> Jonathan, if negative numbers take so much > explanation, maybe we should just get rid of them. > It's starting to affect our GDP and national > security. I am a fan of your syntactical methods, > even though I have not yet mastered them. Perhaps we > need to "formalize" them, that is, get some notation > for them. Then we can formalize the consequences as > well.
I did have a go at formalising the 'axioms' of natural integer logic at the link in my previous post.
Also I suggest you have a look at source documents rather than buy pre-digested stuff. Enjoy chapter 2 of Euler's 1821 'An introduction to the elements of Algebra' for his explanation of plus and minus signs. then chapter 13 is about negative powers, which you are also asking about.
Euler was part of the subsequent problem as he stated negative numbers were less than zero. Like everyone before him he used the example of debt, which of course is counted with positive numbers. The unit debt is negative and the unit wealth (or reserves) is positive.
US debt is $16 trillion. There is no negative sign in the previous statement as the object is debt. It's bigger than zero!
China's reserves are likewise mesured in the trillions. There is no positive sign needed as the object is reserves (wealth).
When numbers describing units/objects descend to zero, they almost always bounce up again with an inverse unit.
Negative numbers are the realm of pure or abstract math and very handy. Yet for the average person, negative numbers need not, and often do not exist.
The numbers remain just that, numbers describing the nouns that implicitly carry the sign!