Properties of the Real Numbers

• Addition Properties

• Associative Property

• $$a+(b+c)=(a+b)+c$$

• Commutative Property

• $$a+b=b+a$$

• Identity Property

• $$a+0=a$$

• Inverse Property

• $$a+(-a)=0$$

• Multiplication Properties

• Associative Property

• $$a \cdot (b \cdot c)=(a \cdot b) \cdot c$$

• Commutative Property

• $$a \cdot (b \cdot c)=(a \cdot b) \cdot c$$

• Identity Property

• $$a \cdot 1=a$$
• Inverse Property
• $$a \cdot \frac{1}{a}=1$$

• Distributive Property

• $$a \cdot (b + c) = a \cdot b + a \cdot c$$

Significance of Associativity and Commutivity

The associative and commutative properties of both addition and multiplication show that the order in which addition and multiplication operations are performed does not matter. When you have mathematical expressions combining addition and multiplication operations that is a different matter, of course.

Significance of the Identity Properties

The identity properties establish the importance of the numbers one and zero, and they are very useful for all kinds of arithmetic and algebra applications.

Significance of the Inverse Properties

The inverse properties for addition and multiplication serve to define the operations of subtraction and division respectively.

Significance of the Distributive Property

The distributive property is the basis for many elementary arithmetic number manipulation techniques and is very important in algebra as well.