Subject: Re: [HM] A question around zero
From: John Harper (John.Harper@MCS.VUW.AC.NZ)
Date: Thu Jan 20 2000 - 18:06:18 EST
On Thu, 20 Jan 2000, Karen Dee Michalowicz wrote:
> Take the set of integers
>
> ..., -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
>
> if I start counting "every other number" beginning with an even number,
> such as,
> -8, -6, -4, -2,
>
> I come to zero and then go on to
>
> 2, 4, 6, 8, 10 , etc.
>
> Since zero is in this set of "every other number", it is even.
I would prefer to do it this way: there was a time when zero was unknown
(Dionysius Exiguus had never heard of it, so 1 BC was immediately
followed by 1 AD in his system, which is why this year is the last year
of the 20th century not the first of the 21st.) Someone, sometime,
somewhere in Asia (and [HM] reaches people who know a lot more about the
details than I do) realised zero would be useful and had to decide how
to think of it him/herself and how to describe it to other people.
When a new entity in mathematics is defined one must also define its
properties, and it's a good idea to make them consistent with the
properties of related entities if possible. In this case, nonzero
integers obey the rule (odd number - odd number = even number) and
1 - 1 = 0 implies that either 0 ought to be even or there will be an
annoying exception in that rule. (We put up with annoying exceptions
when we have to, as we do with zero in the case of division, where
x/y = z/y implies x = z unless y = 0.)
John Harper, School of Mathematical and Computing Sciences,
Victoria University, Wellington, New Zealand
e-mail john.harper@vuw.ac.nz phone (+64)(4)463 5341 fax (+64)(4)463 5045
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