does .9 repeating equal one?

[SlowwHand]

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?

[KrazyHorse]

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.

[KrazyHorse]

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).

[alofatti]

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.

[Kirnwaffen]

I think you guys have just discouraged me from ever taking a course in number theory...

[KrazyHorse]

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...)

[KrazyHorse]

Number theory is different. This is Real Analysis (which number theory is based on, partly...)

[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?

[Asher]

Are you a math grad student or something, alofatti?

[KrazyHorse]

Is that case? I'm not sure (been 3 years since I did Analysis III...)

Let me think.

[alofatti]

"Are you a math grad student or something, alofatti?"

No, I am not. I study Computer Science.

[Asher]

Which branch? Theory, software engineering, etc?

[Felch]

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.

[Asher]

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.

[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.