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# 1.999 repeating = 2

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 Voting Style: Open Point System: 7 Point Started: 1/15/2008 Category: Science Updated: 14 years ago Status: Voting Period Viewed: 14,814 times Debate No: 1822
Debate Rounds (5)

42 comments have been posted on this debate. Showing 1 through 10 records.
Posted by beem0r 14 years ago

You've admitted that .333... is equal to the fraction 1/3.
However, the same logic that has made you conclude that .999... contains some nonzero infinitesimal could be applied to .333... and 1/3.

However, the notation in which 0.999... is written is completely valid in the reals. There is no separate component. It's just a number.

Also, .999... is also 1 in the hyperreals. Please check whatever source you think has told you otherwise, as they are miserably wrong.

So to conclude:
1. 0.999... is a valid decimal representation in the reals.
2. Just using simple thought, the only possible difference between 0.999... and 1 would be an infinitesimal.
3. There are no nonzero infinitesimals in the reals.
4. There is no mathematical difference between 0.999... and 1 in the reals.

The extent to which they are different is the same as that of 1/3 and 2/6. Same number, two representations.
Posted by kennard 14 years ago
One thing that beem0r said was that "In the reals, an infinitesimal is zero". I checked this on several places on internet and I think there is a misunderstanding here. The number zero is the only infinitesimal which is also a real number, but this doesn't mean that a non-zero infinittesimal equals zero in the real numbers line. It would be like saying that the imaginary number equals zero in the real line (I should clarify that in the real line, the "b" in a+bi form of a complex number is zero, but not the i itself).

Now that I have read some more on hyperreal numbers, I think that 0.999 repeating, and numbers that have repeating nines after the decimal are not real numbers at all, they are hyperreal numbers. First of all it should be realized that hyperreal numbers are an extension of the real line in the same way that complex numbers are an extension of the real line. In other words, all the real numbers are a subset of the hyperreal numbers and of the complex numbers. However, as we know, not all complex numbers are real numbers. In the same way, not all hyperreal numbers are real numbers. In the hyperreal number line 2 and 1.999 repeating are two different numbers, this means that 2 cannot equal to 1.999 repeating in the real line because 1.999 repeating does not exist in the real line.
This is analogous with complex numbers. For example, 1 and 1+i, are different numbers in the complex plane. However, we cannot say that in the real line, 1 and 1+i are equal just by claiming that the imaginary part in 1+i does not exist in the real line. The complex number 1+i is not part of the real numbers at all because it has an imaginary part. Similarly, 1.999 repeating and all the repeating nines numbers are not part of the real numbers at all because they have a non-zero infinitesimal component with them, and non-zero infinitesimals do not exist in the real line.
Posted by Einstein 14 years ago
I actually think the discussion of real number analysis on Wikipedia cleared it up for me - such as the Dedekind cut analysis.
Posted by beem0r 14 years ago
Perhaps I shouldn't have been so harsh about what you said about the inability to add/subtract/mult/div decimal representations of numbers. I should have said "I find Con's supposed reason quite spurious" or something to that extent. I acknowledge that you at least attempted to back up your claim, and I apologize for stating otherwise earlier.

>>The last thing I have to say about the argument is that two REAL numbers are only equal if their difference equals ZERO. However the difference between 2 and 1.999 repeating is not zero, it is an infinitesimal.<<

In the reals, an infinitesimal is zero. Ergo, the difference is zero. I thought I was clear about this previously, my apologies if not. I can often be too long-winded or non-sequential in my arguments.

http://en.wikipedia.org......
http://en.wikipedia.org...

A lot of skeptics like yourself there, using arguments quite similar to yours, and it seems they are unable to hold up their side with people with perhaps more knowledge and interest than myself. There's a whole page dedicated to people making arguments against .999... = 1, and I believe all their objections have all been answered quite well.

Well, I can't say I didn't enjoy our battle, so thanks. Be seeing you.
Posted by kennard 14 years ago
Well beem0r, I am sorry for having said that you should wait until you finish college to write a definition of infinity that makes sense. But I felt you offended me first by saying that I was just saying stuff and not backing it up, when I obviously made several arguments that make sense. I understand that you might not have been completely convinced by them, but don't claim that I am just saying "That's just the way it is". For example you wrote:

" Also, con is not backing up his "you can't add/subtract/mult/div repeating decimals."
Pro asked why you can't, con said "That's just how it is, you can't do it. "

The last thing I have to say about the argument is that two REAL numbers are only equal if their difference equals ZERO. However the difference between 2 and 1.999 repeating is not zero, it is an infinitesimal. So 1.999 repeating is a representation of the theoretical "closest" number to 2 in the same way that infinity is a representation of the theoretical "largest" number. It would seem to me that accepting 2 = 1.999 repeating is the same as accepting that their exist a real number which is larger than any other real number.

