The commutative property holds in addition and multiplication.
In mathematics , operations have different properties. The distributive property , for example, is applied in multiplication and indicates that the number multiplied by the sum of two addends is equal to the sum of the products of each of these addends by the number in question. That is: A x (B + C) = A x B + A x C.
The associative property , applicable in multiplication and addition, indicates that the result of the operations is not linked to the way in which the numbers are grouped. Said in an algebraic expression: (A + B) + C = A + (B + C)
What is the commutative property
Now it is the turn to analyze another of these properties : the commutative property , which indicates that the order of the numbers used in the operation does not alter the result of the operation .
The commutative property appears in addition and multiplication and defines the possibility of adding or multiplying numbers in any order, always obtaining the same result:
A + B = B + A o A x B = B x A
Functioning in addition and multiplication
First of all, let's see how the property works in addition. If we have the values A = 5 and B = 7 , we will obtain the following equivalence from the commutative property:
5 + 7 = 7 + 5
12 = 12
In the case of multiplication, the reasoning is the same. Working with the same values as in the previous example, therefore, we will obtain this equivalence:
5 x 7 = 7 x 5
35 = 35
The commutative property contributes to the resolution of equations with unknowns.
Usefulness of the commutative property
Knowing the commutative property when performing additions and multiplications is very useful, especially when solving equations with unknowns, since it takes away the burden of maintaining a particular order for each of its addends and factors. Let us 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
Note 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 rooting and potentiation, as well as constants (fixed values, as opposed to variables) and divisions that cover all or part of a term.
When trying to solve an unknown, it is essential to know all the properties of the operations involved in the equation to avoid making mistakes. Let us not forget that mathematics is an exact science and that, in general, its use leads us to achieve a single possible value; In other words, making a small mistake is enough to invalidate the rest of the work.
Incompatible operations
On the other hand, it is also very important to know that the commutative property does not hold in subtraction, division, potentiation and establishment . It is enough to reverse the order of any simple equation that includes one of these operations to appreciate this incompatibility.
In the following examples you can see how dangerous it can be to try to apply the principles of the commutative property outside of addition and multiplication: 12 – 8 = 4 , while 8 – 12 = -4 ; 4 / 2 = 2 while 2 / 4 = 0.5 ; 3 raised to the eighth power is equal to 6561 , and is far from 8 cubed , which results in 512 .