



 
Justin, The reason that in the usual extension of the real numbers by "infinity" and "minus infinity" you cannot divide one infinite quantity by another has nothing to do with different sizes of infinity. Rather, it is the same as the reason why you cannot divide zero by zero. Division is meant to be an inverse to multiplication  that is, dividing 6 by 3 should be the same the same as answering the question "what do you multiply by 3 to get 6"? This is only meaningful if there is a unique correct answer! As multiplying any finite number by 0 gives 0, the question "what do you multiply by 0 to get 0?" does not have a single answer  thus there is no good value to give to 0/0. Similarly, multiplying any positive number by infinity gives infinity. Thus there is no good value to give to infinity/infinity. In the much more complicated extension used in nonstandard calculus, if N is infinite and k is not 1, then kN is an infinite number but not equal to N; and infinite numbers _can_ be divided by other infinite numbers. The answer may be "standard", infinite, or infinitesimal. RD
 


Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. 