What is the commutative property and What is the associative property?

Answer 1

Binary operations can have commutative and associative properties. Binary operations are essentially rules that tell you how to combine two elements to make a third (they need not all be different). Addition, subtraction, multiplication and division are the more common ones. Exponentiation, taking logarithms, etc are less well known.

Commmutativity implies that

a * b = b * a

Associativity implies that

(a * b) * c = a * (b * c) and so either can be written as a * b * c

Addition and multiplication of numbers are associative as well as commutative whereas division is neither. However, multiplication of matrices is not commutative.