>>10940835>>10939568>>10939066geometric reason:
We can take numbers to represent transformation: take your favorite object (a car or a banana or whatever) and each number represents a scaling of the object. means make it three-times larger, means make it three times smaller. means make it three times larger and also reflect it along some fixed axis or plane. multiplication now represents composition of these scalings: if you make the banana two times larger and then three times larger , it will get six time larger than the original size etc. is a neutral element, because it literally says "do nothing". means that if you make it three times larger and then three times smaller, nothing happens. represents just the reflection, without changing size. clearly says that two reflections cancel each other out. etc. etc. you see that everything checks out. we define to be: for it means do the scaling succesively -times, for it means do the scaling succesively -times. you can easily check that the usual arithmetics for exponentation like now hold for these purely geometric reasons. as for the zero, should literally mean "do zero times", which means "do nothing". and doing nothing is .
