>>13677181Adding integers is commutative and associative. As is adding reals, complex numbers, vectors, functions, etc.
Multiplying matrices or combining rotation/reflections (a symmetry group) are both extremely common, important noncommutative operations. Very few interesting operations aren't associative, but knot theory and numerical analysis of floating-point operations provide two examples. The cross product is neither.
Being able to categorize is important because we can generalize results. There's a whole field of commutative algebra which is fairly different from more general group/ring theory.
