D) NEW METHOD OF SQUARE ROOTING

Let's start learning this new method by trying to analyze 'how the square root of 2' comes out as 1.4142, without the traditional method of the long-hand division.

 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 create a table of list of squares from 1.0 up to 2.0 using the Two-Digit SSQ

2.0²   .   04.00   .
1.9²   .   03.61   .
1.8²   .   03.24   .
1.7²   .   02.89   .
1.6²   .   02.56   . 
1.5²   .   02.25   .      Take note that 2.00 is in-between 2.25 and 1.96
1.4²   .   01.96          We then choose 1.96 as the value nearest but less than 2.00
1.3²   .   01.69   . 
1.2²   .   01.44   .  
1.1²   .   01.21   . 
1.0²    .  01.00   .                                     

The first digit that we written (1._ _ _ _) needed four blanks to be filled up. We can say that the √2 is about "one point something". 

So, we write down 4 as the next digit

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

To get the third digit, we must create another table, this time between the values of 1.40² and 1.50² using the Three-Digit SSQ

1.50²   .   02.25'00
1.49²   .   02.22'01
1.48²   .   02.19'04
1.47²   .   02.16'09
1.46²   .   02.13'16
1.45²   .   02.10'25
1.44²   .   02.07'36
1.43²   .   02.04'49
1.42²   .   02.01'64
1.41²   .   01.98'81    We choose 1.9881 as the value nearest but less than 2.00 
1.40²  .    01.96'00


So, we write down 1 as the third digit

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

 
To find out the next two remaining digits, we must create another sets of tables for the Four-Digit SSQ and Five-Digit SSQ (see Multi-Digit SSQ)

For the fourth digit, we can say that 2.00 is in-between the values 1.410² and 1.420²


1.420²   .   02.01'64'00
1.419²   .   02.01'35'61
1.418²   .   02.01'07'24
1.417²   .   02.00'78'89
1.416²   .   02.00'50'56
1.415²   .   02.00'22'25
1.414²   .   01.99'93'96      We choose 1.999396 as the value nearest but less than 2.00 
1.413²   .   01.99'65'69
1.412²   .   01.99'37'44
1.411²   .   01.99'09'21
1.410²   .   01.98'81'00




So, we write down 4 as the fourth digit

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

 
TIP:

If you wish to get the values of 1.411² up 10 1.419² , try the Groupee SSQ


Example:
 
1.411²  = 01.'96'01'21
14x22 =  .     3'08     .
..............01.'99'09'21


For the fifth digit, we can say that 2.00 is in-between the values 1.4140² and 1.4150²

1.4150²  .   02.00'08'10'25
1.4149²  .   02.00'19'42'01
1.4148²  .   02.00'16'59'04
1.4147²  .   02.00'13'76'09
1.4146²  .   02.00'10'93'16
1.4145²  .   02.00'0810'25
1.4144²  .   02.00'05'27'36
1.4143²  .   02.00'02'44'49
1.4142²  .   01.99'99'61'64   We choose 1.99996164 as the value nearest but less than 2.00 
1.4141²  .   01.99'96'78'81
1.4140²  .   01.99'93'96'00   


So, we write down 2 as the fifth digit

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