Anyways, I must admit that you seem like a smart guy, and even though I honestly think that the arguments that I have made are stronger I am pretty sure by now that I won't be able to convince you regardless of what I say.
Posted by beem0r 14 years ago
Wikipedia is generally a great one-stop source for information. It gathers information from a variety of referenced, reliable sources. Sure, sometimes a claim isn't referenced, but the reader is generally made aware of this.

If a page's contents are debated, like the .999... page I linked, there are usually an abundance of references used. Pages with much less disagreement, such as kennard's division by zero page, don't tend to use quite as many, since no one demands sources on widely-known information.

Wikipedia _can_ be changed, but any user can look at the changelog to see what was changed when. If some troll keeps changing the page, it will be obvious.

Wikipedia is probably the best one-stop source for information anywhere on the internet. It's a great source, but sometimes you have to chack the references and/or history if something seems off kilter.
Posted by kcougar52 14 years ago
Just to let everyone who reads this know, wikipedia is not a source. it can be easily changed and even if it is fixed, it can be changed again.
Posted by beem0r 14 years ago
Just as infinity is literally not a real number, neither is an infinitesimal.

You say:
infinity = 1/infinitesimal

We are talking about _REAL_ numbers. Not surreal numbers, not hyperreal numbers, not extended real numbers. We're talking about reals here.

Neither infinity or infinitesimals are reals.

Infinitesimals are treated as 0, infinities are treated as undefined. That is how the real numbers treat these two concepts.This is why you have never gotten either of these as an answer to a problem, nor can they be input into equations. As you yourself said,

"it is well known that since infinity is not a real number, then algebraic operations break down with it. Similarly, since an infinitesimal is not a real number then algebraic operations break down with it as well."

Get that? NOT a real number. It is 0 if we're using reals.
The difference between the two is 0.

Also, 1.9... is not an infinitesimal. It is a real number, so there is no reason one cannot use arithmetic on it. Therefore, the proof given by Pro in round 2 is completely valid.

I really thought that would be enough, but at least now you can't claim that I should 'wait until I graduate college,' since it's held by mathematicians that .999...=1.
Posted by kennard 14 years ago
Well, I read the wikipedia link that says that 0.999 repeating equals 1 and I think it is wrong. I know that this sounds like a big claim but I can back this up.

Zero is a real number, all mathematical operations can be performed with it (except dividing by zero).

Some people might think that 1/0 equals infinity but this is wrong, 1/0 is undefined.
(Reference: http://en.wikipedia.org...).

Infinity is not a real number, it is just a symbol used to indicate that the real numbers grow unbounded. In other words, infinity is a symbol that indicates that the real numbers will keep on going forever.

Most people don't have much trouble accepting this, but they have trouble accepting the fact that there are infinitely many numbers between 0 and 1. So how do we represent this type of infinity? We represent it with infinitesimals. The symbol for infinity and the symbol for infinitesimal are related in the following way:

1/infinity = infinitesimal

In other words, in the same way that infinity represents how real numbers get larger and larger forever, infinitesimals represent how two real numbers get closer and closer forever. Let me rephrase this statement because it could be misinterpreted: In the same way that infinity represents how you can always get a number that is larger, an infinitesimal represents how you can always get a number that is closer.

Also, it is well known that since infinity is not a real number, then algebraic operations break down with it. Similarly, since an infinitesimal is not a real number then algebraic operations break down with it as well.

Since

infinity = 1/infinitesimal

then we can conclude that an infinitesimal can never equal zero, because then we will end up with the equation that is undefined, namely:

infinity = 1/0

So 2 - 1.999... = infinitesimal
Hence, 1.999 repeating represents how you can always get a number that is closer to 2, but it itself is not equal to two.
Posted by Rob1Billion 14 years ago
Kennard, his link says specifically that 0.999... is equal to 1, and that it is a long-held belief by math scholars. I would think that the matter is settled then?
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