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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