I like that you can solve x^{2} = x in the 10adics and get four answers.
0, 1, ...212890625, and ...87109376.
Bad math
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Re: Bad math
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Re: Bad math
gmalivuk wrote: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.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.
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 gmalivuk
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Re: Bad math
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.
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.
Re: Bad math
snowyowl wrote:I like that you can solve x^{2} = x in the 10adics 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 10adics 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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 MartianInvader
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Re: Bad math
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!
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Re: Bad math
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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Re: Bad math
There are indeed infinitely many solutions to x^{2} + 1 = 0 in the quaternions, namely x = b*i + c*j + d*k where b^{2} + c^{2} + d^{2} = 1.
Back to the point, how big of a problem is it that there are zero divisors in the 10adic integers, if we're looking to set up equivalence relations between the 10adics 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 splitcomplex 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.
Back to the point, how big of a problem is it that there are zero divisors in the 10adic integers, if we're looking to set up equivalence relations between the 10adics 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 splitcomplex 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.

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Re: Bad math
If ...212890625=1, then ...212890626=2, and therefore 4=...212890626^{2}=...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.
Likewise, if ...212890625=0, then ...212890626=1, and should be idempotent. It isn't, so ...212890625 =/= 0.
Re: Bad math
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 10adics are, as above, not a field. There are zero divisors and numbers without inverses. Even if we look at the padics, with p prime so we have a field, this is distinct from the real numbers.
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