Understanding the Associative and Commutative Properties of Addition and Multiplication Examples Included

In math, the associative and commutative properties are laws applied to addition and multiplication that always exist. The associative property states that you can re-group numbers and you will get the same answer, and the commutative property states that you can move numbers around and still arrive at the same answer.

These properties are helpful when working with the order of operations, as they help us reach rework and rearrange problems while still reaching the correct answer.

TL;DR (Too Long; Didn’t Read)

The associative and commutative properties can still apply to subtraction and division, but it is important to frame them in terms of addition and multiplication to avoid any arithmetic mistakes (e.g. dividing by 3 is the same as multiplying by 1/3).

What is the Associative Property?

The associative property comes from the words “associate” or “group.” It refers to grouping of numbers or variables in algebra. You can re-group numbers or variables and you will always arrive at the same answer. We typically group numbers and variables with parentheses in math, and the associative property allows us to regroup numbers without changing the value of an expression.

The associative property of addition:

((a+b) + c = a + (b + c))

(textit{Ex:} (4+3)+6 = 4+(3+6) = 13)

The associative property of multiplication:

((a times b)times c = a times (b times c))

(textit{Ex:} (7 times 2) times 2 = 7 times (2 times 2) = 28)

In some cases, you can simplify a calculation by multiplying or adding a different order of the numbers to find easier patterns or combinations of variables.

‌Example of the Associative Property:‌ What is 19 + 36 + 4?

(19+36+4 = 19 + (36+4) = 19 + 40 = 59)

What is the Commutative Property?

The commutative property in math comes from the words “commute” or “move around.” This rule states that you can move numbers or variables in algebra around and still get the same answer.

This equation defines the commutative property of addition:

(a + b = b + a)

(textit{Ex:} 4 + 3 = 3 + 4 = 7)

This equation defines commutative property of multiplication:

(a times b = b times a)

(textit{Ex:} 7 times 3 = 3 times 7 = 21)

Sometimes rearranging the order makes it easier to add or multiply as well, especially with larger numbers.

‌Example of the Commutative Property:‌ What is 2 × 16 × 5?

(2 times 16 times 5 = 2 times 5 times 16 = (2 times 5) times 16 = 10 times 16 = 160)

Other Considerations

All of our examples above used positive whole number (also called natural numbers or positive integers), but these properties work for operations on any fractions, decimals, negative numbers. This includes rational numbers like 1/2 or 5/8, and it also works for all real numbers (which includes irrational numbers like pi and e).

It becomes equally important to identify the algebraic expressions and situations where these properties do not apply. The associative and commutative properties only work with operators of the same type, so you can only swap values that are all added or multiplied together.

Some other properties of addition and multiplication not covered in these scenarios include:

‌The Distributive Property of Multiplication‌

(a(b + c) = ab + ac)

‌The Identity Property of Multiplication‌

(a times 1 = a)

‌The Identity Property of Addition‌

(a + 0 = a)

There are additional properties and conventions for exponents, mixed numbers, logarithms, operations in trigonometry, and much more.

Additional Practice Problems for Students

This exercise works through a combination of associative and commutative properties:

(text{Evaluate:})((6+4) + 2 = ?)(6 + (4 + 2) = ?)(4 + (2 + 6) = ?)((2+4) + 6 = ?)

What is this equation equal to:

(text{Evaluate:})(6 times (2 times 9) times (5 times 5) = ?)

Find the unknown value for ‌x‌:

(2 + (x +8) = (4+2) + 8)(x = ?)

Find the unknown value for ‌x‌:

((2 times 3) times x = (4 times 2) times 3)