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Before the construction of the complex numbers, people thought you couldn't take the square root of a negative number. Then came along of the definition of the imaginary unit

$$i^2 = -1$$

and now we've got a whole new system with all-new properties and a few non-properties as well (like traditional ordering).

I was wondering if I could construct a unit that allows distance to be negative. Consider the definition

$$\left| j\right| = -1$$

Would this be possible? Could a whole new number system, with all-new properties (and perhaps non-properties) be constructed? What kinds of consequences would result from this definition?

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Probably not, because $\lvert\cdot\rvert$ is a norm, and thus should be positive definite. –  Stefan Nov 30 '12 at 21:57
You ask about distance, but use the notation for modulus. The two concepts are related, but distinct. First thing to do is get the distinction straight in your head, so you can ask a coherent question. –  Gerry Myerson Nov 30 '12 at 22:01
Remember: at the beginning, there were only positive numbers. Then someone thought: what about a number that is not positive? and added $-$ and defined absolute value to be $-x$ for $x$ negative. Then similar story with complex numbers. Now you are not satisfied with that. You can use quaternions, but there $j^2=i^2=-1$. You can use octonions but they're just too crazy. –  tohecz Nov 30 '12 at 22:28
Actually, in the beginning there were only positive numbers, and then someone thought: what happens when I take away a bigger number from a smaller number? and thus negative numbers were born. –  chharvey Dec 1 '12 at 0:43
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2 Answers

You'd have more luck considering generalizations of algebraic norms. For example, the field norm of an algebraic number can be negative. While norms in this sense are multiplicative (like the standard norm on the real/complex numbers), the connection between small field norms and "smallness" of an algebraic integer is not clear to me. Certainly (when restricted to algebraic integers), small norms (i.e. $\pm 1$) correspond to algebraic units. Thus we may view algebraic integers with small norms as those "close" to units. In addition, we inherit a poset structure on algebraic integers from divisibility of norms.

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Certainly, systems with "negative distance" exist. In some sense, the time-space (or space-time) is an example of such system, where the distance of two points $(t_1,x_1,y_1,z_1)$ and $(t_2,x_2,y_2,y_2)$ is given by $$\left(\frac1{c^2}(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2\right)^{1/2}.$$ However, don't ask me for dietails, I'm not a physicist.

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To be picky, it's rather the "distance squared" that might be negative. –  Hans Lundmark Nov 30 '12 at 22:39
Yeah, that's true. I really don't remember the precise tricks and stuff around this. –  tohecz Nov 30 '12 at 22:46
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