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[Graknorke]

Ninjedit:
Fuck it, Vector said it better.

Infinities are dumb :v

[Putnam]

Yeah, you'd be very surprised at how awful people are with the whole concept. I only try to talk about it when I know someone is completely wrong because people like Vector have obviously done more actual work involving it.

Almost every confusion about infinity can be seen in arguments against the fact that 0.9999... (infinitely repeating) == 1, so if you ever want to see the kind of misconceptions that come up, look some of that up.

[Helgoland]

I usually blame that one on people not being taught the difference between concepts and notation.

[UXLZ]

A1, B1, A2, B2, A3, B2

= \

I still don't get how someone could mistake finite for meaning 'countably infinite' though...
I guess if English isn't the first language it's somewhat plausible, if still strange.

0.99999 infinitely repeating approaches one but doesn't equal it. It's just like an asymptote, right?

[Graknorke]

0.(9) is equal to 1.

[Vector]

Ninjedit:
Fuck it, Vector said it better.

I prefer Helgoland's exposition, myself

Yeah, you'd be very surprised at how awful people are with the whole concept. I only try to talk about it when I know someone is completely wrong because people like Vector have obviously done more actual work involving it.

I'd expect this from a neophyte taking their first proof-based course, but this is a second-semester graduate math course at UC Berkeley :/

A1, B1, A2, B2, A3, B2

= \

It's a repeating sequence

I still don't get how someone could mistake finite for meaning 'countably infinite' though...

We were re-introducing the measure-theoretic construction of the integral and I guess he left his smart pants at home that day. Just one of those weapons-grade math errors.

0.99999 infinitely repeating approaches one but doesn't equal it. It's just like an asymptote, right?

I thought you were talking about a number? Who introduced sequences into the argument?

[i2amroy]

0.99999 infinitely repeating approaches one but doesn't equal it. It's just like an asymptote, right?

0.(9) is equal to 1.

This right here. It has to do with the fact that there is no such number as .000000....1, so since you can't put another number in between .9999999... and 1 they must be equal.

[flabort]

ARRG.
FIX THE GODAMN TOASTER I AM THIS CLOSE TO QUITTING AND STOP PATRONIZING ME YOU WERE COMPLAINING ABOUT THAT MESS BUT WHEN I SAID I DID IT YOU COMPLETELY CHANGED YOUR TUNE WHAT MAKES ME A BETTER EMPLOYEE THAN ANYONE ELSE?

Sorry. It's just that I know I'm doing a horrible job and nobody seems to be able to see that.

[Putnam]

0.99999 infinitely repeating approaches one but doesn't equal it. It's just like an asymptote, right?

0.(9) is equal to 1.

This right here. It has to do with the fact that there is no such number as .000000....1, so since you can't put another number in between .9999999... and 1 they must be equal.

Rather more rigorous is real numbers are dense (between any two non-equal real numbers, there are an infinite amount of real numbers) and there are no real numbers between 0.(9) and 1, so they must be the same number in the real number system.

[UXLZ]

I may be misunderstanding things greatly, but couldn't you, through application of that logic, come to the conclusion that all real numbers are equal?

[Vector]

0.99999 infinitely repeating approaches one but doesn't equal it. It's just like an asymptote, right?

0.(9) is equal to 1.

This right here. It has to do with the fact that there is no such number as .000000....1, so since you can't put another number in between .9999999... and 1 they must be equal.

Rather more rigorous is real numbers are dense (between any two non-equal real numbers, there are an infinite amount of real numbers) and there are no real numbers between 0.(9) and 1, so they must be the same number in the real number system.

You should probably note that it doesn't make sense to say such and such set is "dense." It can only be dense within the context of another set, i.e. the rationals are dense in the reals because the reals are their topological closure. I think you're thinking instead of the property of the continuum (that between any two reals, there's another real number).

My favorite proof of this property is observing that the sequence .9, .99, .999, etc. is a Cauchy sequence tending to 1 and .(9), and then remembering that the reals are complete.

I may be misunderstanding things greatly, but couldn't you, through application of that logic, come to the conclusion that all real numbers are equal?

Only if you can demonstrate that there's no number between any two real numbers, which is axiomatically false.

[Tawa]

Er, how doesn't 0.00000...1 exist? I am confused. :|

[Vector]

Er, how doesn't 0.00000...1 exist? I am confused. :|

.(0)1 is not a real number because you cannot have an infinite number of zeroes, ended by a 1. There's another set of numbers that have that property (I don't remember what they're called anymore... something something set theory transfinite blah blah blah), but the real numbers do not.

[Tawa]

.(0)1 is not a real number because you cannot have an infinite number of zeroes, ended by a 1.

Ok, but... what exactly is the logic behind that bit?

[Vector]

.(0)1 is not a real number because you cannot have an infinite number of zeroes, ended by a 1.

Ok, but... what exactly is the logic behind that bit?

Because in this case, the infinite we assume that we're talking about--by definition--is the countable kind of infinite we were all talking about earlier. So, it's not like "no, you inherently cannot have infinite zeroes and then a one," but rather "we define the real numbers to act in such and such a way because they have certain nice properties."