Difference of Commutative and Associative Math

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Commutative and associative relate to properties in mathematics. The root word "commute" means to exchange; thus in mathematics, the commutative property involves switching the placement of values. "Associate" means to combine. When using the associative property in mathematics, you associate a different set of values; these adjustments do not change the value of the expression. There is a commutative and associative property of addition and multiplication.

Commutative Property of Addition

  • The commutative property of addition involves switching two terms that are separated by an addition symbol. For example, the expression 2a + c is the equivalent of c + 2a. Notice the commutative property is performed to switch the order of the expressions. This proves to be useful when attempting to combine like terms in the following expression: 2a + c + 3a. Using the commutative property, the algebraic expression can be written as c + 2a + 3a so that the like terms are paired together if you wish to combine them (c + 5a).

Commutative Property of Multiplication

  • The commutative property of multiplication involves switching two terms that are separated by a multiplication symbol. For example, the expression 5 6 is the equivalent of 6 5. Notice the commutative property is performed to switch the order of the expressions. This property is useful when deciding how much of a particular object to use to cover a shape (e.g., rectangle). For example, if you plan to rotate the original layout plan for a garden by switching the length with the width, you may do so without requiring more fertilizer to cover the area. The product or total area has not changed when the dimensions are switched.

Associative Property of Addition

  • The associative property of addition involves combining two terms that were not originally associated together. For example, the expression (2a + b) + 3a is the equivalent of (2a + 3a) + b. Notice the associative property is often performed to combine like terms. The terms 2a and 3a can be combined to form an equivalent expression, 5a.

Associative Property of Multiplication

  • The associative property of multiplication involves multiplying two terms that were not originally associated together. For example, the expression (2 324) 2 is the equivalent of (2 2) 324. Notice the associative property is often performed to lessen the use of mental strain when multiplying various quantities using mental math. For example, it is easier to multiply 2 * 2 first, then multiply that product by the large quantity, rather than have to multiply a large quantity twice when a calculator is unavailable.

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