Nth Complexity wrote:
Dirk Van de moortel wrote:
By the way, zero is usually taken to be both positive and
negative.

Hahahahahahahahahahahahahahahahahahahahahahahahaha hahahahahahahaha!
And you expect to teach OTHERS?!

I'm not certain Dirk is wrong. Most websites (like Wolframs) imply that
0 is neither positive or negative, but I don't think it's possible to
prove this (at least I can't, perhaps someone else can comment).

If you consider commutative rings (e.g. integers) or ordered fields
(e.g. reals) there is the additive identity axiom:

There exists y such that for all x, x + y = x

This is entirely insufficient to imply a single unique additive
identity y, although this seems to be the universal interpretation. 'y'
is usually called zero and given symbol 0.

Considering the integers Z, you can split Z into two sets P and N such
that:
1. For all x in Z, P contains x iff N contains -x; AND
2. P is closed under addition and multiplication.

Trivially, P is the set of positive integers and R is the set of
negative integers.

Where does 0 lie in here?

Well if it lies in BOTH P and Q there are no contradictions at all. But
this implies that 0 occurs twice in the integers (otherwise it couldn't
be placed in any of P or Q). Again, this is not strictly prohibited by
additive identity axiom. You could say there are two additive
identities 0+ and 0-, positive and negative respectively. A cursory
analysis their arithmetic reveals no axiomatic contradictions.

Based on this, I would need to say that 0 by itself does not exist.
Rather you have positive 0+ and negative 0-.

Is there an error here?


REMARK: The widely used terminology related to all this is:
"Positive integers" = 1,2,3, ...
"nonnegative integers" = 0, 1, 2, 3, ...
"negative integers" = -1, -2, -3, ...
"nonpositive integers" = 0, -1, -2, -3, ....





-- Nth Complexity --
-- Have A Nice Day! --
http://www.insurgent.org/~kook-faq/s...query=Uncle+Al
"The teaching of science and mathematics must be purged of its
authoritarian and elitist characteristics, and the content of these
subjects enriched by incorporating the insights of the feminist,
queer, multiculturalist and ecological critiques." -- A.D.S.





--
Sent by nth_complexity from yahoo included in com
This is a spam protected message. Please answer with reference header.
Posted via http://www.usenet-replayer.com