How do you explain associative property of real numbers?

Answer 1

First of all there is no such thing as the associative property in isolation. The associative property is defined in the context of a binary operation.

A binary operation is a rule for combining two elements (numbers) where the result is another (not necessarily different) element. Common mathematical binary operations are addition, subtractions, multiplication and division and the associative property does not apply to subtraction or division.

Having established that it only makes sense to talk about the associative property in the context of an operation, the associative property of real numbers, with respect to addition, states that, for any three real numbers, x, y and z,

(x + y) + z = x + (y + z)

That is to say, the order in which the OPERATIONS are carried out does not matter. As a result, either of the above sums can be written, without ambiguity, as x + y + z.

Thus associativity is concerned with the order of the operations and not the order of the numbers.

Also, note that the order of the elements on which the operator acts may be important. For example, with matrix multiplication,

(X * Y) * Z = X * (Y * Z) but X * Y ≠ Y * X

So matrix multiplication is associative but not commutative.