Rational Numbers

Rational Numbers

A rational number is any number that can be represented by the ratio of two integers.

 

Examples:

, , ,

      The horizontal bar is called a vinculum and acts both as a grouping symbol (serves as parenthesis) and to indicate division. These numbers are also called fractions, where the number above the vinculum is called the numerator and the number below the vinculum is called the denominator. Many times we'll say '6' over '4', meaning, .

 

Properties:

A. Unlimited Representation

Every rational number can be represented in an unlimited number of ways.

Example: = = = ...

 

Example: 1 = = = = ...

(The previous example is used heavily to manipulate rational numbers, more on this later.)

 

B. Perfect Rationals are the Integers

      A perfect rational is a fraction whose denominator is 1.

 

 

Example: , , , ...

 

 

which are integers 1, 2, 3, 4, ...

 

C. Changing forms

      Recall that multiplying any number by 1 (the multiplicative identity) does not change that number. This result holds for rational numbers as well. In other words recall that

1 = = = = ... = = = ...

Examples:

Write so that the denominator is 16

    Well, 4 * 4 = 16, and multiplying any number by 1 does not change that number, the version of 1 we need is .

 

So, we have: * =

 

Write so that the denominator is 21

 

      3 * 7 = 21, we have * =

Write so that the numerator is -10

 

      5 * -2 is -10, we have *=

 

D. Writing Rationals in their simplest form

      Factor the numerator and the denominator into their prime factors, then use the previous result to simplify. This operation is usually called canceling common factors.

Examples:

      = = * *

               = 1 * 1 *

               =

 

 

      = = * *

 

                  = 1 * 1 *

               =

Operations:

 

E. Multiplication   (first since addition requires multiplication)

      When multiplying two or more rational numbers together, the result is the product of the numerators over the product of the denominators.

Example:

 

     

      Since a rational has an unlimited number of forms, we usually write them in their simplest form, that is, the form with the smallest possible numerator and smallest possible denominator. A systematic way to do this will follow shortly.

We would write as for the answer, although, technically, both are correct.

Example:

 

         * * = =

Example:

        * * = = =

 

 

Example:

          3 * = * =

Example:


       * 5 = * =

 

F. Division

     To divide rational numbers (fractions) you invert the denominator then multiply. 

Example:   Here we invert 2/9 to 9/2 first

                 

Example:    In this example we recall 12 is 12/1, then we invert

                   

 More on inverting and multiplying:

     Let's take another look at the first example:

 

original division of two fractions
written in terms of a fraction over another
now multiple by the equivalent of 1 in terms
of the denominator  9/2
 
the denominator becomes 1
 
18/18 = 1
 
and we have the result from inverting and multiplying

 

Here are some useful identities, the third and fourth are practice manipulating fractions and the last is a numeric example.

 

'a' can be factored out
1/b  can be factored out
placing 'a' into the denominator
 
placing 'b' into the numerator
 
move 4 into the denominator
move 2 into then numerator

G. Adding Rational Numbers

 

      There are two ways to add rational numbers, the first is to convert them so that they all have the same denominators, the second is using multiplication and addition.

 

 

Example: using same denominators

 

          + = * + *

                             = +

 

                             =

                             =

 

Example: Same as above but using multiplication and addition

 

+ = = =

 

 

      In the first case we use the fact that multiplying any number by 1 does not change that number, and in particular, only changes the form of the rational number.

                        = 1 as does = 1.

Our goal is to make the denominators the same. Once the denominators are the same, we add the numerators together. The common denominator we seek is the lcm of the denominators, and in this example ,the lcm of 3 and 7 is 21.

      The second case is really the same as the first but it does not necessarily use the lcm of the denominators. In this second case, notice that the second step, , can be split into two fractions,

and which are and respectively.

 This decomposition uses the distributive law of multiplication over addition which is used heavily in algebra to factor expressions. In a sense we are combining steps.

Example: + +

 

method 1: lcm = 20 we have then,

 

             * + * + *

 

                    =

method 2:

             

 

                    

                     = * =

 

More on method 2:

      Method 2 is how addition is defined for rational numbers in courses covering the theory of mathematics. Sometimes we refer to this method as cross multiplying when two fractions are involved, since we we multiply the numerator of the first fraction with the denominator of the second fraction then add the product of the second fraction's numerator and the first fraction's denominator. We then write this sum over the product of the denominators. 

 

   

Multiply a by d  and multiply b by c and add them together.

Divide this sum by the product of b and d.

If there's a minus sign between the first two fractions, then there will be a minus sign in the numerator as well.

 

F. Mixed Numbers

      A mixed number is a number represented by an integer and a proper fraction. A proper fraction is a fraction where the numerator is less than the denominator.

Example: 2 read, two and one third. This form looks like multiplication but is NOT multiplication, it is actually: ( 2 +), with

the + sign and the parenthesis omitted, but understood to be there.

 

      To write this number as a fraction (improper fraction), remember that 2 is the same as then  combine this with . We need a denominator equal to 3.

               1 =, * = .

So, we have

           + = =

                       2 =

 

      Using cross multiplication we have ( 2 * 3 + 1 ) divided by 3.

Note: the vinculum is a grouping symbol, it groups the sum 6 + 1.  This fraction could be written as = * (6 + 1), and using the distributive law of multiplication over addition we have

          * (6 + 1) = * 6 + * 1 = +

                             = +

 

                             = 2 * +

 

                            = 2 + = 2

      It is important to note that the denominator 3 in this last example divides each term of the sum in the numerator. Again, this is nothing more than the distributive law of multiplication over addition.  However, not understanding this causes numerous errors later in algebra simplifying expressions.

      To write an improper fraction as a mixed number, divide the numerator by the denominator; the integer is the quotient, the proper fraction will have the remainder as its numerator.

Example:


                

     27 divided by 6 is 4 with remainder 3 , so

     we have 4 ,

     3 over 6 can be reduced to 1 over 2 so we have

     the answer: 4

                        = 4

 

Example:

 

             75 divided by 25 = 3 (no remainder),

             so the answer is 3.

 

G. Negative mixed numbers

 

        Consider: — 3

 

         Remember the implied + sign and parenthesis?

— 3 IS        — ( 3 + )

 

— 1 ( 3 + ) = — 1 * 3 + — 1 *

= — 3 + — = +

=

 

            — 3 =

Example:

 

           - 4 + 1

 

       +

             +

           

             

                

 

NOW, careful here, -31 divided by 10 is -3 remainder -1, so

we have -3 as the answer. --> -1 *( 3 +)

In other words, momentarily ignore the – sign, think of the result as a positive improper fraction, change it into a mixed number, then put the – sign back.

 

Operations with mixed numbers.

 

Addition, Subtraction:

      Two methods:

          1)  convert all numbers to improper fractions first.

          2)  separate the numbers into integer and fractional parts first.

Example:

                        

Multiplication, Division:

    First change every mixed number into an improper fraction, then carry out the multiplication and/or division as shown earlier.

Example: