Why is the Multiplicative Identity Positive and Not Negative?

[Jonny_trigonometry]

yes, I understand that I'm not talking about ordinary multiplication on the reals as the rest of the world knows it.

hmmm... ok, I'm startin to get it... but! what if -1 is the multiplicative identity?

how far back do the rules go? does it stop at the multiplicative identity? or is there a reason that the MI must be positive 1?

 

[matt grime]

There is a reason that 1 is the mult ident and that is becaue the operation is * the multiplcation operator as we know it where n*m means add m up n times (n, m are positive integers) and which is extend to the rest of the integers as we invented them.

The element that is the identity with respect to some opereation is dependent on the operation.

can we talk about addition since that is simpler?

Take Z the integers with the usual operations of addition denoted as + , then defnie a new opertaion & where

x&y=x+y-1

then -1 is the identity with respect to this "addition". See, it can be done, but you are attempting to think of our declaration of identities (an inverses) as independent of an operation.

 

[Zurtex]

yes, I understand that I'm not talking about ordinary multiplication on the reals as the rest of the world knows it.

hmmm... ok, I'm startin to get it... but! what if -1 is the multiplicative identity?

how far back do the rules go? does it stop at the multiplicative identity? or is there a reason that the MI must be positive 1?

The multiplicative identity is defined such that if we let e be the multiplicative identity:

e*x = x = x*e

Just as the additive identity, say f, is defined as:

f+x = x = x+f