Schoenfeld wrote:
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?!


[IGNORE PREVIOUS POST - SYMBOLS GOT MIXED UP DUE TO CARELESS EDITING]


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 N is the set of
negative integers.

Where does 0 lie in here?

Well if it lies in BOTH P and N 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 N). 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, ....