H) HELPFUL TIPS IN USING ESR

 
ESR EXPRESS

If you think that you already mastered this ESR Method, then it would be much easier to speed up your way of taking the square roots of numbers by doing some kind of modifications. This technique is much effective if the number problem involved is in small value (single or two-digit numbers such as √3, √7, √13, √37, √88,... etc), but as you become familiar with this ESR-Express, then you can probably apply it also for large valued numbers. 

Let's try again to take the square root of two...


...1......             .   (Put a decimal point after 1)
 √2.00'00'00'00        

_______________________________
1.5²     = 2.25
4/3 {           
...........   1.---
2/3 {
1.0²     = 1.00  
..........    3.25 / 2
..........    1.---   
________________________________
 
1) Find the middle square (2.25) and add it to the square of 1 (that is, 1.00)
2) Divide the sum by 2
 3) But this time, don't complete the digits involved in the quotient. The value, 1 followed by a point and ' ---' will be enough. The 'one point something' is good enough to give you an idea that the value you taken is less than 2.00
4) There's also no need to write down 1.25² ( quarter-square).. Simply write down 1.-- and on it's left side, the notation 4/3 above 1.--- and write down the notation 2/1 on the lower left side.
5) Encircle 4/3, which is the area you're are looking for. Do the sampling  
6) Once you located the 'required' digit ( at this time, the digit 4), write 4 next to 1 in your  'answer line'. Do this every time you came up with the required digit to complete your answer.



...1.  4                  .  <<< This is your 'Answer Line'
 √2.00'00'00'00       
 _______________________________________


Let's advance a little...
 ...
 ...
 ...

1.42²  = 1.96'04
14x4 =       5'6   .
(Do not write anything here)
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7) As much as possible, check if the sum will become 'more than' 2.00. Skip this stage by not writing any value for the supposed to be the 'sum'

Take Note:
In doing the sampling, pick first the higher digit (4 first before 3, or 2 first before 1) and if the supposed sum become greater than the number problem (2.00), then try the next lower digit number.  



  
ESR-EXPRESS FOR SQUARE ROOT OF 7

...                         . 
 √7.00'00'00'00 

Step 1:
The number problem given (that is "7"),  is in-between 9 (which is the square of 3) and 4 (which is the square of 2). We can't choose 9 because it is over or beyond 7. We then prefer 4. Therefore the first digit for our answer will be 2. (Reason: The square root of 4 is 2 or √4 = 2)

...2.                        .  (Don't forget the decimal point)
 √7.00'00'00'00 

Step 2:
Find the middle square value. Since it is between 2² and 3², the middle square will then be 2.5², which is equivalent to 6.25


LOWER QUARTER- SQUARE

If you will notice, in taking the square root of 2, the middle square is above or greater than the given number problem (2.00). That is the reason why we choose 1 (or 1.00) as the next lower value ( that supposed to be, or must always be 5 units  below the middle square). We call that quarter square below the middle square as the lower quarter-square. 

UPPER QUARTER-SQUARE

Now, in this this given problem (taking the square root of 7), the middle square value of 6.25 (which is the square of 1.5), is much lower than 7 (or 7.00). So we must consider the next higher value to that middle square ( that supposed to be, or must always be 5 units above the middle square). We will then call that quarter square above the middle square as the upper quarter-square. 


Step 3: 
Perform the Square Averaging Method for 3² and 2.5². 

3.0² = 9.00
2.5² = 6.25
........ 15.25 / 2
.......... 7.---

 INSERT the value ' 7.--- ' in-between 9.00 and 6.25

3.0² =  9.00
.......... 7.--    (Insert Here) 
2.5² =  6.25  
......... 15.25 / 2
........... 7.---

Step 4:
Notice that the quarter square value 7.--- ( seven point something), is considered above or greater than the given number problem (7.00). So, write down between that quarter square and the lower value the notation " 7/6 ".


