ℚ − ℤ
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@boomzilla said:
You're leaving stuff out, so I'm not sure if I'm reading your mind correctly, but it's totally possible to have two series have the same limit where the series are different. Since I didn't cite any wiki articles, I don't even know which one you're talking about.
Sorry, I didn't even notice that those were google links and clicked through on muscle memory. That's basically what I'm saying though, the series implied by 1.̅0 and 0.̅9 are distinct, but 1.̅0 and 0.̅9 are numbers defined as the limit of those series, which is why they are equal.@boomzilla said:
Did you know that glass is a super cooled liquid?
I thought that was an urban legend?
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@Faxmachinen said:
@boomzilla said:
You're leaving stuff out, so I'm not sure if I'm reading your mind correctly, but it's totally possible to have two series have the same limit where the series are different. Since I didn't cite any wiki articles, I don't even know which one you're talking about.
Sorry, I didn't even notice that those were google links and clicked through on muscle memory. That's basically what I'm saying though, the series implied by 1.̅0 and 0.̅9 are distinct, but 1.̅0 and 0.̅9 are numbers defined as the limit of those series, which is why they are equal.1.0 and 0.9 don't imply any series. They are equal because they are the same number. Quirks of base 10 representation of numbers give us two representations of the same number. I think you're referring to the sequence of digits, which is generally a different concept than "series" or "sequence" in mathematics. It may seem like I'm being a pedantic dickweed about this, but that's really the only way to be about math.
@Faxmachinen said:
@boomzilla said:
Did you know that glass is a super cooled liquid?
I thought that was an urban legend?You're wrong!
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@boomzilla said:
1.0 and 0.9 don't imply any series. They are equal because they are the same number. Quirks of base 10 representation of numbers give us two representations of the same number. I think you're referring to the sequence of digits, which is generally a different concept than "series" or "sequence" in mathematics. It may seem like I'm being a pedantic dickweed about this, but that's really the only way to be about math.
Exactly. In this case it's important to be a pedantic dickweed about it, because the misconception of 0.̅9 being a series or other kind of process is the source of the whole confusion.
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@boomzilla said:
@Faxmachinen said:
@boomzilla said:
Did you know that glass is a super cooled liquid?
I thought that was an urban legend?You're wrong!
Did you know that granite is a super cooled liquid?Did you know that oxygen is a super heated liquid?
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@PJH said:
@rad131304 said:
Crap - see what happens when you don't use that degree you paid so much money for?@theheadofabroom said:
Oh look - someone's wrong on the internet again.@boomzilla said:
Technically, no it isn't 1, .@theheadofabroom said:
The set of rational numbers is a subset of the set of real numbers, and both sets are infinite, but the set of reals has to be bigger, as there would be something left over if you were to subtract the rationals from the reals.
Oh, I get it now. You were writing about ℚ and ℤ, but you were thinking about ℝ.
Did you know that 0.9 equals 1?
Well of course, as 0.9 can be thought of as notation for Σ(9E-x) for x between 1 and infinity, which is trivially 1
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@rad131304 said:
Crap - see what happens when you don't use that degree you paid so much money for?
Bah...you should be using radians instead.
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@boomzilla said:
@rad131304 said:
either way it's dimensionless ... which seems somehow fitting.Crap - see what happens when you don't use that degree you paid so much money for?
Bah...you should be using radians instead.
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@boomzilla said:
Did you know that glass is a super cooled liquid?
It's usually characterised as a "glass"; it's a solid, but not crystalline or a compound. They're disordered like a liquid, but the disorder is locked in place; glasses don't flow on any normal timescale. (Of course, under the right conditions then even crystalline/compound materials like rock can flow; that's how the Earth's mantle works.)
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@boomzilla said:
1.0 and 0.9 don't imply any series. They are equal because they are the same number. Quirks of base 10 representation of numbers give us two representations of the same number. I think you're referring to the sequence of digits, which is generally a different concept than "series" or "sequence" in mathematics. It may seem like I'm being a pedantic dickweed about this, but that's really the only way to be about math.
I vaguely remember that there's a consistent axiom system which makes them different, something to do with categorization of infitesimals.However, you're right that in any normal mathematics they're the same: pick any finite value you want, and you can find that the difference between 1.0 and 0.9 is smaller than that. Since you can't measure the difference between them, they're the same.
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@dkf said:
I vaguely remember that there's a consistent axiom system which makes them different, something to do with categorization of infitesimals.
?
ℚ − ℤ
