Why does commutivity of real numbers exist for multiplication ie why a*b=b*a ?
It boils down to the natural numbers being commutative plus the way the set of real numbers is constructed (in a few steps) from N.
But why is multiplication in N commutative? How's that for a prediction. And to show some predictability here's what my favourite source of maths comes up with:
The entirely unenlightening answer is because that's how we defined them.
Historically speaking, the motivation behind the real numbers was measurement in geometry. For example, the product of two sides of a rectangle is the area of that rectangle, and the "rectangle with sides of length a and b" is congruent to the "rectangle with sides of length b and a", so commutativity was desired.
Separate names with a comma.