Subject: Re: [HM] A question around zero
From: Ed Wall (ewall@umich.edu)
Date: Fri Jan 21 2000 - 23:30:24 EST
Unfortunately Ed Wall did not ask a question about "the evenness or
oddness of zero"; however, he has received a number of 'answers'
about this. My question, slightly edited, is repeated below just in
case anyone wants to pursue it:
> My impression is that conceptually even and oddness (in a few
> variations) substantially predated zero and the negative integers.
> So my question is: it known when and how evenness and oddness were
> extended from the natural numbers to the integers? There seems to
> have been substantial dislike of the negative integers (one source
> mentions into the 1800s) so an early extension might have been to
> the positive integers (although zero being even seems most useful
> if the negative integers are included).
I have found the coincident discussion of the negative integers quite
interesting in regard to this question - especially the bit about the
number line in the work of Wallis and Newton taken together with
Karen Michalowicz's mention of the alternation of odd and even. I
chanced to look at a number theory book from the early part of the
1900s tonight and noticed that evenness and oddness were defined by
alternation first on the natural numbers and then 200 'sections'
later on the integers.
Anyway, thanks for the responses. I am at a bit of a standstill
right now since most texts (historical and mathematical) take the
extension of evenness and oddness to the integers more or less for
granted. Is it worth looking at Brahmagupta's Brahmasphuta Siddhanta
and, if so, is there a reasonable translation (my Sanskrit is quite
rusty - 20 years some)?
Ed Wall
Jean-Luc Gautero wrote:
> Ed Wall asked a question about the evenness or oddness of zero.
> I don't understand the question, because I thought that "even"
> and "odd" were the same as, in french, "pair" and "impair": that
> is, that an even number was a multiple of 2, and an odd number
> not a multiple of 2 (2n+1, where n is an integer): in that case,
> 0 is clearly even: 0=2*0. So, how do you define an even number?
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