The Commutative property is one of those properties of algebraic operations that we do not bat an eye for, because it is usually taken for granted. The commutative property has to do with the order of the operation between two operands, and how it does not matter which order we operate them, we get the same final result of the operation.
The commutative property is a one of the cornerstones of Algebra, and it is something we use all the time without knowing. It is even in our minds without knowing, when we use to get the "the order of the factors does not alter the product".
First of all, we need to understand the concept of operation. In mathematical terms, an operation "\(\circ\)" is simply a way of taking two elements \(a\) and \(b\) on a certain set \(E\), and do "something" with them to create another element \(c\) in the set \(E\).
So then, when you take two elements \(a\) and \(b\) in a set, you operate them with the "\(\circ\)" operation and you get \(c\). You write this mathematically as \(a \circ b = c\).
Definition: An operation \(\circ\) is commutative if for any two elements \(a\) and \(b\) we have that\[ a\circ b = b \circ a\]
Note that not all operations satisfy this commutative property, although most of the common operations do, but not all of them. Indeed, addition and multiplication satisfy the commutative property, but subtraction and division do not.
Very that the common subtraction "\(-\)" is not commutative.
Indeed, let us consider the numbers: \(8\) and \(4\). Observe that:\[ \large 8 - 4 = 4 \]
whereas\[ \large 4 - 8 = -4 \]
So then, \(8 - 4\) is not equal to \(4 - 8\), which implies that the subtraction "\(-\)" is not commutative.
Let us define the following operation:\[ \large a\circ b = ab+a+b \]
Is this operation commutative?
Observe that\[ a \circ b = ab+a+b\]
On the other hand, we also get that\[ b \circ a = ba+b+a = ab + a + b\]
because both the common addition and multiplication are commutative. So then, we can see that \(a \circ b = b \circ a\). Hence, the operation "\(\circ\)" is commutative.
More About Commutativity
Commutativity is one property that you probably have used without thinking many, many times. You get it since your elementary school years, like a lullaby: "the order of the factors does not alter the product". And I guess it works because it sticks. If they told you "the multiplication is a commutative operation", and I bet you it will stick less.
One important thing is to not to confuse associativity with commutativity. When we refer to associativity, then we mean that whichever pair we operate first, it does not matter. That is not the same as saying that the order of the operation does not matter, which is the property of associativity.
Why is the commutative property important?
The commutative property is very important because it allows a level of flexibility in the calculation of operations that you would not have otherwise. There are mathematical structures that do not rely on commutativity, and they are even common operations (like subtraction and division) that do not satisfy it. So, commutativity is a useful property, but it is not always met.