# Properties of Real Numbers: Addition and Multiplication Properties

This packet helps students understand and use different properties of addition and multiplication. All students should know and understand these properties (for instance, the Commutative Property of Addition states that 4 + 5 is the same as 5 + 4 and the Identity Property of Multiplication states that 6 x 1 = 6). Moreover, many standards-based tests also require students to be able to identify these properties by name. This worksheet helps students learn the functionality and the names of important properties of addition and multiplication.

The properties covered in this packet are the Commutative Properties of Addition and Multiplication, the Associative Properties of Addition and Multiplication, the Identity Properties of Addition and Multiplication, the Inverse Properties of Addition and Multiplication.

For reference:

The **Commutative Properties of Addition and Multiplication** state that order does not matter when adding or multiplying numbers. The Commutative Law does not apply to subtraction or division. So:

$3 + 4 = 4 + 3$ and $5 \times 6 = 6 \times 5$

but

$6 - 4 \neq 4-6$ and $10\div 5 \neq 5 \div 10$

The **Associative Properties of Addition and Multiplication** state that the way that you group numbersdoes not matter when adding or multiplying numbers. The Associative Law does not apply to subtraction or division. So:

$3 + (4 + 2) = (3 + 4) + 2$ and $(5 \times 6) \times 4 = 5 \times (6 \times 4)$

but

$6 - (4 - 2) \neq (6 - 4) - 2$ and $(10\div 5) \div 2 \neq (10 \div (5\div2)$

The **Identity Property of Addition **(also called the Identity Property of 0) is that any number plus 0 is itself:

$3 +0 = 3 $

The **Identity Property of Multiplication **(also called the Identity Property of 1) is that any number times 1 is itself:

$3 \times 1 = 3 $

The **Additive Inverse **is the number you add to a number to make it equal to 0.

$3 + -3 = 0 $

So the additive inverse of 3 is -3.

The additive inverse of any number is the opposite (positive or negative) of that number.

The **Multiplicative Inverse **is the number you multiply a number by to get 1.

$3 \times \dfrac{1}{3} = 1 $

The multiplicative inverse of 3 is $\dfrac{1}{3}$.

The multiplicative inverse of any number is its reciprocal.

For more on mathematical properties see Dr. Math's property glossary.

*Match each property with each example.*

**Simple**: $1+0=1$

**Advanced**: $(4+6)+5=(5+4)+6 $