 != 1/3
0.666 != 2/3

quote:

Originally posted by Joss Weatherby so what is 0.999...+0.999...?

1.999999999999 or 2

I stopped caring about numbers once I got an accountant.

quote:

Originally posted by aquila 0.333 != 1/3 0.666 != 2/3

.3bar, or .3repeating = 1/3
.6bar, etc = 2/3
.9bar /= 3/3

techmology is wack

Re: 0.999...

quote:

Originally posted by Joss Weatherby http://en.wikipedia.org/wiki/0.999 Argued with a couple people about this today. Basically the idea is that in a real number system 0.9999999 with 9 extending for infinity, is equal to 1. They are the same number. I understand the mathematical concept with real numbers, but I prefer the infinitesimal number system when it comes to this. I feel it is far more logical to assume that 0.999... never reaches 1, it just infinitely gets closer and closer, but never becomes 1. This I think is far more representative of physical world realities. Any math people have any opinion on this?

Knowing you, I'm pretty sure they were talking about something completely un-related, then you told them all about 0.9999, they tried to join in on the conversation a few times, but after the third correction, they chose to stand there with a puzzled look on their face until you just kinda stopped talking and went away.

good for you!

Alfi is infinitely approaching intelligence...

Re: 0.999...

quote:

Originally posted by Joss Weatherby http://en.wikipedia.org/wiki/0.999 Argued with a couple people about this today. Basically the idea is that in a real number system 0.9999999 with 9 extending for infinity, is equal to 1. They are the same number.

that would not be a real number if the 9 extends to infinity. There is nothing to argue about, they are not the same. any kid that took intro to calculus could show you the proof. What might seem trivial is the reason why we have spaceships, um lazer beams, submarines. It is a pretty basic concept that everyone with a high school education should get, not mathematicians.

Re: Re: 0.999...

quote:

Originally posted by Looney4Clooney that would not be a real number if the 9 extends to infinity. There is nothing to argue about, they are not the same. any kid that took intro to calculus could show you the proof. What might seem trivial is the reason why we have spaceships, um lazer beams, submarines. It is a pretty basic concept that everyone with a high school education should get, not mathematicians.

That is my view too, infinitesimals. Essentially it can't be a number because it is constantly changing, its always going to be more or less than itself, never equal, and not any other number.

Re: Re: 0.999...

quote:

Originally posted by Looney4Clooney that would not be a real number if the 9 extends to infinity. There is nothing to argue about, they are not the same. any kid that took intro to calculus could show you the proof. What might seem trivial is the reason why we have spaceships, um lazer beams, submarines. It is a pretty basic concept that everyone with a high school education should get, not mathematicians.

So, you're saying that 0.9999.... is not an element in the set of all real numbers? You might want to think that one through again. If you so sure about that one, I would love to see the proof that it is not an element of the set of all real numbers. That's like saying pi isn't a real number....

quote:

Originally posted by Meat187

Re: Re: Re: 0.999...

quote:

Originally posted by Joss Weatherby That is my view too, infinitesimals. Essentially it can't be a number because it is constantly changing, its always going to be more or less than itself, never equal, and not any other number.

All irrational and rational numbers are real numbers. 1/3 = 0.3333...
and it is the quotient of 2 integers, so it is rational, and therefore in the set of real numbers. Irrational numbers, like pi and radical 2, are not equal to the quotient of any integers, but they are still real numbers.

Re: Re: Re: 0.999...

quote:

Originally posted by OurManFlint So, you're saying that 0.9999.... is not an element in the set of all real numbers? You might want to think that one through again. If you so sure about that one, I would love to see the proof that it is not an element of the set of all real numbers. That's like saying pi isn't a real number....

my bad. definitions escape me as it has been a decade since i have touched math. The point i was making was that 0.9999... is not ever equal to 1.

Re: Re: Re: 0.999...

quote:

Originally posted by Joss Weatherby That is my view too, infinitesimals. Essentially it can't be a number because it is constantly changing, its always going to be more or less than itself, never equal, and not any other number.

that is not my view. It is a real number, it is not equal to 1. It can be defined by an equation which does not change. I had to go recheck what those definitions meant as it has been years. But no, i do not agree.

Yea in real numbers they are, but not in infinitesimal numbers where the idea is that you can have an infinitely smaller and smaller fraction of a number, always approaching but never reaching the next number.

The same can be said of pi, pi can be infinitely calculated out, producing more and more accurate results. It doesn't mean that pi is 3.14160 or 3.14158 its 3.14159... forever. Its never below or above that set of numbers its just pi.

For some reason this actually seems more like a philosophical issue than any sort of math issue, since in the real world of percentages of physical things you 1 is definitely not equal to 0.999... It might very quickly become something that is not even apparent in any result set if you are at 99.999...% of something, but its still not 100%, ever, its a physical impossibility. Its either whole or its not whole.

but the fact that it lies between 0 and 1 makes it a real number.

quote:

Originally posted by Looney4Clooney but the fact that it lies between 0 and 1 makes it a real number.

It never returns NaN in any equation?

Re: Re: 0.999...

quote:

Originally posted by Looney4Clooney that would not be a real number if the 9 extends to infinity.

Actually, it would be -- one casual definition of a "real number" is just that it is any number that can be represented by an infinite decimal (even if, as in the case of integers, that infinite decimal would just be something like 1.000000...).

quote:

Originally posted by Joss Weatherby Essentially it can't be a number because it is constantly changing, its always going to be more or less than itself, never equal, and not any other number.

The mistake here is in thinking of a number as being created by a "process" such as adding numbers one by one after the decimal point. A number isn't an algorithm, it's just an abstract mathematical object that is always the same thing independent of representations or means of calculation.

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