To change means to commute. Consequently, if we talk about the commutative property of a mathematical operation, this means that in this operation it is possible to change the elements that intervene in it.
The commutative property occurs in addition and multiplication, but not in division or subtraction. Therefore, if I add two addends by changing their order, the final result is the same (30 + 10 = 40, which is exactly equal to 10 + 30 = 40). The same happens if I add three numbers or more. In relation to multiplication, the commutative property also holds (20 × 10 = 200, which is the same as 10 × 20 = 200).
The commutative property indicates that the order of the numbers used in the operation does not alter the result of said operation. The commutative property is shown in addition and multiplication and defines the possibility of multiplying or adding the numbers in any order, always achieving the same result.
Knowing the commutative property when doing additions and multiplications is very useful, especially when solving equations with unknowns, since it removes the burden of maintaining a particular order for each of its addends and factors. Let's not forget that the examples presented above reflect the simplest possibilities, since the following equation could also be given to demonstrate the effectiveness of the commutative property in both operations:
(A x C + Z / A) x B + D + E x Z = D + B x (Z / A + C x A) + Z x E
We must bear in mind that in this case the commutative property can be applied so that we obtain several equivalences, since by including addition and multiplication, the possible number of combinations increases. A much more complex equation could have operations such as root and empowerment, as well as constants (fixed values, as opposed to variables) and divisions that cover a whole term or part of it.
In popular language, it is often said that the order of the factors does not alter the product, that is, it does not affect the final result. This colloquial expression is applicable in those contexts in which we can change the order of something and this change does not affect the objective we want to achieve (for example, when it is indifferent to start placing something starting from one place or another). What is interesting about this way of speaking is the fact that it implies a mathematical dimension of reality, specifically the commutative property.
