Rational Numbers
     
      Definition
      Comparing rational numbers
         Equivalent fractions
         Reducing or simplifying fractions
      Adding and subtracting rational numbers or fractions
      Multiplication of rational numbers
         Reciprocal fractions, multiplicative inverse
      Division of rational numbers
      Simplifying complex or compound fractions
Definition
A rational number is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. The set of all rational numbers is denoted as
Q = { a/b | a, b Zb is not 0 }.
A rational number or a fraction a/b denotes the result of dividing a by b, i.e., a/b = a b, b  is not 0. 
All integers are rational numbers as they can be written as a fraction with a denominator of 1. 
Terminating decimals and recurring decimals are rational numbers as they can be written as fractions.
Not all decimals are rational numbers, since not all decimals can be written as fractions. 
A rational number is positive if its numerator and denominator are both positive integers or both negative integers. A rational number is negative if its numerator and denominator have different signs, that is, 
Examples:     
Comparing rational numbers
A positive rational number a/b = r corresponds to a point P(r) of the real number line, symmetrically regarding the origin O, an opposite number a/b corresponds to a point P(r). A rational number r1 is less than a rational number r2 if lies to the left of r2 on the number line, written r1 < r2.
  
Two rational numbers, and are equal, i.e., if a d = b c.  
If the numerators a and c both are integers and the denominators b and d both are natural numbers then; 
  if a d < b c  or if a d > b c.  
Examples:   since (-4) (-10) = 5 8 since 3 35 = (-7) (-15)
    since 5 7 > 8 4 since (-9) 3 < 5 (-5)
Equivalent fractions
Convert a rational number or fraction to an equivalent fraction by multiplying the numerator and denominator by the same nonzero number.
Equivalent fractions are different fractions that represent the same number.
Examples:
Reducing or simplifying fractions
To reduce or simplify a fraction to lowest terms, divide the numerator and denominator by their greatest common divisor (gcd) or (greatest common factor - gcf).
Examples:
Adding and subtracting rational numbers or fractions
Find the least common denominator, write equivalent fractions, then add or subtract the fractions. Reduce if necessary.
Examples:  
             12 = 2 2 3,      36 = 2 2 3 3,      48 = 2 2 2 2 3 
                      LCD(12, 36, 48) = 2 2 2 2 3 3 = 144
   
Multiplication of rational numbers
Rational numbers are multiplied by multiplying numerators and multiplying denominators. Change any mixed numbers to improper fractions. Reduce fractions before multiplication.
Examples:
Reciprocal fractions, multiplicative inverse
Two rational numbers different from zero are reciprocal or multiplicative inverse if their product is unity.
Thus, the reciprocal of a fraction
  is since   Example:  
Division of rational numbers
Dividing a rational number or fraction by another fraction is equivalent to multiplying the dividend with the reciprocal of the divisor.
Examples:  
   
Simplifying complex or compound fractions
A complex or compound fraction is a fraction whose numerator and/or denominator are also a fraction or mixed number. Divide the numerator by the denominator.
Examples:
Beginning Algebra Contents A
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