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Base Method for Multiply two number which are close to power of 10

In Vedic Base Method for fast multiplication  sutra 2 “nikhilam Navtascaramam Dasatah” is useful.  The sutra simply means “all from 9 & last from 10”.

Base method is useful to multiply the numbers close to power of 10. Means the number less than or greater than 10,100.1000 etc.

Here we consider power of 10, like  10,100,1000,10000 as a Base.

In Base method two factors are important-

* Base

* Difference (Number – Base)

Now we solve one simple example to understand the Method.

Ex. i) 8 x 9

Step 1: Identify the Base – As numbers are close to 10, base is 10

Step 2: Find the difference –

Difference = Number – base

8 – 10 = -2 & 9 – 10 = -1

Step 3: Write the number and difference as shown below-

Step 4: we divide our answer in two parts LHS and RHS. We differentiate our answer by drawing slanting line.

Step 5: LHS of the answer is sum of number with cross difference.

So,  LHS = 8+(-1) = 7   OR   9+(-2) = 7

In both case the answer is same, so we can apply any one cross sum.

Step 6: The RHS is nothing but product of the difference. The number of digit in RHS is equal to the number of zeros in the base.

So RHS = (-2) x (-1) = 2

If the number of digits in RHS is not equal number of zeros in base, then there are two conditions

1. Number of digit in RHS is less than the number of zeros in base

In such condition we must add zero in left side of RHS

2.Number of digit in RHS is greater than the number of zeros in base

In such condition the extra digit to be added as carry in LHS of answer.

While solving more sums we understand the above conditions.

General form of base method

Case 1: Both the number are lower than the base-

Ex. ii) 92 x 93

Here base = 100

So Difference 1 = 92 – 100 = -08  &  Difference 2 = 93 – 100 = -07

Now we put all the values in general form –

LHS = 92 + (-7) = 85 & RHS = (-8) x (-7) =56

So, 92 x 93 = LHS/RHS = 8556

Ex. iii) 97 x 99

Here base = 100

So, Difference 1 = 97 – 100 = -03 & Difference 2 = 99 – 100 = -01

Now we put all the values in general form –

LHS = 97 + (-1) = 96 & RHS = (-3) x (-1) = 0 ( As number of digit in LHS is less than number of 0 in base)

Answer 97 x 99 = LHS/RHS = 9603

Ex. iv) 992 x 988

Here base = 1000

So, Difference 1 = 992 – 1000 = -008 & Difference 2 = 988 – 1000 = -012

Now we put all the values in general form –

LHS = 992 + (-12) = 980

RHS = (-008) x (-012) =096 ( As number of digit in LHS is less than number of 0 in base)

Answer 992 x 988 = LHS/RHS = 980,096

Case 2: Both the number are higher than the base-

Here we get positive difference and the remaining method is same.

Ex. v) 19 x 12

Here base = 10

So, Difference 1 = 19 – 10 = 9 & Difference 2 = 12 – 10 = 2

Now we put all the values in general form –

LHS = 19 + 2 = 21

RHS = 9 x 2 = 18

Here base =10, so number of digit in RHS should be 1, but our  RHS = 18 so we carryover 1 in LHS                    —-(According to step 6 condition 2)

Hence, LHS = 21 + 1 = 22

RHS = 8

Answer, 19 x 12 = 228

Ex. vi) 112 x 108

Here base = 100

So, Difference 1 = 112 – 100 = 12 & Difference 2 = 108 – 100 = 08

Now we put all the values in general form –

LHS = 112 + 08 = 120

RHS = 12 x 08 = 96

Answer 112 x 108 = 12096

Ex. vii) 1012 x 1096

Here base = 1000

So, Difference 1 = 1012 – 1000 = 012 & Difference 2 = 1096 – 1000 = 096

Now we put all the values in general form –

LHS = 1012 + 96 = 1108

RHS = 012 x 096 = 1152

Here base =1000, so number of digit in RHS should be 3, but our RHS = 1152 so we carryover 1 in LHS                    —-(According to step 6 condition 2)

Hence LHS = 1108 + 1 = 1109

&     RHS = 152

Answer 1012 x 1096 = 1109152

Case 3: one number is less than base & other number is greater than base

Ex. viii) 12 x 07

Here base = 10

So, Difference 1 = 12 – 10 = 2 & Difference 2 = 7 – 10 = (-3)

Now we put all the values in general form –

LHS = 12 + (-3) = 9  &

RHS = 2 x (-3) = (-6)

Ans = 9/-6 = 90 – 6 = 84   (as place value of 9 is 90)

Answer 12 x 7 = 84

Ex. ix) 107 x 92

Here base = 100

So, Difference 1 = 107 – 100 = 07 & Difference 2 = 92 – 100 = (-8)

Now we put all the values in general form –

LHS = 107 + (-8) = 99 &

RHS = 7 x (-8) = (-56)

Ans = 99/-56 = 9900 – 56 = 9844 (As place value of 99 is 9900)

Answer 107 x 92 = 9844

Ex. viii) 992 x 1015

Here base = 1000

So, Difference 1 = 992 – 1000 = (-008) & Difference 2 = 1015 – 1000 = 015

Now we put all the values in general form –

LHS = 992 + 015 = 1007 &

RHS = (-008) x (015) = -120

Ans =   1007/-120 = 1007000 – 120 = 1006880

Answer 992 x 1015 = 1006880

Case 4: When the base is not the power of 10

Here we will use Vedic sutra Anurupyena. It means proportionality. This method useful when the base is not a power of 10, means numbers are not close to 10,100,1000 etc.

In such case we take nearest base as a nearest multiple number of 10 like 30,40, 60, 250, 700 etc.

This is known as working Base.

Ex. 37 x 45

Here, Base = 10, But the difference is big number, so we consider Working base which is close to our number & also Multiplier of 10.

Hence,  Here, Working Base = 50.

Ex. ix) 48 x 46

Here, Actual Base = 100

Working Base = 50(As both the number are close to 50)

Here, working base = Actual base/2

Now remaining steps are same,

Diff. 1= 48-50 = (-2)

Diff. 2 = 46 – 50 = (-4)

Hence, LHS = 48-04 = 44

As we get working base by dividing by 2, so for actual LHS divide the LHS by 2

LHS = 44/2= 22

RHS = (-02 x -04) = 08 (As actual base is 100)

Answer, 48 x 46 = 2208

OR

We can solve same problem by considering Actual base 10

Actual base = 10

Working base = 50 (Actual base x 10)

Diff. 1= 48-50 = (-2)

Diff. 2 = 46 – 50 = (-4)

Hence, LHS = 48-04 = 44

As working base = 5 x Actual base

So Actual LHS =5 X 44 = 220

RHS = (-2) x (-4) = 8

Answer 48 x 44 = 2208

As we can see here the simplicity is less & there are other method also to do multiplications, only to understand the method here I solve one example.

Try This:

i) 998 X 991  ii) 96 X 91  iii) 1009 x 1011

iv) 9987 x 1007  v) 48 x 41 vii) 111 x 107

Find the following link to study more:

i) Criss Cross method for all type of multiplication

ii) Find the Cube & Cube-root of any number by just observing it.