From: vanjac12@... (Van Jacques)
Date: 2004-01-28T19:34:57+09:00
Subject: Re: ruby-math and "why is ** not abelian?"

"Josef 'Jupp' SCHUGT" <jupp@gmx.de> 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,*).

[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?

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.

Van

"Josef 'Jupp' SCHUGT" <jupp@gmx.de> 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