ARITHMATIC – LET’S MAKE IT A FUN (5)

‘Supplementary’ is used in many of the Vedic Math Sutras. The Supplementary of a number is the number which when added to the original number gives us a ‘round’ number, that means 10, 100, 1000, 10000, 100000 or any number made up of ‘1’ and zero(es). For example 47 is added to 53 to find 100, therefore 47 is the Supplementary of 53. Similarly when 61846 is added to 38154 to make it 100,000, 61846 is the Supplementary for 38154. The use of Supplementary is very useful in addition, subtraction and multiplication. To find the supplementary of a number, the following SUTRA of Vedic Math can be very helpful:

ALL FROM 9 LAST FROM 10

The sutra says that such a digit is added individually to all the digits of a number so that all but the last digit comes to ‘9’ and the last as ‘10’, the new number is the supplementary of the given number or we can say that by adding the two numbers, we will get an answer consisting of ‘1’ and ‘0’ or ‘0’es. To understand the concept, let’s discuss the following examples:

2 3 4 1 6 7 3 8 2 9 8 4

7 6 6 8 3 2 6 1 7 1 1 6
1 0 0 0 1 0 0 0 0 0 0 0 1 0 0

In the first example 2+7=9, 3+6=9 and 4+6=10, in the second example 1+8-9, 6+3=9, 7+2=9, 3+6=9, 8+1=9, 2+7=9 and 9+1=10 while in the third 1+8=9 and 4+6=10. When the original number is added to the supplementary so found the answer is always made up of ‘1’ and zeroes.

Let’s try to understand the use of supplementary in Vedic formulae for multiplication. This is understandable that smaller the ‘supplementary’ easier will be the solution. This is not so that if the ‘supplementary’ is larger, it can’t be applied in these formulae but there will be no use of such short cut that is difficult than the ‘long cut’. That is why these formulae are most effective where there is smaller ‘supplementary’, that means for numbers those are near to 10, 100, 1000, 10000, 100000 i.e. 9992, 999989, 91, 985 etc.

First of all, we will discuss squaring with the help of supplementary:

Example: 997 x 997 The answer will be in 3+3=6 (***,***) digits.
997 + 3 = 1000 So supplementary of 997 is 3
First step: 997 – 3 = 994 First three digits 994,***
Second step: 3 x 3 = 9 Last three digits 994,009
So 997 x 997 = 994,004

Example: 99991 x 99991 The answer will be in 5+5=10 (*****,*****) digits.
99991 + 9 = 100000 So supplementary of 99991 is 9
First step: 99991 – 9 = 99982 First five digits 99982,*****
Second step: 9 x 9 = 81 Last three digits 99982,00081
So 99991 x 99991 = 99982,00081

Try to multiply 9995 x 9995 and 9999989 X 9999989.

Now we will discuss multiplying a number by equal number ‘9’:

Example: 36254 X 99999 There will be an answer in 5+5=10 (*****,*****) digits.

First Step: 36254 – 1 = 36254 First five digits 36253,***** (Always deduct ‘1’ )
Second Step: Find the supplement for 36254 (ALL FROM 9 LAST FROM 10)
36254 + 63746 = 100000 Last five digits 36253,63746

So 36254 X 99999 = 36253,63746
Example: 47 X 99 There will be an answer in 2+2=4 (**,**) digits.

First Step: 47 – 1 = 46 First two digits 46,** (Always deduct ‘1’ )
Second Step: Find the supplement for 47 (ALL FROM 9 LAST FROM 10)
47 + 53 = 100 Last five digits 46,53

So 47 X 99 = 46,53

One more to multiply two such numbers are 10, 100, 1000, 10000, 100000 etc.

Example: 993 X 996 There will be answer in 3+3=6 (***,***).

9 9 3 9 9 6

0 0 7 0 0 4

First step: 993 – 4 = 989 First three digits 989,*** (Deduct crosswise)
Second Step: 7 X 4 = 28 Last three digits 989,028

So 993 X 996 = 989,028

Example: 999991 X 999998 There will be an answer in 6+6=12 (******,******) digits.

9 9 9 9 9 1 9 9 9 9 9 8

0 0 0 0 0 9 0 0 0 0 0 2

First step: 999991 – 2 = 999989 First six digits 999989,****** (Deduct Crosswise)
Second Step: 9 X 2 = 18 Last six digits 999989,000018

So 999991 X 999998 = 999989,000018

Example: 9975 X 9988 There will be answer in 4+4=8 digits

9 9 7 5 9 9 8 8

0 0 2 5 0 0 1 2

First step: 9975 – 12 = 9963 First 4 digits 9963,**** (Deduct Crosswise)
Second step: 25 X 12 = 300 Last 4 digits 9963,0300

So 9975 X 9988 = 9963,0300

If there is number having repeated numbers near 100,1000,10000 or any such number, the given number is multiplied by with help of a supplementary.

Example: 95959595 X 17 There will be answer in 8+2=10 (**,******,**) digits.
95 + 5 = 100 So supplementary for 95 is 5.
First step: 17 -1 = 16 First two digits 16,******,**
Second step: 17 X 5 = 85 and 100 – 85 = 15 Last two digits 16,******,15
Third step: 16 + 15 = 31 6 digits in the middle 16,313131,15

So 95959595 X 17 = 16,313131,15

Example: 9898 X 36 There will be answer in 4+2=6 (**,**,**) digits.
98 + 2 = 100 So supplementary for 98 is 2.
First step: 36 -1 = 35 First two digits 35,**,**
Second step: 36 X 2 = 72 and 100 – 72 = 28 Last two digits 35,**,28
Third step: 35 + 28 = 63 6 digits in the middle 35,63,28

So 9898 X 36 = 35,63,28
Such kind of small and large multiplications are helpful in generating confidence for mathematics among the students.

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Wait for much more in the coming articles ………………….