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*To*: Aubrey Jaffer <agj@xxxxxxxxxxxx>*Subject*: Re: My ideas about infinity in Scheme (revised)*From*: Alex Shinn <alexshinn@xxxxxxxxx>*Date*: Mon, 23 May 2005 15:30:02 +0900*Delivered-to*: srfi-70@xxxxxxxxxxxxxxxxx*Domainkey-signature*: a=rsa-sha1; q=dns; c=nofws; s=beta; d=gmail.com; h=received:message-id:date:from:reply-to:to:subject:in-reply-to:mime-version:content-type:content-transfer-encoding:content-disposition:references; b=ism2I002ZT2FbsZMPjJZYQW6JnaG50K75aRW3FD2FqHloUj4KehcYFQS3EprDR3x60gaPXaWV/umtQcjK9KH7jbEkx9NVIhcSsJRUVaNzd8YN7tk9VAE3pJtlODscO6Z0RFHYjcPB+UiflSsfwC88iXLO8VvIzCUyC7epzCmemg=*In-reply-to*: <20050523034054.D625F1B77B4@xxxxxxxxxxxxxxxx>*Reply-to*: Alex Shinn <alexshinn@xxxxxxxxx>*Resent-date*: 23 May 2005 13:03:42 -0400*Resent-from*: agj@xxxxxxxxxxxx*Resent-message-id*: <20050523170342.DBD761B77B4@xxxxxxxxxxxxxxxx>*Resent-to*: srfi-70@xxxxxxxxxxxxxxxxx

On 5/23/05, Aubrey Jaffer <agj@xxxxxxxxxxxx> wrote: > | Date: Sun, 22 May 2005 20:46:53 +0900 > | From: Alex Shinn <alexshinn@xxxxxxxxx> > | > | On 5/22/05, Aubrey Jaffer <agj@xxxxxxxxxxxx> wrote: > | > > | > In hundreds of years of using rational numbers, mathematicians have > | > not discovered 1/0 to be a useful extension to the rational numbers. > | > | Well, mathematically 1/0 isn't real or complex either, as it doesn't > | obey the properties of a field > > 1/0 and -1/0 can be added in a way that preserves the total ordering > of the real numbers. But closure is lost because (+ 1/0 -1/0) is not real (at which point it's no longer a group, much less field). Introduction of 0/0 to regain closure breaks the inverse element property of groups (the identity element is still 1 but 0/0 has no inverse). > | and breaks the fundamental theorem of algebra. > > In which circumstances does it do that? Is it only when 1/0 or -1/0 > is a coefficient in a polynomial? Yes, such a polynomial would have no zeros. > Many ideas about efficiency have been invalidated by the growth of > instruction speed far outstripping growth in L1 cache size. An > article about this is: > <http://swiss.csail.mit.edu/~jaffer/CNS/interpreter-speed> An interpreter is still an order of magnitude slower than a native compiler. Regardless, if there are situations (either explicit from the use of real? or implicit via method polymorphism) where in the middle of a loop you must check against the IEEE-754 infinities, then you will suffer a serious performance loss. -- Alex

**Follow-Ups**:**infinities reformulated [was Re: My ideas about infinity in Scheme (revised)]***From:*Aubrey Jaffer

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