G) HOW ESR WORKS: PART II

Now that we already knew the next digit after 1 (in trying to get the square root of 2 ), the next thing to do is to determine the next digit after "1.4... "

Actually, the basic rules in ESR (Easy Square Rooting) are just limited into the following simple instructions:

1) Find a single digit which index square is nearest but less than the given number problem as our first digit.
2) Find the middle square value. Determine if the middle square is greater or lesser than the given number problem.
3) Find the quarter-square value (either located below or above the middle-square). 
4) Determine which next digit to consider by sampling the 'known' digits.


In Part I, we did the 'sampling' of digits and we choose 4 as the next digit after 1. The question then is, what to do next?


MIDDLE SQUARE: (Phase 2)

The next thing that we must do is to determine again the middle square between 1.4²  and 1.5².

The middle square between 1.4² and 1.5² is 1.45² 

1.5²    = 2.25
1.45²  = ?
1.4²    = 1.96


Digit Per Digit SSQ

We can determine the value for 1.45  by doing the Three-digit SSQ

1.45²    = 1.96'25
14x10 =      14'0   .
.............. 2.10'25
 

COPY AND PASTE

But there is this more practical way of determining the next middle-square value by a technique which I called "copy and paste" (Sounds familiar?). How it works?

 We already knew the next digit after 1 is the digit 4 (The partial answer as 1.4... ). The equivalent square value for 1.4²  is 1.96. 

Activity 1: Write down 1.4² on the left side of the equal sign  and its corresponding equivalent value on the right side of the equal sign.

1.4²  = 1.96

Activity 2: Copy the digits on the left side of the equal sign, except the square sign ( ² ) and the decimal point (if any). Put it under 1.96. 

1.4²  = 1.96
............. 14

Activity 3: Add 14 and 1.96

1.4²  = 1.96
.........     14 .
...........2.10

Activity 4: Paste 25 next to the sum that we got from adding 14 to 1.96 (Optional)

1.4²  = 1.96
..........    14 .
.........  2.10'25

If you will notice, the technique that we used in determining the 'first' middle-square is based on Two Digit SSQ Ending in Five. Now that we have  "1.4... " as our partial answer, determining the 'second' middle-square needs a different approach and this technique (Copy and Paste) is the most effective and practical way we can apply.


NEXT QUARTER-SQUARE

Again, after determining that the middle-square value  (2.10) is above 2.00 (the given number problem), we can now say that the next digit to consider (the third digit), is below the digit 5. Still, we need to know the 'Quarter-Square Value' to cut down our search from four digits into two candidate digits. 

Actiivity 1: Add the upper limit (2.10) to the lower limit (1.96)

1.45²  =  2.10
1.4²    =  1.96  .
............. 4.06

Activity 2: Divide the sum by 2.

1.45²  =  2.10
1.4²    =  1.96  .
............. 4.06  ./ 2
............  2.03

Take Note: At this point, there's no need to subtract six. Why? If you complete the value for 1.425²,  what you got is:

1.425²  = 1.96'06'25              (Applying Groupee SSQ)
14x50² =      7'00      .
.............  2.03'06'25

On the other hand, if we complete the representation for both 1.45² and 1.4², what we have is:

1.45²  = 2.10'25
1.40²  = 1.96'00  .
..........  4.06'25 / 2
..........  2.03'12 remainder 1

With the value 2.03'12, we can apply "the always minus six" notation

20312 - 6 = 20306  (Take note 20312 is the floating value for 2.0312) 
1.425²  = 2.03'06'25  (Paste the digits 2 and 5)

Applying the Square Averaging Method...

1.45²   = 2.10
4/3<< 
1.425² = 2.03    (Insert the quarter square value here)
 2/1<<              (either 2 or 1 is the next digit)
1.4²     = 1.96  .
............  4.06 ./ 2
              2.03 


SAMPLING (Part 2)

Again, we determined that the quarter-square value of 2.03 is above or higher (greater), than the given number problem (2.00). So, we must consider the digits below it and that are the digits 2 and 1. Now it's time to do the Sampling, the second time around...

1.42²  = 1.96'04
14x4 =       5'6   .
............ 2.01'64   (This value is over 2.00, not accepted)

1.41²   = 1.96'01
14x2    =     2'8   .
............<1.98'81 >  (This value is accepted)


Now, we determined the third digit as 1.... 

.. 1. 4  1                
 √2.00'00'00'00

 
FOURTH AND FIFTH DIGITS

Looking for the fourth and fifth digits, all you have to do is to repeat the pattern for getting the middle square (phase 2), that is the 'Copy and Paste' technique. To know the quarter values, do the 'Square Averaging' Method

1.41²   =  1.96'01
14x2    =      2'8   .
............ <1.98'81 >  (This value is accepted)
..........          1'41  . 
1.415²=    2.00'22      (Above 2.00)  
4/3 <<                      (either 3 or 4 is the next digit)
1.4125²  =  1.99'51 
2/1 << 
1.41²     =   1.98'81  . 
..............   3'99'03 / 2
..............   1.99'51



SAMPLING (Part 3)


1.414²  =  1.98'81'16
" x 8    =        112'8   . 
............< 1.99'93'96 > (This value is accepted)




We choose 4 as the fourth digit...

1.414²  =    1.98'81'16
" x 8    =          112'8   . 
............   < 1.99'93'96 >
..............           14'14   
1.4145²  =   2.00'08'10    (Above  2.00)
 4/3 <<
1.41425² =   2.00' 01'03
 2/1<<                               (either 1 or 2 is the next digit)
1.414²     =   1.99'93'96  
................    4.00'02'06 / 2
..................  2.00' 01'03      


SAMPLING (Last Part)

1.4142²  =  1.99'93'96'04
" x 4      =          5'65'6          
............. < 1.99'99'61'64 >  (This value is accepted)


The final digit will then be 2....

.. 1. 4  1   4   2    
 √2.00'00'00'00