It is not one ninth

Title : It is not one ninth

Fiction

It is not one ninth

Billy was given homework with converting repeating or terminal decimals into fractions and converting fractions into repeating or terminal decimals

The back of his textbook gave answers to the odd numbered problems as examples but he was required to answer the even numbered problems

He was supposed to write a short answer as a fraction or a repeating or terminal decimal in a similar format to the back of the book but instead he wrote essay answers for the repeating decimals and short answers for the terminal decimals

Here is an example of the kind of answers he gave

( 1 / 9 ) - 0.1 = ( 1 / 9 ) - ( 1 / 10 ) =
( 10 / 90 ) - ( 9 / 90 ) = 1 / 90

1 / 9 - 0.11 = 1 / 90 - 1 / 100 = 1 / 900

0.2 - ( 1 / 9 ) = ( 2 / 10 ) - ( 1 / 90 ) = ( 18 / 90 ) - ( 10 / 90 ) = 8 / 90

0.12 - ( 1 / 9 ) = ( 12 / 100 ) - ( 100 / 900 ) = ( 108 / 900 ) - ( 100 / 900 ) = 8 / 900

0.112 - ( 1 / 9 ) = ( 112 / 1000 ) - ( 1000 / 9000 ) = ( 1008 / 9000 ) - ( 1000 / 9000 )
1.112 - ( 1 / 9 ) = 8 / 9000

0.1112 - ( 1 / 9 ) = ( 1112 / 10000 ) - ( 10000 / 90000 )
0.1112 - ( 1 / 9 ) = ( 10008 / 90000 )  - ( 10000 / 90000 ) = 8 / 90000

0.1 < 1 / 9 < 0.2
0.11 < 1 / 9 < 0.12
0.111 < 1 / 9 < 0.112
0.1111 < 1 / 9 < 0.1112

0.111...2 = limit as N approaches infinity of ( 1 / 9 ) + [ 8 * ( 10 ^ - N ) / 9 ]
0.111...1 = limit as N approaches infinity of ( 1 / 9 ) - [ ( 10 ^ - N ) / 9 ]

No matter how many ones you put behind the decimal point it is always less than one ninth

So although repeating ones after the decimal point gets you closer to one ninth it is never equal to one ninth no matter how many ones you put

If you want to represent one ninth as a decimal of a finite number of digits using all ones for every digit after the decimal point gives you the closest possible result and putting all ones except for the second digit being a two gives you the second closest possible result

So although it is the best approximation for one ninth as a decimal a zero followed by a decimal point followed by a string of ones is only as close to one ninth as you can represent it but never equal to one ninth

Here is another example

0.6 - ( 2 / 3 ) = ( 18 / 30 ) - ( 20 / 30 ) = - 2 / 30

0.66 - ( 2 / 3 ) = (  198 / 300 ) - ( 200 / 300 ) = - 2 / 300

0.666 - ( 2 / 3 ) = ( 1998 / 3000 ) - ( 2000 / 3000 ) = - 2 / 3000

0.7 - ( 2  / 3 ) = ( 21 / 30 ) - ( 20 / 30 ) = + 1 / 30

0.67 - ( 2 / 3 ) = ( 201 / 300 ) - ( 200 / 300 ) = + 1 / 300

0.667 - ( 2 / 3 ) = ( 2001 / 3000 ) - ( 2000 / 3000 ) = + 1 / 3000

0.6 < 2 / 3 < 0.7

0.66 < 2 / 3 < 0.67

0.666 < 2 / 3 < 0.667

0.666...7 = limit as N approaches infinity of ( 2 / 3 ) + [ ( 10 ^ - N ) / 3 ]
0.666...6 = limit as N approaches infinity of ( 2 / 3 ) - [ 2 * ( 10 ^ - N ) / 3 ]

No matter how many sixes are in a row after the decimal point it is always less than two thirds so a string of sixes is not two thirds but only approaches two thirds up to as close as you want without ever equalling it.  

Furthermore if the last digit is changed from a six to a seven it is always closer to two thirds then if the last written digit is a six therefore two thirds would always be more closely approximated not by a string of sixes but by a string of sixes with the last digit replaced with a seven therefore it is not best policy to represent two thirds as a string of sixes but as a string of sixes with the last digit replaced with a seven if your goal is to write a number as to close to two thirds as possible in decimal format with a certain number of digits.

Here is how the teacher responded

Billy's teacher reading this was about to mark the paper with a big F because the answer in the back of the book is one ninth and Billy's answer disagreed with the answer in the back of the teacher's edition of the textbook

But then Billy's teacher remembered Billy was a clab person and diagnosed with a learning disability

She suddenly decided to give Billy an A plus instead

Now Billy's teachers photocopied all the homework she graded and also gave the students a copy of the homework and told them to keep it so that if there are mistakes in the grading process they can go back to the original

One day Billy's teacher found out that Billy and Billy's parents had political views she disagreed with

She went back and told Billy I am sorry I made a mistake grading that paper earlier you actually got an F

Her obligation to give good grades to Cklab people and people with learning disabilities did not apply to people with politically incorrect opinions after all

The system only punished people who gave bad grades to Cklab people if the Cklab people were on the correct side politically

If the Cklab people were on the wrong side politically she would be rewarded for giving them bad grades because after all they betrayed all other Cklab people by having opinions, ideas, values or beliefs that do not represent the official positions of the media

The primary reason after all for giving Cklab people special immunity is because they were doing what the media wanted them to do to a greater extent than the Shining people so they were generally given the benefit of the doubt unless proven to express wrong viewpoints