Real Numbers

    This topic introduces the real number system in an informal and intuitive manner. First we define the following sets of numbers: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and the real numbers. Then we state and illustrate how to use the (axioms) properties of the real numbers. We follow this by detailing many more properties of the real numbers such as, cancellation properties, distance properties, and absolute value properties.  

Definition (Natural Numbers) The set of numbers is called the natural numbers.

Definition (Whole Numbers) The set of numbers is called the whole numbers.

Definition (Integers) The set of numbers is called the integers.

Definition (Rational Numbers) The set of numbers



are called the rational numbers.

Example (Rational Numbers) Determine which of the following are rational numbers:



    Solution. They all are rational numbers. However, is not a rational number.

    Every rational number can be written as in decimal form with a repeating decimal pattern. For example, or say . Rational numbers are characterized as those decimals that have a repeating decimal pattern. This leads us to our next definition.

Definition (Irrational Numbers) Irrational numbers are those decimals that do not have a repeating decimal pattern.

Example (Irrational Numbers) Determine which of the following are irrational numbers:



    Solution. All of them are irrational except,  

Definition (Real Numbers) Real numbers consist only of rational and irrational numbers. The real numbers are denoted by .

Example (Real Numbers) Determine which of the following are real numbers:



    Solution. Since a square root of negative real number is not a real number, and are not real numbers. Since division of a real number by zero is not a real number, is not a real number. Finally,  since the fourth root of negative real number is not a real number, is not real a number.   

Definition (Properties of the Real Numbers) Let and be a real number. The following axioms are called the properties of the real numbers:

    (i) The commutative law for addition is
    
    (ii) The associative law for addition is
    
    (iii) The additive identity law for addition is
        
    (iv) The additive inverse law for addition is  
            
    (v) The commutative law for multiplication is
    
    (vi) The associative law for multiplication is
    
    (vii) The multiplicative identity law is
        
    (viii) The multiplicative inverse law is
                
    (ix) The distributive law is  
    

Example (Properties of the Real Numbers)

(a) Use the properties of real numbers to solve the equation for

    Solution. The table illustrates how to solve for using the properties of real numbers.
    


(b) Use the properties of real numbers to factor

    Solution. The table illustrates how to factor using the properties of real numbers.
    

    

Properties of the Real Numbers

Proposition (Negatives) Let be a real number.

    (i) The negative of is
    
    (ii) If is a positive number, then is a negative number.
    
    (iii) If is a negative number, then is a positive number.
    

Proposition (Zero Products) Let and be real numbers.

    (i)
    
    (ii) if then either or
    
    (iii) if and , then
    

Proposition (Properties of Negatives) Let and be real numbers.

    (i)   and
    
    (ii)
    
    (iii)
    
    (iv)
    

Proposition (Signs) Let and be real numbers.

    (i) If and have the same sign then and are positive numbers.
    
    (ii) If and have the opposite signs then and are negative numbers.
    
    (iii)
    
    (iv)
    
    (v)
    
    (vi)
    

Proposition (Quotient Properties) Let , , and be real numbers. Then, the following equalities hold given only non-zero denominators.

    (i) if
    
    (ii)
    
    (iii)
    
    (iv)
    
    (v)

    (vi)
        

Proposition (Cancellation Properties) Let , , and be real numbers.

    (i) if and only if
    
    (ii) if , then:   if and only if
    

Absolute Value

Definition (Absolute Value) The absolute value of a real number , denoted by is defined as follows: when and when

Example (Absolute Value)  

(a) Rewrite without using absolute value and simplify the result.

    Solution. Since we see that

(b) Rewrite without using absolute value and simplify the result.

    Solution. Since we see that

(c) Rewrite if , without using absolute value and simplify the result.

    Solution. Since we see and so
    
(d) Rewrite without using absolute value and simplify the result.

    Solution. Since we see   

Ordering The Real Numbers

Definition (Inequalities of Real Numbers) If and are real numbers, is less than if is positive. The order of and is denoted by the inequality This relationship can also be described by saying that is greater than and writing The inequality means that is less than or equal to

Definition (Law of Trichotomy) The Law of Trichotomy states that for any two real numbers precisely one of the three relationships holds: or

Definition (Bounded Intervals of Real Numbers)  The following are bounded intervals of the real number line.

    (i) means
    
    (ii) means
        
    (iii) means
    
    (iv) means

Definition (Unbounded Intervals of Real Numbers)  The following are unbounded intervals of the real number line.

    (i) means
    
    (ii) means
        
    (iii) means
    
    (iii) means

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Cite this as:
Real Numbers
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/real-numbers.html