Algebraic Properties

Learning Goal:

The student will be able to summarize properties and make connections between real number operations.

The Commutative Property

The ORDER in which you add or multiply two numbers does not affect the result.

Commutative Property of Addition

a + b = b + a

3 + 7 = 7 + 3Commutative

Propertyof Multiplication

ab = ba

7●3 = 3●7

The Associative Property

When you add or multiply any three real numbers, the GROUPING (or association) of the numbers does not affect the result.

Associative Property of Addition

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

5 + (6 + 7) = (5 + 6) + 7

Associative Propertyof Multiplication

a(bc) = (ab)c

2(3●4)= (2●3)4

Note: when values “leave” the parentheses but do NOT affect the value

The Additive Identity Property

1. If 0 is added to any real number, the sum is equal to the given number.

2. When you multiply any given real number by 1, the product is equal to the given number.

Note: Think of a mirror image.

Identity Property of Addition

a + 0 = a

6 + 0 = 6

Identity Propertyof Multiplication

a ● 1 = a

2 ● 1= 2

Distributive Property

• When you distribute something, you separate it into portions.

• The distributive property states that for any numbers a, b, and c:a(b + c) = (ab) + (ac)a(b – c) = (ab) – (ac)

Example: -4(5 + 3) = (-4 x 5) + (-4 x 3) -20 + -12 = -32

-(3 – 2) = (-1 x 3) –(-1 x 2) -3 – (-2) = -1

The Inverse Property

The sum of a number and its opposite is zero.

The product of a number and it’s reciprocal is 1.

Inverse Property of Addition

a + -a = 0

2 + -2 = 0Inverse Property of

Multiplication

a ● 1/a = 1

2 ● ½ = 1

The Zero Property

When you multiply a real number by 0, the resulting product is zero.

Zero Property

a ● 0 = 0

Try this….1. x + 9 = 9 + x  is an example of which property?

4. -22 + 22 = 0  is an example of which property?

2. x + ( y + 3 ) = x + ( 3 + y ) is an example of which property?

3. ( 5 y ) • ( 1 ) = 5 y is an example of which property?

Commutative Property for Addition

Commutative Property for Addition

Identity Property for Multiplication

Inverse Property for Addition

Try this Cont.….5. 5 • 1 = 5 is an example of which property?

8.   is an example of which property?

6. ( x y ) z = x ( y z ) is an example of which property?

7. 10 • 0 = 0 is an example of which property?

Identity Property for Multiplication

Associative Property for Multiplication

Zero Property for Multiplication

Inverse Property for Multiplication

The Closure Property

Closure Property of Addition

the sum of any two real numbers is also a real number and there is a unique (only one) solution

Example:

The sum of any two even numbers is always even

Example:

If you multiply a whole number by 2, your product is always even

Closure Property of Multiplication

the product of any two real numbers is a real number and there is a unique (only one) solution

More about Closure…

Let's check out this question.  If you divide two even numbers (from the set of even numbers), is the quotient even?

When you find even ONE example that does not work, the set is not closed under that operation.  The even numbers are not closed under division.

•12 / 6 = 2 Yes, 2 is even.•24 / 2 = 12 Yes, 12 is even.•100 / 4 = 25 NO, 25 is not even!