21-09-2002 09:17									#31
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quote: Originally posted by Frogger Man, the number 1 is the most interesting number out there.

I know 1 is the lonliest number. Does that count?

	21-09-2002 09:18									#32
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quote: Not necessarily. One could easily define a map between two sets as he wants to. I am not obliged to send 0.9999... to 1. In terms of cardinality, both sets have the same cardinal. There is a bijection between these sets (which does not send 0.999... and 1 to 1 because that would not be inyective)

You've now said the same thing 4 times. I agree that the sets are bijective, butread what I wrote: I said that the standard map is not a bijection.

	21-09-2002 09:22									#33
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quote: Well, in metrics spaces "closed" is weaker than "complete" (i.e., a space which is complete is closed), but you are right: both sets are closed (in fact, they are complete)

Under the standard metric for real numbers, all closed sets are complete (can't remember what this property is called).

	21-09-2002 09:25									#34
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Quoted from Frogger:
"You've now said the same thing 4 times. I agree that the sets are bijective, butread what I wrote: I said that the standard map is not a bijection."

I have to said this 4 times because you told me that I was "absolutely and fundamentally wrong here".
I thought that you were implying that the first thing was wrong, my excuses because of the misunderstanding.

	21-09-2002 09:28									#35
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I think you guys have just discouraged me from ever taking a course in number theory...

	21-09-2002 09:30									#36
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I responded too quickly. I scanned what you read, but the part about showing injection and surjection seemed badly written...and I assumed you were trying to prove that the standard map was bijection (especially since just in my last post I'd clarified what I was discussing)...

Let's please stop now...

Unless you can tell me what it's called to have the property that all closed subspaces are complete (like Rⁿ is...)

	21-09-2002 09:31									#37
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Number theory is different. This is Real Analysis (which number theory is based on, partly...)

	21-09-2002 09:32									#38
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Quoted from Frogger:
"Under the standard metric for real numbers, all closed sets are complete (can't remember what this property is called)."

That is because the real number system is complete and each closed set of a complete set is in itself complete.
But what about it?

	21-09-2002 09:33									#39
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Are you a math grad student or something, alofatti?

	21-09-2002 09:33									#40
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Is that case? I'm not sure (been 3 years since I did Analysis III...)

Let me think.

	21-09-2002 09:36									#41
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"Are you a math grad student or something, alofatti?"

No, I am not. I study Computer Science.

	21-09-2002 09:37									#42
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Which branch? Theory, software engineering, etc?

	21-09-2002 09:42									#43
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I like the way Squid put it, but it all seems like some sort of logical lapse.

I recognize Frogger's expertise, but I don't understand what he's saying. As I understand it 0.99... < 1, and if x<>y then x!=y.

Bear in mind I'm a history major, and make no claim that I'm correct in this. It just seems to make sense.

	21-09-2002 09:44									#44
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Felch: I'm the same way, I'm looking at it more from a strictly logical point of view instead of mathematical.

The best way I've had it explained is 1/3 is 0.3 repeating.
1/3 + 1/3 + 1/3 = 1, but if 1/3 is 0.3 repeating only, then 1/3 + 1/3 + 1/3 = 0.9 repeating. But it's not, it's 1.

	21-09-2002 09:44									#45
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That is a somewhat difficult question to answer since there are not branches of study over here.
We study general concepts from computer organization + theory + software engineering (aghh!!) and then we choose some some optative courses. That would, perhaps, qualify as specializations.
To be honest with you, I am not perfectly sure about what would be my specialization, I have taken extra courses in some logic themes, game theory and (very little) of computer graphics.

	21-09-2002 09:47									#46
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quote: Originally posted by alofatti Quoted from Frogger: "Under the standard metric for real numbers, all closed sets are complete (can't remember what this property is called)." That is because the real number system is complete and each closed set of a complete set is in itself complete. But what about it?

Just realised something...

This is not true for unbounded subsets of reals (including whole set).

Take x_n = n^1/2

This is Cauchy sequence but not convergent.

You require compactness on the part of the space to demonstrate that closed subsets are complete.

Then Bolzano-Weierstrass states that every bounded subset of reals is compact...

Compactness can be defined either sequentially (every sequence in set has a convergent subsequence) or topologically (every covering of the set has a finite subcovering)

Ah...the good old stuff is beginning to flow back now...

	21-09-2002 09:48									#47
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quote: Originally posted by alofatti That is a somewhat difficult question to answer since there are not branches of study over here. We study general concepts from computer organization + theory + software engineering (aghh!!) and then we choose some some optative courses. That would, perhaps, qualify as specializations. To be honest with you, I am not perfectly sure about what would be my specialization, I have taken extra courses in some logic themes, game theory and (very little) of computer graphics.

Ah, cool.
I'm doing the software engineering ( ) and computer graphics mostly, with a strong concentration in logic (5 logic courses in the Philosophy department, gotta love it).

Sounds like you're more into the theory aspect more? The math behind everything?

	21-09-2002 09:49									#48
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Asher I'm with you on that rejection of the .3333... times 3 equaling .9999.... The main problem with it is that it pretends that 1/3 = .33..., whereas .33... is just an approximation, I think. I'm probably wrong about that, but that's the problem I see with it.

