E) MORE PRACTICAL WAY OF SQUARE ROOTING

As to the last discussion (or topic), I shown you the list of possible squares that if, one at a time, we use the method of SSQ, can be presented into a table and study which among the possible squares is the rightful or precise value to consider. But of course, this is a very slow process.

But there is a more practical way of doing it. Okay, let's start from the very beginning...



Question:  What is square root of  2 ?

......
√2  = ?
 
 Step 1: Instead of  putting two zeroes after 2 ( see this ), let's put 8 zeroes grouped by in twos

 ..........           .
√2.00'00'00'00

Step 2 : Find a number that when "being multiplied by itself" will bring out a product  that is near but lesser to that asked number
 ..1......             .   (Put a decimal point after 1)
 √2.00'00'00'00      
   we choose 1 because 1 x 1 = 1 ( or 1² = 1)

Now let's find the square root of 2 using ESR



SQUARE ROOT OF TWO USING ESR METHOD


.. 1. 4  1   4   2     (Put a decimal point after the first digit '1')
 √2.00'00'00'00 

(A)

1.5²     = 2.25
4/3<<              (either 4 or 3 is the next digit)
1.25²   = 1.56
2/1 <<
1.0²     = 1.00    
..........    3.25 / 2
..........    1.62   ( 162 - 6 = 156 -> 1.56)

(B)

1.4²  =      1.16
1x8   =         8  . 
........       <1.96 >     (This value is accepted)
....... .           14   . 
1.45²   =     2.10  (Above 2.00)

(C)

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

(D) 

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) 
..........         1'41  . 
1.415²  = 2.00'22  (Above 2.00)

(E)

1,415²    = 2.00'22 
4/3<<                        (either 4 or 3 is the next digit) 
1.4125²  =  1.99'51 
2/1<<                            
1.41²       = 1.98'81  
................. 3.99'03 / 2
................. 1.99'51

(F)

1.414²   = 1.98'81'16
" x 8       =    1'12'8   . 
............ < 1.99'93'96 >    (This value is accepted)
..............         14'14  . 
1.4145²  = 2.00 08'10   (Above 2.00)

(G)

1.4145²   = 2.00 08'10
4/3<<                              
1.41425² =  2.00'04'03
2/1<<                                  (either 2 or 1 is the next digit)
1.414²      = 1.99'93'96    
................   4.00'02'06 ./ 2
.................. 2.00'04'03

(H)

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


COMPARISON 

Comparing ESR to the traditional "Long-Hand Division", we avoided (if not eliminated), the process of subtraction. We also avoided the process of trying to round-off numbers to become divisible (See this). ESR is based on the "process of locality", meaning, we try to determine exactly where the next digit to consider is located by doing some kind of a different technique. Which of the two methods do you prefer (ESR or Long-Hand Division)? Are you interested to know how ESR works?