Invisible wrote:
>>> OK, so I can't spell very well. I think we've established that one. It's
>>> actually a miracle I can read or write at all! :-P
>> 
>>   Given the subject of this thread, shouldn't you try to better yourself
>> at this instead of just using the "I can't spell very well" defense? ;)
> 
> And who says I'm not trying?
> 
> Either way, laughing at me isn't helping anything, is it? :-P
> 
You have an important point there. What if those kids who say "I'm not good
at math" were once laughed at when giving a silly math answer?

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scott wrote:
> It makes things later in life, assuming you go on with studying maths, a
> lot
> easier.  At university I used to cringe at the people who used their
> calculators so often to do what they really should be doing in their head.
> When you are tackling a large problem it is a huge advantage do be able to
> do at least the basics very quickly in your head.

And it's very easy to lose practice. I used to be good at maths at school.
Now (like 2 years later), I sometimes take a few seconds to solve those
silly "6+7" anti-spam tests. Probably I wouldn't take too much time to get
my brain in practice again, but...

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Invisible wrote:
>>> It is not immediately clear to me that the quotient of two 
>>> polynomials is necessarily a polynomial.
>>
>> Nope, same as dividing two integers doesn't always give an integer 
>> answer, you sometimes end up with fractional parts in the answer.
> 
> Right. So there should also be a polymonial modulus operator?

Indeed there is.  It is a useful operation in abstract algebra, 
particularly in extending sets of numbers so that they contain elements 
with useful algebraic properties.  For instance, if you consider 
polynomials over the real numbers modulo X^2-1, you get the same 
structure as the complex numbers (you can do a similar thing for 
quaternions).

In terms of real-world uses, iirc the polynomial modulus is valuable in 
in information coding.  For example modular multiplication of 
polynomials is a step in the AES cypher and a CRC checksum is 
essentially a polynomial modulus.

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>> Right. So there should also be a polymonial modulus operator?
> 
> Indeed there is.

Ooo...

> For instance, if you consider 
> polynomials over the real numbers modulo X^2-1, you get the same 
> structure as the complex numbers (you can do a similar thing for 
> quaternions).

Aren't quaternions non-associative?

> In terms of real-world uses, iirc the polynomial modulus is valuable in 
> in information coding.  For example modular multiplication of 
> polynomials is a step in the AES cypher and a CRC checksum is 
> essentially a polynomial modulus.

...and *this* is why I can't implement AES or CRC.

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Invisible wrote:
>> For instance, if you consider polynomials over the real numbers modulo 
>> X^2-1, you get the same structure as the complex numbers (you can do a 
>> similar thing for quaternions).
> 
> Aren't quaternions non-associative?

Non-commutative you mean, but yes.  The quaternion representation using 
modular polynomials considers polynomials of two variables (x and y) and 
real coefficients where multiplication between x and y is 
non-commutative.  So the polynomials inherit their non-commutativity 
from their variables.

Note that it's not actually important to say what sorts of numbers x and 
y would actually represent -- we're just dealing with polynomials in 
their purest form, where *all* we know about x and y are they have 
operators like addition, subtraction, and multiplication which satisfy a 
few basic mathematical properties: 
http://en.wikipedia.org/wiki/Ring_(mathematics)

> ...and *this* is why I can't implement AES or CRC.

It's not so difficult as you might think, it's just takes some getting 
used to the math.  If you're interested in the implementation rather 
than the math end of things, Wikipedia has a typically verbose article 
on the subject: http://en.wikipedia.org/wiki/Computation_of_CRC

There's also some C code dealing with the polynomials used in AES at the 
bottom of this page: http://en.wikipedia.org/wiki/Finite_field_arithmetic


Also forgot to mention that (if I recall correctly) ideas related to 
this play an important role in some of the operations in computer 
algebra systems.

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>> Aren't quaternions non-associative?
> 
> Non-commutative you mean, but yes.

Oh. Maybe I'm thinking of hypercomplex numbers then? I recall that it's 
not possible to generalise the complex field to 3 or 4 dimensions 
(although I have no idea why), so all the generalisations people have 
come up with fail to be fields.

By complete coincidence, I spent my afternoon today reading about magmas 
and groups and rings and quasigroups and semigroups and groupoids and 
rings and skew rings and splitting fields and fields of factors and 
polynomial rings and that whole zoo of other things... Apparently 
"vector field" doesn't mean what you'd expect at all! :-D

I've spent a while looking at group theory. It's not "hard", it's just 
that there are lots and lots of terms to learn, mostly with unintuitive 
names, and the whole system of groups constructed from groups 
constructed from groups gets confusing quite fast. Similarly, the stuff 
with rings and fields and so forth isn't "hard", there's just lots of 
stuff to remember...

> Also forgot to mention that (if I recall correctly) ideas related to 
> this play an important role in some of the operations in computer 
> algebra systems.

Mathematica directly uses the commutivity and associativity of various 
operators, not to mention distributivity. Indeed there are built-in 
methods for telling Mathematica that your new function is associative 
and/or commutative. (Distributive requires explicit transformation rules...)

-- 
http://blog.orphi.me.uk/
http://www.zazzle.com/MathematicalOrchid*

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Orchid XP v8 wrote:
> I've spent a while looking at group theory. It's not "hard", it's just 
> that there are lots and lots of terms to learn, mostly with unintuitive 
> names, and the whole system of groups constructed from groups 
> constructed from groups gets confusing quite fast. Similarly, the stuff 
> with rings and fields and so forth isn't "hard", there's just lots of 
> stuff to remember...

Yeah, once you get used to it it's not to bad.  I think that proving 
things in it is a bit trickier though, as it seems to involve deep leaps 
of logical intuition more than other areas such as analysis.  It's sort 
of fun like that though, as you can actually do interesting math without 
having to acquire a ton of background knowledge.


>> Also forgot to mention that (if I recall correctly) ideas related to 
>> this play an important role in some of the operations in computer 
>> algebra systems.
> 
> Mathematica directly uses the commutivity and associativity of various 
> operators, not to mention distributivity. Indeed there are built-in 
> methods for telling Mathematica that your new function is associative 
> and/or commutative. (Distributive requires explicit transformation 
> rules...)


bases (a useful took for computer algebra systems), but indeed being 
careful about how you deal with the basic algebraic properties like 
commutativity and associativity is more fundamental to such programs.

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Orchid XP v8 wrote:
> By complete coincidence, I spent my afternoon today reading about magmas
> and groups and rings and quasigroups and semigroups and groupoids and
> rings and skew rings and splitting fields and fields of factors and
> polynomial rings and that whole zoo of other things... Apparently
> "vector field" doesn't mean what you'd expect at all! :-D

And I've yet to figure out how to solve t out of

(X0 + Xd * t - Xc)^2 + (Y0 + Yd * t - Yc)^2 + (Z0 + Zd * t - Zc)^2 = Sr^2

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