Commutative, Associative and Distributive Properties of Real Numbers
Commutative Property of AdditionIf 'a' and 'b' represent any real numbers, then
a + b = b + a Example 1: Is this statement true or false? 2 + 3 = 3 + 2 2 + 3 = 5 3 + 2 = 5 Therefore, this statement is true according to the Commutative Property of Addition. |
Let's look at this example: Is this statement true or false? 3 - 2 = 2 - 3 Let's look at the first part of the equation 3 - 2 = 1 Now let's look at the second half of the equation 2 - 3 = -1 The solutions are not the same so therefore, this statement is false. |
The Commutative Property of MultiplicationIf 'a' and 'b' are real numbers, then a * b = b * a Example 1: Is this statement true or false? - 4 * 5 = 5 * -4 - 4 * 5 = - 20 5 * - 4 = - 20 Therefore, this statement is true according to the Commutative Property of Multiplication. |
Let's look at another example. 10 / 5 = 5 / 10 10 divided by 5 is 2. 5 divided by 10 is 1/2. The answers are not the same so therefore, this statement is false. This shows us that division is not commutative. |
The Associative Property of AdditionIf 'a' , 'b' and 'c' are any real numbers, then
( a + b ) + c = a + ( b + c ) This property is called associative property of addition. It shows that when we are given three numbers to add, we get the same answer when we add the first two numbers together first as when we add the last two numbers first. Is this statement true or false? ( 2 + 3 ) + 4 = 2 + ( 3 + 4 ) ( 2 + 3 ) + 4 = ( 2 + 3 ) = 5 5 + 4 = 9 ( 3 + 4 ) = 7 2 + 7 = 9 Both solutions to this problem is the same. Therefore, this statement is true by the associative property of addition. |
Let's look at this example: Is the following statement true or false: ( 4 - 3 ) - 2 = 4 - ( 3 - 2 ) The first half of the equation: ( 4 - 3 ) = 1 1 - 2 = - 1 Now let's look at the second half of the equation. ( 3 - 2 ) = 1 4 - 1 = 3 These answers are not the same, so therefore this statement is false. This shows us that there is no associative property of subtraction. |
The Associative Property of MultiplicationIf a, b and c represent any real number, then
( a * b ) * c = a * ( b * c ) Is the following statement true or false? ( 2 * 3 ) * 4 = 2 * ( 3 * 4 ) ( 2 * 3 ) = 6 6 * 4 = 24 ( 3 * 4 ) = 12 2 * 12 = 24 Since both sides of the equation are equal, then the statement is true by the associative property of multiplication. |
Now let's look at this example: ( 20 / 10 ) / 2 = 20 / ( 10 / 2 ) Is this statement true or false? First half of the equations: 20 divided by 10 = 2 2 / 2 = 1 Second half of the equation: ( 10 / 2 ) = 5 20 / 5 = 4 The answers are not the same so therefore there is no associative property of division. |
In Summary:
The commutative properties guarantee that the sum or product of two numbers remains the same even if the order of the numbers is changed.
The associative properties guarantee that the sum or product of three numbers remains the same whether you operate on the first two numbers first or the second two numbers first.
If you want to multiply several factors simultaneously, here is a tip:
Simply multiply the absolute values of the factors and determine the correct sign of the answer. The answer will be either:
Positive if there is an even number of negative factors
Negative if there is an odd number of negative factors
The associative properties guarantee that the sum or product of three numbers remains the same whether you operate on the first two numbers first or the second two numbers first.
If you want to multiply several factors simultaneously, here is a tip:
Simply multiply the absolute values of the factors and determine the correct sign of the answer. The answer will be either:
Positive if there is an even number of negative factors
Negative if there is an odd number of negative factors
