Originally Posted by
HallsofIvy What rules of math are you referring to? If you mean things like "you can't take the square root of a negative number" or "you can't divide by 0", there are no "rules of math" that say you can.
The set of real numbers forms a "complete ordered field". The "rules of math" for the real numbers are precisely those of a complete, ordered, field:
1) The real numbers form a commutative group under addition.
_____a) (x+ y)+ z= x+ (y+ z)
_____b) x+ y= y+ x
_____c) there exist a number, 0, such that x+ 0= x for all x
_____d) for every x there exist a unique y such that x+ y= 0
2) xy= yx
3) there exist a number, 1, such that 1x= x for all x
4) For every x except 0, there exist y such that xy= 1
5) x(y+ z)= xy+ xz
6) There exist an order x< y such that
____a) if x< y then x+ z< y+ z
____b) if x< y and 0< z then xz< yz
____c) for any two number x and y, one and only one of
_______i) y= x
______ii) x< y
_____iii) y< x
is true.
7. All Cauchy sequences converge.
Which of those do negative numbers or 0 "defy"?