F. Multiple-Digit Number Group Squaring

Now that you have the idea of squaring numbers 'digit per digit' using SSQ, it is also an advantage if you can do things mentally or in short cuts.

In INTRODUCTION, I required you to remember the "index squares" of the single-digit numbers from 0 to 9. But this time, it will also be an "added advantage" if you can also memorize the squares of other numbers, to at least up to 15.( To be frank, I myself can only memorize the squares of numbers up to 15)

10²  = 100       11²  = 121        12²   = 144        13²  = 169        14²  =  196        15²  = 225

 
In my third article (Three-Digit Number Squaring), I introduced to you to the word "groupee" as the fun way of calling a group of digits considered as one unit. The above numbers from 10 up to 15 can be considered as groupees, They consist of two digits that act as one number. We can no longer say 11 as "one-one" but rather, as, "eleven", neither 10 as "one-zero" but rather as "ten".

In SSQ, there are instances (if you know the technique), that we can divide a multiple-digit number into groups of digits (groupees) as a "short way" of squaring  numbers. To further understand what I'm talking about, it is important that I must first, show you an example


Question:

124²  = ?


There are two ways of getting the square value of 124


Digit Per Digit SSQ Approach

(Example 1)
 

124² = (Blank)
12..²  = 01'04..
1x4   =      4         . 
124² =   01'44'16
" x 8  =        9'6   .
.............01'53'76 

Groupee SSQ Approach

If you look closely at the given number 124, you may notice that it consist of a groupee (12) and a single digit 4. So, we can simplify our way of squaring that number by...

(Example 2)

 124²  =  01'44'16 
12x8    .        9'6   .          
 .............01'53'76                         


Groupee Index Squares    

10²  = 01'00       
11²  = 01'21        
12²  = 01'44        
13²  = 01'69        
14²  = 01'96        
15²  = 02'25


If you are gifted enough, that you can memorize the squares of numbers more than the listed numbers above, then it will be really an easy thing for you to do "groupee squaring"  

(Example 3)

412²  = ?

If you will notice, the given number 412, is again, a combination of a single digit followed by a groupee...

 412²  = 16'01'44
 4x24 = .    96     . 
.............16'97'44

Again, if you look at them closely, 124 and 412 are somehow, related, in a sense that we use the same index squares (16 and 01'44), except their positions are interchanged


Adjust SP Two Digits to The Left


412²  = 16'01'44 ← PSL
 4x24 =.    96^^ ← SP ( you might notice that 6 of 96 is aligned to second 1 of 16'01'44) 
.............16'97'44


It is very important that you 'be more careful' in aligning the sub-product  to the PSL when doing 'groupee squaring'. Take note that in Example 2, the last digit is a single digit 4, so, we simply adjust the SP one digit to the left. While in Example 3, the last digit is not exactly the single digit 2 but the groupee '12', which happens to be a two-digit groupee. In that case, adjust the last digit of the SP, two digits to the left.


(Example 3)

1,413²  = ?


1413²   = 01'96'01'69
14x26  =        3'64     .
...............01'99'65'69




 

E. Squaring Multiple-Digit Numbers

Four-Digit Number Squaring

Question: What is the square of 2,384?

2,384² = ?

Following the rules we just learned...

2,384²   = (leave this space blank) 
23..²     = 04’09.. ← PSL 1 
+2x3x2=   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)  
238..²  = 05’29'64   ← PSL 2   
23x16  = .   36'8  .   ← SP 2  (8 of 36'8 is aligned to 6 of 05'29'64)
2384²=   05'66'44'16   ← PSL 3 (new PSL) 
" x 8 =.        1'90'4   .  ← SP3 (new SP) (4 of 1'90'4 aligned to 1 of 05'66'44'16)
..............05'68'34'56   ← T-Sum

Write in the 'blank space' above 23..² , the actual number value of  2,384²  

2,384²   = 5,683,456   

You might noticed that the same procedures are applied in squaring a multiple-digit number such as in the four-digit number 2384. The only difference is that as the digits increases, so do the way we do the SSQ method.

As a summary:

1) Do the two-digit squaring for first two digits 2 and 3.
2) The partial sum become part of the new PSL (PSL2) and we insert the next digit 8 along with its equivalent index square 64.
3) Add the SP2 to  PSL2 to get the new partial sum.
4) The partial sum become part of another new PSL (PSL3) and we, again insert the next digit 4  along with its equivalent index square 16.
5) Add the new Sub-product (SP3) to the PSL3 to get the Total Sum (T-Sum)

The " Notation

You might also noticed that instead of writing, 238 x 8 at the left side of the SP3 line (look the yellow shade below), we simply indicated the " notation. We do that so that we can save time and effort in repeating to write numbers.Just makes it sure that you properly multiply the "all to the left" of the last digit, to the double value of the last digit itself ( remember Dou-LAL?) 

23..²     = 04’09.. ← PSL 1 
+2x3x2=   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)  
238..²  = 05’29'64   ← PSL 2   
23x16  = .   36'8  .   ← SP 2  (8 of 36'8 is aligned to 6 of 05'29'64)
2384²=   05'66'44'16   ← PSL 3 (new PSL) 
" x 8 =.        1'90'4   .  ← SP3 (new SP) (4 of 1'90'4 aligned to 1 of 05'66'44'16)
..............05'68'34'56   ← T-Sum

Underlined Last Digit

You might also noticed that the 'involved' last digit is always 'underlined'. As a beginner, you are required to do this but as you become familiar with the method of Systematic Squaring (SSQ), there's no need to do this. Just make sure that you're doing the right thing.  


23..²     = 04’09.. ← PSL 1 
+2x3x2=   1'2    . ← SP1  (2 of 12 is aligned to second 0 of 04'09)  
238..²  = 05’29'64   ← PSL 2   
23x16  = .   36'8  .   ← SP 2  (8 of 36'8 is aligned to 6 of 05'29'64)
2384²=   05'66'44'16   ← PSL 3 (new PSL) 
" x 8 =.        1'90'4   .  ← SP3 (new SP) (4 of 1'90'4 aligned to 1 of 05'66'44'16)
..............05'68'34'56   ← T-Sum




Exercise: (Do It Yourself)

Now that you have the clear idea of how SSQ works, try to square the following given numbers:

1) 45,896

2) 398

3) 978,675