For the discussion of math. Duh.

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snowyowl
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I like that you can solve x2 = x in the 10-adics and get four answers.

0, 1, ...212890625, and ...87109376.
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Spambot5546
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gmalivuk wrote:
Spambot5546 wrote:But saying "we can extrapolate from this and apply a finite value to that divergent series" isn't the same as "this series actually adds up to that". Any convergence test and even the definition of the limit of a series say it doesn't. Common sense does, too, but common sense has no place in mathematics.
The only reason you think common sense applies in the case of convergent series is because you've had enough exposure to standard definitions that it feels natural to you. But just look at the vehemence with which people claim 0.9999... < 1 to see how uncommon that sense really is.

Oh, lord, I used to be that guy. Boy, I hope that experience isn't in any way reflective of this one.
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gmalivuk
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The difference as I see it is that the people claiming 0.999... isn't 1 are totally comfortable writing and mathematically manipulating something like "0.999..." without understanding what that string of characters is defined to mean by standard mathematics.

Folks who don't like divergent series, at least, tend to avoid treating "1+2+3+..." as any particular numericalvalue in the first place. It is different to deny that a thing has a sensible meaning at all than it is to deny that it means what is logically implied by definitions you've already accepted.
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mike-l
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snowyowl wrote:I like that you can solve x2 = x in the 10-adics and get four answers.

0, 1, ...212890625, and ...87109376.

Normally we only use prime p, specifically to avoid things like this (4 solutions to a second degree polynomial implies that there are zero divisors)

The 10-adics are a ring but not a field. Another common example of rings are nxn matrices, which also have more than 2 idempotents.
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Wait, too many solutions to a polynomial implies there are zero divisors? I don't think that's true... what about the quaternions, where there are like 6 solutions to x^2+1=0?
Let's have a fervent argument, mostly over semantics, where we all claim the burden of proof is on the other side!

jestingrabbit
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MartianInvader wrote:Wait, too many solutions to a polynomial implies there are zero divisors? I don't think that's true... what about the quaternions, where there are like 6 solutions to x^2+1=0?

Well, it implies the violation of some field axiom or another.
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cyanyoshi
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There are indeed infinitely many solutions to x2 + 1 = 0 in the quaternions, namely x = b*i + c*j + d*k where b2 + c2 + d2 = 1.

Back to the point, how big of a problem is it that there are zero divisors in the 10-adic integers, if we're looking to set up equivalence relations between the 10-adics and the real numbers? Is there a reason we shouldn't say that ...212890625 equals 1 (or 0), like how 0.9999... equals 1? This reminds me a little of the split-complex numbers, where you could replace every instance of j with ±1 in many expressions, but you would lose some interesting underlying structure in the process.

arbiteroftruth
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If ...212890625=1, then ...212890626=2, and therefore 4=...2128906262=...212890626*2. This is not the case, so ...212890626 clearly behaves differently from 2, so it can't be equal to 2, and by extension ...212890625 =/= 1.

Likewise, if ...212890625=0, then ...212890626=1, and should be idempotent. It isn't, so ...212890625 =/= 0.

mike-l
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MartianInvader wrote:Wait, too many solutions to a polynomial implies there are zero divisors? I don't think that's true... what about the quaternions, where there are like 6 solutions to x^2+1=0?

Sorry, I implicitly meant for commutative rings. The general statement is, as JR says, that a degree n polynomial has at most n roots in a field, so if there are more it isn't a field.

Cyanyoshi

Well the 10-adics are, as above, not a field. There are zero divisors and numbers without inverses. Even if we look at the p-adics, with p prime so we have a field, this is distinct from the real numbers.
addams wrote:This forum has some very well educated people typing away in loops with Sourmilk. He is a lucky Sourmilk.