Vedic Math - Base Multiplication Part-2

In the previous article, we learnt multiplication by taking base. We discussed following three cases:
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

And now, we shall discuss rest of the two cases here:
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.

Before going further, let us take few examples of larger numbers with the same method, which we have discussed in previous article.

Example: 6848 x 9997

 6848    -3152
 9997    -0003
----------------
 6845  |  9456     (Refer to previous article for details of approach)
----------------

Example: 87654 x 99995

Let us discuss the fourth case:
D. Numbers are not near the base number, but are near a multiple of the base number, like 20, 30, 50 , 250 , 600 etc.

As you know that this method is applicable in all the cases, but works best when numbers are close to the base. Here, we shall apply the same sutras  “All from 9 and the last from 10” and "Vertically and Crosswise"  discussed earlier and also the sub sutra i.e. "Proportionately".

Take example as 207 x 213 . Here the numbers are not near to any of the bases that we used before: 10, 100, 1000 etc. But these are close to '200'. So, when neither the multiplicand nor the multiplier is near to the convenient power of 10 then we can take a convenient multiple or sub-multiple of a suitable base (as our 'Working Base'). And then perform the necessary operation by multiply or divide the result proportionately. Like in this example, we take 100 as a 'theoretical base' and take multiple of 100 i.e. 200 (100 x 2) as our 'working base'.

207 x 213
  207    +007
  213    +013
  ------------
= 220  |  091
  ------------
= (220 x 200) | 91
= 44000 + 91
= 44091

As they are close to 200, therefore deviations are 7 and 13 as shown above. From the usual procedure (refer to previous article), we get, 220 | 91. Now since our base is 200, we multiply the left-hand part of the answer by 200 and add it to the right-hand part. That is, (220 x200) + 91 {(Left side x base) + right side}. The final answer is '44091'.

Let us take one more example: 406 x 417
Here Working base is 400 ,which is (100 x 4)

   406    +06
   417    +17
   ------------
 = 423  |  102  
   ------------
 = 423 x 400  |  102
 = 169200  |  102
 = 169200 + 102
 = 169302

Another example is, 46 x 43. We can solve it in two ways:
First, we take 10 as a theoretical base and take multiple of 10 i.e. 50 (10 x 5) as our working base.
Second, we take 100 as a theoretical base and take sub-multiple of 100 i.e. 50 (100/2) as our working base.

 46    -4
 43    -7
-----------
 39  |  28
-----------
From our usual procedure, we get 39 | 28. Now,
First way (10 x 5 = 50):
1. Multiply the left-hand part of the answer by 50 (39 x 50)and get 1950 as the first part.
2. Now add 1950 to 28 and get 1978.
3. Final answer is '1978'

Second way (100/2 = 50):
1. Divide the left-hand part of the answer by 2 and get 19.5
Note:-Here 39 being odd, its division by 2 gives us a fractional quotient or decimal number.So, number after decimal(i.e. 5) of the left hand side is carried over to the right hand (as 50).
2. Left-hand part is now '19' and right-hand part is 78(i.e. 50 + 28)
3. Final answer is: 1978

In the above discussed two examples, we can also apply the second method. Like in 207 x 213 , we can also take working base: 1000/4 = 250 or 1000/5 = 200. And in 406 x 417 , we can also take working base: 1000/2 = 500. We have to choose the best convenient multiple or sub-multiple.

