Beyond + and -
#18
I tried to think on something similar recently: I was thinking about a number "line" that is 2-dimensional, but I couldn't imagine anything that wasn't equivalent to the complex plane. The more I think about them, the more I am made uneasy by the idea of negative numbers: Why should a "sign" flip at some arbitrary point on the number line? Why can't the numbers just decrease in value without ever reaching zero? Are the properties of our number system determined by the way that humans naturally think, or is the converse true? The more I try to push beyond the current number system, the more I realize that I am a slave to ideas like plus and minus, for whatever reason.

Regarding the topic of the OP, if we can think of + and - as states of a number, then perhaps a number could be a superposition of those 2 states (a.k.a. the physicist's approach). This can perhaps be expressed by replacing every single-valued number with a 2D vector, where the i and j components correspond to the + and - components of the number. Then we would have to redefine the basic arithmetic operations on these 2D numbers. I am not sure what would happen if the value of one of the components changes sign; Does it somehow contribute to the other component? How do you express vector operations with vectors for which one component can spill over into another?

Edit: So I think that the 3-sign system could be expressed in terms of complex numbers. Say we define a complex number, c, with a fixed magnitude. Then decreasing the real component requires an increase in the imaginary component. Now throw in an additional type of imaginary number (call it "j") and thus complex numbers give us the needed functionality if we interpret +i as -1 and +j as §1 (where I'm using "§" to represent the 3rd sign). In this system +1, -1, and §1 point in different directions. Sure + and - are no longer opposite directions, but I'm not sure you can have opposite directions in a 3-sign system.


Messages In This Thread
Beyond + and - - by Benny - 08/16/2012, 05:35 PM
RE: Beyond + and - - by JmsNxn - 08/17/2012, 01:25 AM
RE: Beyond + and - - by Benny - 08/17/2012, 11:58 AM
RE: Beyond + and - - by tommy1729 - 08/17/2012, 03:18 PM
RE: Beyond + and - - by Benny - 08/19/2012, 05:24 PM
RE: Beyond + and - - by tommy1729 - 08/21/2012, 11:04 PM
RE: Beyond + and - - by tommy1729 - 08/22/2012, 11:04 AM
RE: Beyond + and - - by Benny - 08/22/2012, 12:09 PM
RE: Beyond + and - - by tommy1729 - 08/22/2012, 02:05 PM
RE: Beyond + and - - by Benny - 08/22/2012, 06:34 PM
RE: Beyond + and - - by tommy1729 - 08/22/2012, 10:07 PM
RE: Beyond + and - - by tommy1729 - 08/22/2012, 10:20 PM
RE: Beyond + and - - by Benny - 08/23/2012, 12:08 PM
RE: Beyond + and - - by tommy1729 - 08/23/2012, 02:12 PM
RE: Beyond + and - - by Benny - 08/23/2012, 03:33 PM
RE: Beyond + and - - by tommy1729 - 08/23/2012, 04:12 PM
RE: Beyond + and - - by Benny - 08/23/2012, 07:41 PM
RE: Beyond + and - - by hixidom - 05/28/2014, 01:40 AM
RE: Beyond + and - - by JmsNxn - 05/28/2014, 04:33 PM
RE: Beyond + and - - by hixidom - 05/28/2014, 04:46 PM
RE: Beyond + and - - by tommy1729 - 05/28/2014, 10:44 PM



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