The commutative property is an ancient idea in mathematics that still has numerous uses today. Essentially those operations that fall under the commutative property are multiplication and addition. When you add 2 and 3 together, it doesn’t really matter in which order you add them. Similarly when you multiply 2 and 3 together, you’re going to get the same results whether you say 2 times 3 or 3 times 2.

These facts express the basic principals of the commutative property. When the order of two numbers in an operation does not affect results, then the operation may be commutative. The concept of this property has been understood for millennia but the name of it wasn’t used much until the mid 19th century. Commutative may be defined as having a tendency to switch or substitute.

In basic math classes, students may learn about the commutative property as it applies to multiplication and addition. Even in the later primary grades students may be studying the commutative property of addition with formulas like a + b = b + a. Alternately they may quickly commit to memory that a x b = b x a. Students often learn a related property called the associative property, which also concerns order in multiplication and addition. Usually the associative property is used to show that order of more than two digits using the same operation (addition or multiplication) will not affect outcome: e.g., a + b + c = c + b + a and is also equal to b + a + c.

Some operations in math are called noncommutative. Subtraction and division fall under this heading. You cannot change the order of a subtraction problem, unless the digits are equal to each other, and get the same results. As long as a does not equal b, a – b is not equal to b – a. If a and b are 3 and 2, 3 - 2 equals 1 and 2 – 3 = -1. 3/2 is not the same as 2/3.

Many students learn the commutative property at the same time they learn the concept of order of operations. When they understand this property they can understand whether a math problem needs to be solved in a certain order or whether order can be ignored because the operation is commutative. While this property may seem fairly basic to understand it does underpin much of what we know and assume about the nature of mathematics. When students studied more advanced math, they will see more complex applications of the property in action.