Multiplication by 9, 99, 999 , 9999 and so on

Actually, multiplication by 9, 99, 999, 9999 and so on series seems easy and simple. This can be achieved by just two operation of subtraction and addition by shifting the positions to the right equal to the number of 9 used to multiply.

The rule that works for all the cases is 

Subtract (multiplicand-1) from multiplier (which is the series of 9), then add the resultant with (multiplicand-1) by shifting the 'n' positions to the right, where 'n' is the minimum number of   multiplier or multiplicand, n=min(no.  of multiplicand digits, no. of multiplier digits) in the series. 

Consider a simple example:

$8 \times 9=72$,

1.  Multiplicand-1 = 8-1=7,
2.  Subtract 7 from 9, 9-7=2,
3. Shift 2 to the right one position and add one zero after 7 and perform bit-wise addition  as 

                                             02

                                             70

                                                          

                                             72

$88 \times 9=792$

1.  Multiplicand-1 = 88-1=87,
2.  Subtract 87 from 9, 9-87=-82,
3. Shift -82 to the right one position and add with 87  i.e 

                                       (-08) 2

                                          87  0

                                                     

                                     (79)    2

$888 \times 9=7992$

1.  Multiplicand-1 = 888-1=887,
2.  Subtract 887 from 9, 9-887=-882,
3. Shift -882 to the right one position and add with 887  i.e 

                                     (-088) 2

                                        887  0

                                                    

                                    (799)   2
$8 \times 99=792$
       

1. Multiplicand-1= 8-1=7,
2. Subtract 7 from 99 = 99-7=92,
3. Shift 92 to the right by one and perform the addition as  

                               700

                               092

                                             

                               792