Basically I was taught that the decimal places have meaning, and a number with zero in the ones place and nothing greater than that (e.g. 0.999...) is less than 1, and if numbers are greater than or less than each other, they are not equal.

I'm guessing Frogger is cross-posting with me some explanation as to why I'm wrong, so you all can probably disregard this.

	21-09-2002 09:51									#49
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quote: Originally posted by Felch X I like the way Squid put it, but it all seems like some sort of logical lapse. I recognize Frogger's expertise, but I don't understand what he's saying. As I understand it 0.99... < 1, and if x<>y then x!=y. Bear in mind I'm a history major, and make no claim that I'm correct in this. It just seems to make sense.

1-0.9 = 0.1

1-0.99 = 0.01

1-0.999 = 0.001

etc.

The numbers get closer and closer to 1, right (therefore the difference between 1 and them gets closer to zero). Now, real numbers have the nice property that if a doesn't equal b, a-b doesn't equal zero. So what does 1-0.999999.... equal? It's greaer than or equal to zero...but it's less than 0.1, and less than 0.01, and less than 0.001, and less than 0.0000000001, etc.

The only number that satisfies these requirements is 0. Therefore a-b = 0 and therefore a=b

Consider this your 30 second introduction to limits...

	21-09-2002 09:51									#50
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Quoted from Felch:
"I recognize Frogger's expertise, but I don't understand what he's saying. As I understand it 0.99... < 1, and if x<>y then x!=y."

Well, it is like you said it before, it is somewhat of a logical trickery. The thing is that 0.99... is not < than 1, IF we are talking about the real numbers.
You could actually redefine the meaning of "<" so that 0.999... is < than 1, but then you would be talking of a different number system which is, in some aspects, not "as good" as the real system. For example, it would allow for "lagoons" in the continuum, between 0.999... and 1 you would not have any number, something that does not happen in both the rational and real numbers.

	21-09-2002 09:54									#51
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quote: Originally posted by Felch X Asher I'm with you on that rejection of the .3333... times 3 equaling .9999.... The main problem with it is that it pretends that 1/3 = .33..., whereas .33... is just an approximation, I think. I'm probably wrong about that, but that's the problem I see with it. Basically I was taught that the decimal places have meaning, and a number with zero in the ones place and nothing greater than that (e.g. 0.999...) is less than 1, and if numbers are greater than or less than each other, they are not equal. I'm guessing Frogger is cross-posting with me some explanation as to why I'm wrong, so you all can probably disregard this.

1/3 is exactly equal to 0.33333...

But the way to show this is true in the first place is exactly the same as the way to show that 0.9999...=1

	21-09-2002 09:55									#52
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Are you guys the fellows thats been running the elections down in Florida?

They have trouble counting too.

One man, one vote. Or is that one man, .999999999~ votes?

	21-09-2002 09:55									#53
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Okay, but I was taught (In an American school) that limits are not reached. For example 1/x will approach zero, but never reach it.

It seems like this is all bullshit people make up so they can get research grants. Mathematicians need to grow some cajones and fess up that they're just bullshitting like we in the Humanities do.

Now I have to get to work writing up a proposal that will convince UMBC to send me to Amsterdam to research the effects of decriminalizing cannabis.

	21-09-2002 09:56									#54
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I'm going to start discussing the Baire Catewgory theorem soon...

I need to take a serious pure math course. It's been 1.5 years now, and I forgot how fun it is...

	21-09-2002 09:59									#55
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quote: Originally posted by Felch X Okay, but I was taught (In an American school) that limits are not reached. For example 1/x will approach zero, but never reach it. It seems like this is all bullshit people make up so they can get research grants. Mathematicians need to grow some cajones and fess up that they're just bullshitting like we in the Humanities do. Now I have to get to work writing up a proposal that will convince UMBC to send me to Amsterdam to research the effects of decriminalizing cannabis.

That's cool, Felch...but while 1/x may never equal 0, it approaches it in a way that mathematicians have taken to terming "the limit of 1/x as x approaches infinity is 0"

Same in the number problem. The limit of 0.9999... as the number of nines approaches infinity is 1. And since the decimal system allows infinite strings (for some very good reasons) then we're allowed to create a beast with an infinite number of 9s...even if we could never hope to actually write it down.

	21-09-2002 10:01									#56
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Aight, so saying .9...=1 is simply a shorthand?

	21-09-2002 10:07									#57
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Quoted from Frogger:

quote: Just realised something... This is not true for unbounded subsets of reals (including whole set). Take xn = n^1/2 This is Cauchy sequence but not convergent.

xn = n^1/2 is not a Cauchy sequence, it is not bounded (every Cauchy sequence is bounded).

	21-09-2002 10:11									#58
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As a practial matter if something is priced at $999.99 is that not $1000.00? Would you consider the two numbers different when you payed for something?

	21-09-2002 10:14									#59
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Yup...

Mixed up convergence and uniform convergence...



I'm not on the ball tonight.

	21-09-2002 10:14									#60
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quote: Originally posted by Frogger Yup... Mixed up convergence and uniform convergence... I'm not on the ball tonight.

We all love you regardless.