Chapter

Before we proceed with the actual subject matter, let us learn the concept of ‘Purak’ or complement.

Finding ‘Purak’: To find the Purak of a given number we simply subtract the number from its base 10.  This then is called ‘Purak” or the complement of the given number.

Example:

Number                                  Base                Complement              (Purak)

8                                              10                    8 - 10                         = -2

6                                              10                    6 – 10                         = -4

5                                              10                    5 – 10                         = -5

7                                              10                    7 – 10                         = -3

Addition       

In the vedic system of addition we use the complements of the digits.  As shown above the complement of any given digit is always of the opposite sign, hence whenever we need to add a digit, its complement is added with its proper sign.  Let us illustrate addition:

   .

  9

   .

+8

___

23

___     

The dot above the numbers signify 1 (1 dot is equivalent to 1)

Step 1:  Starting from the bottom, we find that 6 + 8 = 14 which exceeds 9 and hence complement of 8 (ie - 2) is added to 6.  The result is 6 – 2 = 4 and a dot is kept over 8.

Note: Whenever we use the complement we place a dot above the number.

Step 2:  Now resultant 4 from step 1 is added to 9 giving 13 which again is greater than 9.  So again we add the complement of 9 (- 1) to 4 giving us 3 and a dot is placed over the 9.

Step 3:  We now write the resultant number 3 as the unit digit in the answer.

Step 4:  One dot is equivalent to 1 so now we count the number of dots which are two in this case so we place 2 in the tenth place of the answer giving us the final answer 23.

Note that nowhere in the addition the digit 9 is exceeded.

Now, let us examine a three column addition (addition of 3 digit numbers) example :

III        II          I

.          

7          4          6

.                       .

+9        2          2

            .          

+3        8          4

+4        2          5

__________________________________

Step 1:  Starting from the bottom of the first column 5 + 4 = 9, 9 – 8 (Complement of 2) = 1 and 1 + 6 = 7.  Place one dot over the 2.

Step 2:  We write 7 as the unit digit in the answer and carry over 1 (equal to 1 dot ) to the second column.

Step 3:  1 (carried over) + 2 = 3, 3 -2 (Complement of 8) is equal to 1, (1 + 2) = 3 and 3 + 4 = 7.  Place one dot over the 8.

Step 4:  We write resultant 7 at the tenth place and carry over 1 to the third column.

Step 5:  1 (carried over) + 4 = 5, (5 + 3) = 8, 8 – 1(Complement of 9) = 7 and 7 – 3 (Complement of 7) = 4.  Place one dot each on 9 and 7.

Step 6:  Write the resultant digit 4 in the 100^th place and count the number of dots, two in this case, in this column.  Hence, 2 is placed in the 1000^th place. 

Hence the complete answer is 2477.

Exercises:

8 + 6 + 5 = 19

23 + 32 + 78 = 133

786 + 108 + 369 = 1,263

7856 + 3257 + 9987 = 21,100

1235 + 687 + 33 + 7 = 2, 102

Note in the last exercise you must have the same number of digits (687 will become 0687 and 33 will become 0033).  To achieve that you will need to place preceding zeros.

Subtraction:

In the vedic system of subtraction, we use the complements of the digits.  As you know the complement of any given digit is always of the opposite sign, hence whenever we need to subtract a bigger digit from a smaller digit, its complement is subtracted with its proper sign.  Let us illustrate subtraction:

  7 1 0 2

  .    .  .

________

4   4 1 8

________

Rule:  Whenever we need to use the complement of any digit, the adjacent digit is increased by unity (1) which is shown by placing a dot over it.

Step 1:  Since 4 is greater than 2 we subtract its complement from 2  i.e 2 – (- 6) = 8 and place a dot over the 8, which is adjacent to 4.

Step 2:  Now 8 has a dot over it so it become 9 (8 + 1).  This again cannot be subtracted from zero directly.  So we use the complement of 9 which is -1 giving 0 - (-1) = 1.  Therefore a dot is placed over the adjacent digit 6.

Step 3:  6 with a dot over it becomes 7 so we subtract its complement (- 3) from 1.  1 - (-3 )= 4 and a dot is placed over the adjacent digit 2.

Step 4:  2 with a dot over it becomes 3 which is less than 7 therefore we can directly subtract 3 from 7 so 7-3=4.

Step 5:  You can now derive the final answer as 4418.

Example 2:

4  1  7  0 2

  .       .    .

2  5  4  1  3

__________

Step1:  2 - (-1) = 3 and put a dot over the 8

Step2:  0 – (-1) = 1 and place a dot over the 2

Step 3:  7 – 3 = 4

Step 4:  1 - (-4)  = 5 and place a dot over the 1

Step 5:  4 - 2 = 2

Final answer is 25413.  Please note that we have not used the complement in step 3 because 3 is less than 7.

Exercises:

 92 – 69 = 23

156 – 88 = 68

459 – 348 = 111 (Note no compliments were used in this exercise)

786 – 678 = 108

3269 – 1973 = 1296

4789 – 3993 = 0796

10001 – 8888 = 01113

45454 – 27786 = 17668

41902 – 16289 = 25613

55894 – 47987 = 07807