How are addition and multiplication different?

Answer 1

for natural numbers, multiplication can be thought of as shorthand for a particular way that we want to do addition. once we define addition, we come across the need to define the addition of a single number x to itself a certain number of times, say, y times. so the question is, how do you write that? we could always just write

x+x+x+x+x+x+x+ . . . , and make sure we've included enough a's

or even

x+ . . . +x (y times) [simply stating that this happens y times]

but writing it either way each time would produce algebra textbooks of biblical proportions.

so, we use the shorthand x*y and understand it to mean "add x to itself y times" (or equivalently, add y to itself x times).

when we allow for x and y to be any real numbers, then this interpretation is valid only if you are comfortable accepting the notion of doing something pi number of times, or even 1/3 number of times for that matter (the latter being easier to swallow than the former). here, multiplication is just another classic abstractification of an intuitive concept.

OR

as binary operators [a(x,y)= x+y, m(x,y)= x*y], they are clearly different since for most values of x and y, a(x,y) =/= m(x,y) ( except whenever x = y/(y-1) ).