The commutative property applies to addition and multiplication, where the order of numbers does not affect the result. It is taught in basic math and is related to the associative property. Subtraction and division are non-commutative. Understanding this property helps solve math problems.
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 matter in which order you add them. Likewise when you multiply 2 and 3 together, you will get the same results whether you say 2 times 3 or 3 times 2.
These facts express the fundamental principles of the commutative property. When the order of two numbers in an operation does not affect the results, the operation can be commutative. The concept of this property has been understood for millennia, but the name wasn’t used much until the mid-19th century. Commutative can be defined as a tendency to change or substitute.
In basic math lessons, students can learn about the commutative property as applied to multiplication and addition. Even in later elementary grades students can study the commutative property of addition with formulas such as a + b = b + a. Alternatively they can quickly memorize that axb = bx a. Students often learn a related property called the associative property, which also affects order in multiplication and addition. Usually the associative property is used to show that the order of more than two digits using the same operation (addition or multiplication) will not affect the result: for example, a + b + c = c + b + a and is also equal ab + a + c.
Some operations in mathematics are called non commutative. 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 achieve the same results. As long as a doesn’t equal ab, a – b doesn’t equal ab – a. If a and b are 3 and 2, 3 – 2 equals 1 and 2 – 3 = -1. 3/2 is not equal to 2/3.
Many students learn about the commutative property at the same time they learn about the order of operations. When they understand this property, they can figure out whether a math problem must be solved in a certain order or whether the order can be ignored because the operation is commutative. While this property may seem basic enough to understand, it underlies much of what we know and assume about the nature of mathematics. When students have studied more advanced mathematics, they will see more complex applications of the property in action.
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