How to GRAPH THE POINTS of some REPEATING DECIMALS on the NUMBER LINE

To Graph the POINT of any DECIMAL NUMBER on the NUMBER LINE is relatively easy. This Article will show HOW TO GRAPH the POINT of a REPEATING DECIMAL on the NUMBER LINE.

Things You'll Need

• Pencil and
• Graph paper.

Instructions

1. 

   The Number ( 0.333333...) is defined as a Recurring or Repeating Decimal. The first Dot between the zero-digit Number ( 0 ), and the first three-digit Number ( 3 ), is THE DECIMAL POINT. The three Dots that follow the last Number ( 3 ), signifies that the number, three ( 3 ), continues indefinitely or just keep on repeating. In order to graph this point on the Number line we need to change this Repeating Decimal to a Rational Number so that it can be easily graph on the Number Line. Please click on the Image to the left to see a method of accomplishing this.

2. 

   Since we do not know what this Rational Number will be, we choose a variable to represent this Rational Number. We choose X to be that variable and set X = 0.333333... .Please click on the Image to the Left for a better view.

3. 

   We now look to see how many digits are repeating. In this problem we see that one digit is repeating, and that digit is the digit ( 3 ). We need to stop this repeating process. To do so we Multiply both sides of the Equation by the number ( 10 ), that is,... 10(X) = 10(0.333333...).
   We now have the Equation...( 10X = 3.333333... ). Please note that when a Decimal Number is multiplied by the numbers ( 10 ), ( 100 ), ( 1000 )
   ..., the Decimal Point moves from its original position to the Right, one place, two places, three places ..., respectively. Please click on the Image to the left for more details.

4. 

   We will now Subtract the first Equation ... ( X = 0.333333... ) from the second Equation ... ( 10X = 3.333333... ), resulting in a third Equation ... ( 9X = 3 ). We solve for X by dividing both the Left side and the Right side of this third Equation by ( 9 ), which gives us
   X = ( 3/9 ) which is equal to ( 1/3 ) when reduced. Please click on the Image to the left for more details.

5. 

   We now try to graph the Repeating Decimal ... ( 0.12121212... ).
   The process of changing this Repeating Decimal is the same, except that this Decimal has two digits repeating those digits being ... ( 12 ).
   So we set X = 0.12121212..., and Multiply both sides of the Equation by
   100. which results in the Equation ... 100(X) = 100( 0.12121212... ),
   that is equivalent to ... ( 100X = 12.12121212... ). We Subtract the First Equation ... ( X = 0.12121212... ) from the second Equation ( 100X = 12.12121212... ), and get the third Equation ... 99X = 12.
   To solve for X, we divide both sides of the Equation by ( 99 ) and get X = ( 12/99 ) or ( 4/33 ) when reduced to lowest terms. Please click on the Image to see the Graph of the point ( 4/33 ).

Tips & Warnings

• To change any Repeating Decimal to a fraction, count how many digits are repeating, depending on the number of repeating digits then that is the number of zeros that we attached to the number ( 1 ). That is if there are three repeating digits then we attach three zeros to the number ( 1 ) making it ( 1000 ), we then Multiply both sides of the Equation by ( 1000 ).

• A Short Cut to writing some Repeating Decimals as Rational Numbers is to Divide the number of repeating digits by the same Number of repeating ( 9 )s.

• That is for example ... , ( 0.888888... ) = ( 8/9). also ( 0.345345345... ) = ( 345/999 ) etc.

• We some times get a Decimal in which only apart of the digits repeat, for example ( 0.4577777... ), in this problem only the digit 7 is repeating, we do the following ... , Set ( X = 0.4577777... ) then Multiply both sides of the Equation by ( 100 ) so as to separate the non-repeating digits from the repeating digits, we now have the Second Equation ...( 100X = 45.77777... ). To stop the digit ( 7 ) from repeating we Multiply both the Left side and the Right side of the Second Equation ( 100X = 45.77777... ) by ( 10 ) and get the Third Equation ( 1000X = 457.77777... ), Subtracting the Second Equation from the Third Equation we get the Fourth Equation ( 900X = 412 ). To solve for X we Divide both sides of the Equation by ( 900 ) and reduce to lowest terms. The result is ... X = ( 412/900 ) or in lowest terms X = ( 103/225 ).