From: timsuth@... (Tim Sutherland) Date: 2004-01-28T20:54:59+09:00 Subject: Re: ruby-math and "why is ** not abelian?" In article <70ae81fd.0401280234.cb1f380@posting.google.com>, Van Jacques wrote: >"Josef 'Jupp' SCHUGT" 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 2004 SDKACM President Software Developers' Klub - the University of Auckland ACM Student Chapter http://www.sdkacm.com/