See for reference: http://www.math.ucla.edu/~tao/resource/general/121.1.00s/vec... Addition is only defined on two elements of the space, multiplication is only defined for multiplying an element by a real number, which results in an element. These satisfy the unambiguous distributive axiom, real * (el_1 + el_2) = real * el_1 + real * el_2. Since addition only applies to elements, real * el_1 and real * el_2 must be computed into elements first before the sum. While this doesn't strictly, logically imply the same for real * real + real * real, it's at least suggestive (hence the e.g.) and you can turn to the Peano Arithmetic axioms and definitions of addition and multiplication and do the same thing for natural numbers. Or you can just say the ordering implied by assuming PEMDAS, but just saying that alone is boring and doesn't really expose the elegance in notation/algebraic transformation gained from such a convention.

(For those who don't get the reference, every so often some variant of "a + b * 0 + c = ?" will show up in the public's attention and receive very emotional arguments about how it must be 0, or must be a+c, or must be c, or it must be ambiguous and therefore undefined. Similarly to the logic puzzle programmers can just point to the problem expressed in Python for the correct answer...)