Vedic Math - Base Multiplication

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Today, we are going to learn the GENERAL formula of multiplication. This is simple and easy technique and good part is that it is applied to almost all the cases. This method works better when numbers are near their base value. After going through the below discussed method, you will see that multiplication tables are not required for calculation above 5 x 5. You will be able to do all types of multiplication involving bigger multiplicands and multipliers quickly and easily; like for '789789 × 999997'. All the sutras (formulae) of Vedic Math are short and simple; and with the practice of the techniques, most of the calculations become a playful experience for you.

Following is the sutra that we will follow today:

The formulae (sutras) are : “All from 9 and the last from 10” and "Vertically and Crosswise"
The algebraical expression is :(x+a) (x+b) = x (x+a+b) + ab.

From the title of the article, you can understand that today we shall do the multiplication by taking the base of numbers. So, first we need to be familiar what is 'base'. The term ‘base’ in Vedic Math has a broader meaning than you may be used to. We work in a base 10 number system, but within Vedic Math the ‘base’ is the number you will use as a basis for calculation. The numbers taken can be either less or more than the base considered. The difference between the number and the base is termed as deviation. Deviation may be positive or negative.

Now observe the following table.

Number      Base         Number – Base       Deviation
   13             10                  13 - 10                      3

    7              10                     7 - 10                    -3

   89             100                  89 - 100                -11

 1110           1000             1110 - 1000              110

99998          100000       99998 - 100000            -2

So, the deviation obtained are from "All from 9 and the last from 10" sutra (formula).

Following are the cases which we shall discuss here:
A. Numbers are below the base number
B. Numbers are above the base number
C. One number is above the base and the other number is below it
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc
E. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base

Let us discuss these cases one by one.

A. Numbers are below the base number

Working with Base 10
Let us take an easy and simple example to start this technique. Suppose we have to multiply 6 by 8.
Now the base is 10. Since it is near to both the numbers.
Place the two numbers 6 and 8 above and below on the lefthand side (as shown below). Subtract the base value (i.e. 10 in this case) from both of the numbers and write down the remainders (i.e. 4 and 2) on the right-hand side with their deviation sign (-).

6 x 8   


Left side (4 ways to calculate left side)
a) 6 + (- 2)= 4                   (add top left to bottom right)
b) 8 + (- 4)= 4                   (add bottom left to top right)
c) 6 + 8 - 10= 4                 (add numbers in the left column and subtract from base)
d) 10 + (-4)+(-2)= 4          (add numbers in the right column and the base)

Right side
(-4) x (-2) = 4                   (multiply numbers of right column)

Final answer: 6 x 8= 48

Some more examples of the same rule:

 9 x 9            9 x 8              8 x 7            8 x 8               7 x 9
 9    -1          9    -1             8    -2           8    -2             7    -3
 9    -1          8    -2             7    -3           8    -2             9    -1
---------          ----------        ----------         ----------         -----------
 8     1            7     2            5     6            6     4             6     3
---------          ----------        ----------         -----------        -----------

This comes under the "Vertically and Crosswise" Sutra.

Sometimes there can be a carry figure, so let’s have a look on the next example.
 6 x 6
6     -4
6     -4
---------
2   |   6
       1
---------
But here, as '16' is a two digit number, so you need to carry the 1 over to the left side i.e. '2'. Therefore, left hand side now becomes (2 + 1 = 3).
And Final answer : 6 x 6 = 36

Following are some of the examples:

 8 x 5            7 x 6          6 x 5            7 x 5
 8    -2          7    -3         6    -4           7    -3
 5    -5          6    -4         5    -5           5    -5
--------          ----------       ---------       -----------
3  |    0           3  |    2        1  |    0         2  |    5
      1                   1                 2                  1
--------           ----------      ----------      ------------
 = 40              = 42            = 30            = 35

