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