Re: ruby-math and "why is ** not abelian?"

In article <[e-mail]>, Van Jacques wrote:
>"Josef 'Jupp' SCHUGT" <[e-mail]> wrote in message news:<20040126221508.GD3659@jupp%gmx.de>...
>> Hi!
>> 
>> * Van Jacques:
>> > If + is commutative, and the successor operation to + is *, which
>> > is also commutative;
>> > (a*b = b*a), then why isn't a**b = b**a since ** is successor
>> > operation to * ?
>> 
>> Neither (Float, +) nor (Float, *) is a group. A non-group cannot be
>> an Abelian group.
>> 
>> Josef 'Jupp' SCHUGT
>
>Hi Josef,
>
>Why is " Neither (Float, +) nor (Float, *)" a group?
>
>Is it because of limits on Float? (I just read that all numbers in
>ruby are either int or float--I have never even seen the E-n, as
>1.0E5, notation in ruby, though Float(nul) = 0.0.) Are you speaking of
>the inability to rep very large and small floats,  (machines are not
>infinite), and/or round off error here?
>
>Mathematically, (though Z != ruby Integers, or any machine integers)
[...]
>(Z,+)  and (R,+) and (Q,+) are groups, though
>(Z,*) is not, because of no division, which is what leads to Q, so
>that
>(Q,*) is a group, and of course (R,*).
[...]

(Q, *) and (R, *) are not groups since the `0' element has no (multiplicative)
inverse.

(Q, +, *) and (R, +, *) are rings though.

>[a bunch of nonsense about iterated + and * deleted, as it made no
>sense]
>---------
>But _why_ is 3*2 = 2 + 2 + 2 = 2*3 = 3 + 3 ?
>
>Is it because a + b = b + a  that x*y = y*x?

No. We can have "non-commmutative rings" where x*y!=y*x for some x,y.
(Z, +, *) happens to be a "commutative ring".

See http://planetmath.org/encyclopedia/CommutativeRing.html

>I don't see it, and its starting to make my head hurt.
>
>I must admit I haven't thought about this and don't really have any
>insight into it--maybe its just the properties of numbers--just the
>way things are.
[...]

NB: Sorry everyone for off-topic comments. I will try not to post in this
thread again ;)

Van Jacques and others, http://math.wolfram.com/ and http://planetmath.org/
are useful.

-- 
Tim Sutherland <[e-mail]>
2004 SDKACM President
Software Developers' Klub - the University of Auckland ACM Student Chapter
http://www.sdkacm.com/