** 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

**Answer** = LHS/RHS = 72

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

**Answer**, 92×93=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) = 03 ( 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.

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