Working base 100
Example: 96 x 98

96    -04
98    -02
---------
94  |  08   =   9408
---------

Working base 1000
Example: 998 x 988

998    -002
988    -012
------------
986  |  024    =   986024
------------

B. Numbers are above the base number. Process will be same as described above.

Working with Base 10
Suppose we have to multiply 13 by 14.
Now the base is 10. Place the two numbers 13 and 14 above and below on the lefthand side (as shown below). Subtract  base (10) from each of the numbers and write down the remainders (3 and 4) on the right-hand side with their deviation sign (+).
 13 x 14
 13      +3                          (as before,subtract 10 from both numbers)
 14      +4
-----------
 17  |    2
          1           = 182
-----------

Left side (4 ways to calculate left side)

a) 13 + (+ 4)= 17                       (add top left to bottom right)
b) 14 + (+ 3)= 17                       (add bottom left to top right)

c) 13 + 14 - 10= 17                   (add numbers in the left column and subtract from base)
d) 10 + (+ 3) + (+ 4)= 17           (add numbers in the right column and the base)

Right side
(+3) x (+4) = 12                   (multiply numbers of right column)

Since 12 is  2-digit number, so carry the 1 over to the 7. Therefore, left hand side is now
(17 + 1 = 18). And Final answer : 13 x 14 = 182

Some more examples of the same rule :-

 12 x 11      16 x 12      13 x 15      14 x 12
 12   +2       16   +6       13   +3       14   +4
 11   +1       12   +2       15   +5       12   +2
---------        ---------       ---------        ----------
 13  |   2      18  |   2      18  |   5        16  |  8
                           1               1
---------        ---------        ---------        ----------
 =132          =192           =195          =168

Working base 100
Example: 102 x 107

102     02
107     07
-----------
109  |  14   = 10914
-----------

Working base 1000
Example: 1005 x 1003

1005     005
1003     003
-------------
1008  |  015   =  1008015
-------------

C. One number is above the base and the other number is below it
In this, one deviation is positive and the other is negative. So the product of deviations becomes negative. So the right hand side of the answer obtained will therefore be subtracted i.e. right side answer get subtracted from (left side answer x base )

Working with Base 10
Suppose we have to multiply 13 by 4
The nearest base is 10.
 13 x 4
 13    +3                   (as before, subtract 10 from both numbers)
  4     -6
----------
  7  |  -18    = 7 x 10 - 18  =  70 - 18  =  52
----------

Left side(4 ways to calculate left side)

a) 13 - 6= 7                        (add top left to bottom right)
b) 4 + 3= 7                         (add bottom left to top right)

c) 13 + 4 - 10= 7                (add numbers in left column, subtract 100)
d) 10 + (+ 3) + (- 6)= 7      (add numbers in the right column and the base)
Right side
(+3) x (-6) = -18                 (multiply numbers of right column)
                         __
So, 13 x 4 = 7 | 18
           = (7 x 10) - 18          (right side answer get subtracted from (left side answer x base ))
           = 70 - 18
           = 52  
Final answer : 13 x 4 = 52

Following are some of the examples:-

 12 x 8            14 x 6             13 x 3          15 x 7
 12    +2         14    +4           13   +3        15    +5
  8     -2           6     -4             3    -7           7    -3
-----------         ----------         ---------         ----------
 10  | -4           10  |  -16         6  | -21       12  |  -15
-----------         ----------         ---------         ----------
 =100-4           =100-16         =60-21        =120-15
 =96                 =84                =39              =105

Working base 100
Example: 102 x 97

102     02
 97    -03
-----------
 99  | -06
----------
  = 9900-6
  = 9894

Working base 1000
Example: 1005 x 993

1005      005
 993     -007
--------------
 998  |  -035
--------------
 = 998000-35
 = 997965

We shall discuss about rest of the two cases in next article. Stay tuned!!

Next Article will cover:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.
E. Numbers near different bases like multiplier is near to different base and multiplicand is near to different base.

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