Properties of Real Numbers

• Commutative Property: For all real numbers a and b:

$$a + b\; = b + a$$

$$a \times b\; = b \times a$$

Order doesn’t matter in addition or multiplication of real numbers.

• Associative Property: For all real numbers a and b:

$$(a + \;b) + \;c = a + (b + c)$$

$$(a \times \;b) \times \;c = a \times (b \times c)$$

• Closure Property: For all real numbers a and b:

$$a + \;b\;and\;a\; \times \;b\;are\;real\;numbers$$

• Identity Property: For all real numbers a:

$$a + \;0 = a$$

$$a \times \;1 = a$$

Zero is called the identity for addition, and one is the identity for
multiplication. These numbers leave the number a unchanged when the operation
is performed.

• Inverse Property For every real number a, there is a number – a and, if
  a ≠ 0, there is a number l/a such that

$${\rm{a}}\; + {\rm{ }}\left( { – a} \right){\rm{ = 0}}$$

$$a \times \left( {{1 \over a}} \right) = 1$$

The number – a is called the inverse of a for addition, and l/a is the inverse of
a for multiplication, also called the reciprocal. These numbers take the number a
back to the identity (0 or 1) when the operation is performed.

• Distributive Property For all real numbers a, band c:

$${\rm{a}}\;{\rm{(b}}\;{\rm{ + }}\;{\rm{c) = ab}}\;{\rm{ + }}\;{\rm{ac}}$$

This property is the distributive property of multiplication over addition.