Commutative Property 

Have you ever tried adding any two random numbers, say 2 + 5 and then 5 + 2? Did you get different answers? Of course not — the answer will be the same, which is 7. That’s not just a coincidence; it’s a perfect example of the commutative property in mathematics. It is a common yet important law of mathematics. In this article, we will be discussing this law in minute detail, which ultimately forms a strong base for understanding the complex topics in mathematics. 

1.0Commutative Property Meaning 

The commutative property is the most basic and one of the first rules of mathematics we study while studying operations on numbers. It is a law that states:

"Interchanging the position of numbers within an operation won't change the outcome."

It got its name from the word “commuter”, which simply means to travel about or change positions. In the context of mathematics, it means you can change the order of the numbers and still obtain the same result, in addition and multiplication. 

2.0Commutative Property of Addition

The commutative property of addition states that when any two numbers are added, the result of these numbers will remain the same, even if we change the order of the numbers. In other words, this property simply tells that changing the order of numbers being added does not affect the sum. Mathematically, take any two numbers, say a and b, then the rule of addition will be: 

$a+b\models b+a$

To understand this better, take a commutative property of addition example, say a is 4 and b is 6, then according to the rule: 

$\begin{array}{rl}6+4& =4+6\\ 10& =10\end{array}$

Note that the commutative property of addition holds true not only for a pair of two numbers but also for more than two numbers. 

3.0Commutative Property of Multiplication

Just like addition, the operation of multiplication also follows the same rule of commutativity. Which means, while multiplying any two numbers, the order of the numbers doesn’t affect the result of the operation. In other words, swapping the order of numbers doesn’t actually matter in the multiplication of these numbers. Mathematically, the rule can be expressed as: 

$a\times b=b\times a$

For better understanding, take an example, say a is 5 while b is 2, then according to the commutative property of multiplication: 

$\begin{array}{rl}5\times 2& =2\times 5\\ 10& =10\end{array}$

4.0Operations That Are Not Commutative 

Math operations are not all commutative. Subtraction and division are two operations in mathematics in which the order of operations is important, and altering it does make a difference. 

Commutative Property of Subtraction 

The operation of subtraction does not always follow the rule of commutativity. In the subtraction of any two different numbers, say a and b, the order in which the numbers are being subtracted matters. Mathematically, this rule can be stated as:

$a-b\ne b-a$

However, if both numbers are the same, then in this case, the numbers hold the property of commutativity. For example, the number is a then as per this rule: 

$\begin{array}{rl}& a-a=a-a\\ & 0=0\end{array}$

Commutative Property of Division

Just like subtraction, the operation of division of any two numbers also does not follow the rule of commutativity. Which means if you divide any two random numbers and change the order of the numbers, then the result will also change. Mathematically, this can be expressed for any two numbers, say a and b, as: 

$a÷b\ne b÷a$

However, just like in subtraction, if both numbers are the same, then in this case, the numbers hold the property of commutativity. For example, the number is a then as per this rule: 

$\begin{array}{rl}a÷a& =a÷a\\ 1& =1\end{array}$

5.0Why is the Commutative Property Important?

The following are some reasons why it is important to know the commutative property of different operations in mathematics:

• Saves time while solving equations.
• Aids in rearranging algebra terms.
• Simplifies mental math.
• Form a basis for advanced math subjects. 

6.0Commutative Property, Associative Property and Distributive Property