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The numbers used to measure real-world quantities such as length, area, volume, speed, electrical charges, probability of rain, room temperature, gross national products, growth rates, and so forth, are called real numbers. They include such number as  ,  ,  ,  ,  ,  ,  ,  and  .
The basic algebraic properties of the real numbers can be expressed in terms of the two fundamental operations of addition and multiplication.
Basic Algebraic Properties:
Let  and  denotes real numbers.
(1) The Commutative Properties
(a)  (b) 
The commutative properties says that the order in which we either add or multiplication real number doesn’t matter.
(2) The Associative Properties
(a)  (b) 
The associative properties tells us that the way real numbers are grouped when they are either added or multiplied doesn’t matter. Because of the associative properties, expressions such as  and  makes sense without parentheses.
(3) The Distributive Properties
(a)  (b) 
The distributive properties can be used to expand a product into a sum, such as  or the other way around, to rewrite a sum as product: 
(4) The Identity Properties
(a)  (b) 
We call  the additive identity and  the multiplicative identity for the real numbers.
(5) The Inverse Properties
(a) For each real number  , there is real number  , called the additive inverse of  , such that 
(b) For each real number  , there is a real number  , called the multiplicative inverse of  , such that 
Although the additive inverse of  , namely  , is usually called the negative of  , you must be careful because  isn’t necessarily a negative number. For instance, if  , then  . Notice that the multiplicative inverse  is assumed to exist if  . The real number  is also called the reciprocal of  and is often written as  .
Example:
State one basic algebraic property of the real numbers to justify each statement:
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) If  , then 
Solution:
(a) Commutative Property for addition
(b) Associative Property for addition
(c) Commutative Property for multiplication
(d) Distributive Property
(e) Additive Inverse Property
(f) Multiplicative Identity Property
(g) Multiplicative Inverse Property
Many of the important properties of the real numbers can be derived as results of the basic properties, although we shall not do so here. Among the more important derived properties are the following.
(6) The Cancellation Properties:
(a) If  then, 
(b) If  and  , then 
(7) The Zero-Factor Properties:
(a) 
(b) If  , then  or  (or both)
(8) Properties of Negation:
(a) 
(b) 
(c) 
(d) 
Subtraction and Division:
Let  and  be real numbers,
(a) The difference  is defined by 
(b) The quotient or ratio  or  is defined only if  . If  , then by definition 
It may be noted that Division by zero is not allowed.
When  is written in the form  , it is called a fraction with numerator  and denominator  . Although the denominator can’t be zero, there’s nothing wrong with having a zero in the numerator. In fact, if  , 
(9) The Negative of a Fraction:
If  , then 
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