Is there a mathmatical rational for the choosing of negative exponents to represent fraction reversal or was is it just a matter of convention.
This is obviously the case for positive values of and , because we can simply write the meanings in full:
If we want to continue to use this law for negative powers (whatever they may mean), then we shall be forced to conclude that will have to be defined as . Why? Well, if the law above continues to remain in force, then (and we'll assume for now that ):
But we already know that , by 'cancelling' all the 's in the denominator with of the 's in the numerator.
And therefore we shall insist that and mean the same thing.
If that all seems too complicated, and you want a simpler reason, just look at the pattern of powers of 2 below, and ask yourself what happens if you continue the pattern, dividing by 2 each time: