aguacate wrote:EDIT: Alright, what I'm trying to say is that it would be helpful to define j a little better. I'm assuming that you meant 0 is not an element of the Joshua system, but this was not made clear.
Yeah, that's what I meant. Basically, j fills in the role of zero, but isn't exactly the same.
aguacate wrote:Even if this system turns out to be consistent and what not, while interesting, I don't see how it pertains to the original question if you are going to exclude 0.
Well, it's not fully what I'm asking for, since I realized from the posts and working it out that the reason I can never have a proper 0/0 is that using zero removes information, that can't be regained. So if I replace 0 with something that does retain the information, then I can do stuff like (2-2)/(1-1). Which would equal 2.
buttons wrote:1. a + b = b + a
2. a + (b + c) = (a + b) + c
3. 5 - 3 = 2
Confirm 1 and 2, deny 3 (I think), as 5-3 = 2 +3j.
Thrown in as an axiom is OK. The problem is proving 2j != j by throwing in an axiom. First come the axioms, next come the proofs. If you are modifying the number system as you go, fine, but I got the impression that it was a defense of the current system rather than a reworking.
I haven't written out anything formally yet, since I haven't done this type of math in at least five years, and I'm a bit rusty. Plus as I said I never had defined a number system before.
jestingrabbit wrote:I think you'd probably actually want (a+bj) + (c+dj) = (|a| + |b| + |d|)j, or I could prove that j = 2j I think. And then you could use that for a, b, c and d complex too.
You're right, I do need that.
jestingrabbit wrote:However, if you start with a+b=a+c, then to get where you want to, you want to add the additive inverse of a to both sides ie
But this is out in your system. For instance, say we want to look at the equation x+2 = 5. Then
x + 2 = 5
(x + 2) + (-2) = 5 + (-2)
x+2j = 3
So you end up with b + aj = c + aj. What's wrong with that?
jestingrabbit wrote:but then we're stuck, there's no way we can get x by itself. x=3 is still a solution using the equation for addition you agreed with earlier, but... I think there are serious problems with what you want, but there's little that we actually gain by adopting it.
Hey, I'm still working on defining it, and finding out (and proving if so) that it is internally consistent. Convincing others to use it, or even finding out if there's any use for it comes later
You ought to read up on the field axioms, think about how we manipulate formulae and consider the pros and cons of not working with a given axiom or collection of axioms.
I'm finding it difficult to grok field axioms from the online sources. Do you have any online or in print advice on things to learn from?
Edit: and btw, we haven't even got up to multiplication yet, and when we do, J will be getting a whole lot bigger. What is j*j?
I'm gonna go with this for now:
J*j = (1-1)*(1-1)
= 1 -1 -1 +1
= 2 - 2