Following an example in which you solve it in many ways:
 62 x 48

(1) Working Base 10 x 4 = 40          
        62    +22                                          
        48    + 8                                      
       -----------
        70  |  176                                       
       x40                                                    
       -----------                                          
       2800 |  176                                      
       -----------                                        
       = 2800 + 176                                                                                            
       = 2976          
                                                       
(2) Working Base 10 x 6 = 60        
        62    + 2                                         
        48    -12                                          
       ----------                                      
        50  | -24                                   
       x60                          
       -----------                                    
       3000 | -24                                        
       -----------                                          
       = 3000 - 24                                        
       = 2976                                           

(3) Working Base 10 x 5 =50
        62    +12                                      
        48    - 2                                          
       ----------                                        
        60  | -24                                 
       x50                            
       -----------                                            
       3000 | -24                                          
       -----------                                          
       = 3000 - 24                                          
       = 2976                                               

(4) Working Base 100/2 = 50
        62    +12                                      
        48    - 2                                          
       ----------                                        
        60  | -24                                 
        /2                            
       -----------                                            
        30  | -24                                          
       -----------                                          
       = (30 x 100) - 24                                          
       = 2976

Example: 43 x 47
In this, the base is 100/2 = 50 , so divide the left side by 2 to get the answer.
    43   -7                             (as above we subtract 50 from our numbers)
    47   -3
   ----------
   40  |  21                          

   /2                                     (calculate the left side, then divide the answer by 2)
  -----------
   20  |  21
  -----------
  =(20 x 100) + 21
  =2021
This gives: 43 x 47= 2021

E. Numbers near different bases i.e. 'multiplier' is near to one base and multiplicand is near to other base.
Many of times we need to multiply numbers that are not near to same base but are near to different bases.
In the example given below, you find that one number is close to 1000 and the other is close to 100.

Example: 996 × 97
= 996 × (97 x 10) / 10            (can be rewritten like this)
= 966 x 970 / 10                     (which can futher be written like this)

Let us first solve 966 x 970
996   –004
970   –030
------------
966 |  120
------------
= 966 x 1000 + 120              (multiply left side result by base)
= 966000 + 120
= 966120

So the answer for 966 x 970 is 966120

Now we shall solve the complete equation i.e. will divide it by 10,
  = 966120 / 10
  = 96612

So 996 x 97 = 96612

Example: 10005 x 1002
Here the numbers are close to different bases: 10000 and 1000

So equation can be written as
= 10005 x (1002 x 10) / 10

Now let us solve 10005 X 10020 first

10005     + 005
10020     + 020
-----------------
10025   |   0100
= 10025 x 10000 + 100         (multiply left side result by base)
= 100250000+ 100
= 100250100

So 10005 X 10020 = 100250100
Now let us solve the original equation i.e. will divide the answer by 10
= 100250100/10
= 10025010

So 10005 x 1002 = 10025010

Example: 212 x 104
Rewrite as : 2 x (106 x 104)
            = 2 x 11024
            = 22048

Example: 192 × 44.
Here you can halve 192 and double 44.
= 192 x 44
= (192 / 2) x (44 x 2)              (divide by 2 and multiple by 2)
= 96 x 88

Now solve this equation by procedure defined above
= 96 × 88
= 8448.

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Vedic Math - Base Multiplication

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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Vedic Math - Mathematical magic trick

Let us take a break from calculations and enjoy a small magical mathematics trick.

1. Think a number (any number). To keep it easy, think any number having 1 or 2 digit.
2. Now double this number i.e. multiple by 2.
3. Add 12 to the result
4. Divide the total by 2.
5. Subtract the original number from above result.
6. And you will always get '6' as final result.

Try this one with different numbers, and you will see that the sequence always produces the '6' as result ,no matter which number was originally selected.

For example, if the original number is 15.
1. 2 x 15 = 30
2. 30 + 12 = 42
3. 42 / 2 = 21
4. 21 - 15 = 6

You must be curious that how this magic trick is working. Following is the concept for this magic trick:

Let us take a number be x and the steps performed by us are:
1. 2x
2. 2x + 12
3.(2x + 12) / 2 = x + 6
4. x + 6 - x = 6

So, in last step, we see that whatever number we choose (as x) will finally result in '6' with above calculation. After knowing the formula, now you can change the answer by changing the formula. For example, if you want to change the answer to '4', then you add twice of that '4' i.e. '8' in the second step.

Enjoy this trick!!