11.4 Associative Properties

Associative Properties describe how addition and multiplication of three or more real numbers produce the same result regardless of how the numbers are grouped for combination, provided their order is left unchanged, and how this grouping-independence fails to hold for subtraction and division.

Associativity of Addition

Associative Property of Addition

The associative property of addition states that when three real numbers are added together, the way they are grouped by parentheses does not affect the sum; the first two can be added together before adding the third, or the last two can be added together before adding the first, with the same result either way.

Addition Regrouping

Regrouping under addition means moving the parentheses to combine a different pair of adjacent addends first, and this regrouping can be performed freely on any sum of three or more terms, with the final total remaining unaffected no matter which adjacent pair is combined first.

Signed Addition Associativity

The associative property of addition holds regardless of the signs of the numbers involved, so regrouping a sum containing negative addends still produces the same total no matter which adjacent pair is combined first.

Associativity of Multiplication

Associative Property of Multiplication

The associative property of multiplication states that when three real numbers are multiplied together, the way they are grouped by parentheses does not affect the product; the first two can be multiplied together before multiplying by the third, or the last two can be multiplied together before multiplying by the first, with the same result either way.

Multiplication Regrouping

Regrouping under multiplication means moving the parentheses to combine a different pair of adjacent factors first, and this regrouping can be performed freely on any product of three or more factors, with the final result remaining unaffected no matter which adjacent pair is combined first.

Signed Multiplication Associativity

The associative property of multiplication holds regardless of the signs of the numbers involved, so regrouping a product containing negative factors still produces the same result no matter which adjacent pair is combined first, as long as the overall sign determined by the count of negative factors is tracked correctly.

What Must Stay Fixed during Regrouping

Order Preservation during Regrouping

The associative property only permits changing which adjacent numbers are grouped together; it does not permit changing the order in which the numbers appear. Regrouping and reordering are separate operations, justified by different properties, even though both are often used together when rearranging a longer expression.

Operations That Are Not Associative

Subtraction Nonassociativity

Subtraction is not associative, meaning that changing which pair of numbers is grouped together in a chain of subtractions generally changes the result, since removing a grouped difference first produces a different outcome than removing the individual numbers in the original left-to-right order.

Division Nonassociativity

Division is likewise not associative, meaning that changing which pair of numbers is grouped together in a chain of divisions generally changes the result, since dividing by a grouped quotient first produces a different outcome than dividing by the individual numbers in the original left-to-right order.

Confirming Nonassociativity

Numerical Counterexample to False Associativity

A single numerical example in which regrouping a chain of subtractions or divisions changes the result is sufficient to confirm that these operations are not associative in general, since a true associative property must hold for every possible set of numbers, and one disagreeing case disproves it entirely.