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# What is the Commutative Property?

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.

## Discussion Comments

It's interesting that an idea so basic is part of the foundation of modern mathematics! And how wonderful would it be if all math were this simple? I definitely struggled with advanced math when I was in high school and college, but I didn't have any trouble remembering the commutative property definition or even the associative property!

@JessicaLynn - I can understand that. As adults, the commutative property is pretty much second nature. Everyone knows that the order of the numbers doesn't matter in addition and multiplication, but it does for subtraction and division. But that's definitely something you have to learn at some point!

Anyway, I remember learning this also. My math teacher at that time used a lot of commutative property examples to illustrate this instead of just saying something like: a + b = b + a. Those formulas mean very little to younger students, but concrete examples are much easier to understand, I think.

When I was a kid, I remember having a little bit of trouble learning division and subtraction. The commutative property of multiplication and addition made a lot of sense to me, and was easy to remember. Then subtraction and division came along and threw everything on its head!

I did eventually get it, but I really struggled for awhile. I don't remember what finally did it for me, but then one day it just clicked, and I've never had trouble remembering since then.

I recall learning about the commutative property in math class, and I thought it was so neat at the time. When the teacher started showing us how subtraction cannot work the same way, though, she confused me a little.

Then came algebra and all its specific orders in which you had to solve problems. Things just seemed to be getting more confusing, and my grades became less than ideal.

I wish things could always be as simple as the commutative property. For people like me who have a hard time grasping mathematic concepts, it is a comforting property.

Another way to show associativity is: ( a + b ) + c = a + ( b + c )

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