..... 3.0² =  9.00 
9/8{
................ 7.--    (Insert Here)
7/6 {                     (Either 7 or 6 is the digit we're looking)  
.... 2.5² =   6.25   
............... 15.25 / 2 
...............   7.---

Step 5:
Perform 'SAMPLING'

2.7²    = 4.49
2x14 =   2 8   . 
..........  (Skip)

2.6²    = 4.36
2x12  =  2 4   . 
......... < 6.76 >


Step 6:
Write down 6 as the second digit in our answer

 ..2. 6                      .  (Don't forget the decimal point)
 √7.00'00'00'00 

Step 7: 
Find the next middle-square 

2.6²    = 4.36
2x12  =  2 4   . 
......... < 6.76 >
............    26  .
2.65²   = 7.02

Step 8:
The value 7.02 is greater than 7.00, so consider the lower value 2.6² = 6.76. Perform again the second 'Square Averaging Method' and so on and on...


TRY TO DO IT YOURSELF and check if you come up with the same pattern values below :

..2.  6  4   5   7     .  (Don't forget the decimal point)
 √7.00'00'00'00
_______________________
3.0² =  9.00
.......... 7.--    (Insert Here) 
7/6 }
2.5² =  6.25  
......... 15.25 / 2
........... 7.---
________________________  
 2.7²    = 4.49
2x14 =    2 8   . 
..........  (Skip)

2.6²    = 4.36
2x12 =   2 4   . 
......... < 6.76 >
............    26  .
2.65² =   7.02
4/3 }
............. 6.---
2.60²  =  6.76  . 
........... 13.76 / 2
............. 6.---
________________________

2.64²    = 6.76'16               In the pink shaded area, you will notice that the middle square 
"x8       =   20'8               is below 7.00. we must choose the next higher value above it and that
.........   < 6.96'96 >          is the 2.65². In our previous computation (see blue shaded  
........    .       2 64   .          number above left), we just wrote down 7.02. But to avoid 
2.645² =  6.99'60   .       confusion, simply paste 25 after 7.02
7/6 }                         .            Also, as a guide, draw an arrow going up on the right side of    .............. 7. ---        .       of that pink shaded area to indicate that it is a 'reverse' representation
9/8 }                          .      where the lower 'digits notation' such the 7/6 notation is in the upper
2.65²    =  7.02'25    .    portion (see green shade)
              14. --- / 2   .      
............... 7. ---         .
_________________________

 2.647²   = 6.96'96'49       Suggestion: Leave the space for the sum, "blank" and on the  
"x14     =       3 69 6  .      right side of it, draw a half circle going clockwise with the arrow 
                (Don't Add)     pointing downward to indicate the possible value is 'over'
_________________________


 2.646²   = 6.96'96'36        
"x12     =       3 16 8  .       
                (Don't Add) (Same suggestion)
__________________________

2.645²   = < 6.99'60'25 >      (No need to do SSQ at this time, simply copy the value for   
...............           26'45  .       2.645²  above. Don't forget to paste 25) 
2.6455²  =    6.99'86'70      ( Take Note: The next higher value to 2.6455² is  
.................  6.---                    2.6460² or 2.646²  . It's time to go back and compute for the 
9/8 }                                        sum of 2.646² . That's the reason why I suggest to leave the 
2.646²    =    7.00'13'16         the space for the sum of  2.646²  'blank'.)                  
                   13. ---          / 2        Also by visual check, there's no need to complete to
...................  6.---                    add 6.99'86'70 to 7.001316. Just add the first two digits of
                                                the required values (6.--- + 7.--- = 13.---) and then divide by 2
__________________________

2.6459²  = 6.99'60'25'81                     Take Note: There are instances that the 'sampling'  
" x18      =       47'61'0   .                             procedure will be more than two attempts.
.................. (Leave It Blank) OVER              As a final step, we consider the digit 7 as the
                                                                        final digit to be included in our answer
 2.6458²  = 6.99'60'25'64 
" x 16    =         42'32'0    .
................. (Leave It Blank) OVER

2.6457²  = 6.99'60'25'49 
" x 14    =        37'03'0   . 
..............< 6.99'97'28'49 > 

                (END)  
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