Multiplication in less than 5 seconds: The Base Method or Nikhilam Method Of Multiplication
Like the Criss Cross System of Multiplication, the Base Method, also known as the Nikhilam method, is a wonderful contribution to Vedic Mathematics. When standard multiplication is used, it can often take a long time to calculate the result, making it incredibly helpful in some situations. You can find the multiplication in less than 5 seconds using the method outlined in the Base Method. In order to define this system, Swamiji provided the following Sanskrit sutra:
“Nikhilam Navatascaram Dasatah.” Its translation is “all from 9 and the last from 10.”
To comprehend the other Vedic Mathematics equations, it is essential to study the Base Method. Because a specific number is used as the base in this approach, it is known as the Base Method. Although any integer can serve as the base, powers of 10 are typically used. Numbers like 10, 100, 1000, and 10000 are among the powers of ten. Depending on the numbers in the query, we use a specific base. Suppose we are requested to multiply 95 by 98. Since both of these numbers are closer to 100, the correct base in this scenario would be 100. The correct basis for multiplying 1004 by 1021 would be 1000 because both values are closer to that number.
We will find the answer in two parts — the left hand side and the right hand side. The left hand side will be denoted by the acronym LHS and the right hand side will be denoted by the acronym RHS.
Let us have a look at the procedure involved in this technique of multiplication.
STEPS
(a) Find the Base and the Difference
(b) Number of digits on the RHS = Number of zeros in the base
(c) Multiply the differences on the RHS
(d) Put the Cross Answer on the LHS
These are the four primary steps that we will use in any given problem.
To understand how the system works we will solve different questions.
Find 97 X 99.
STEP A: Find The Base and the difference.
The first part of the step is to find the base. Have a look at example A. In this example the numbers are 97 and 99. We know that we can take only powers of 10 as bases. The powers of 10 are numbers like 10, 100, 1000, 10000, etc. In this case since both the numbers are closer to 100 we will take 100 as the base.
We are still on step A. Next, we have to find the differences.
In this example, the difference between 100 and 97 is 3 and the difference between 100 and 99 is 1, I.e.
Difference between 100 & 97 = 100 – 97 = 3
Difference between 100 & 99 = 100 – 99 = 1
Base:
100
97 – 3
99 – 1
Step B: Number of digits in RHS = No. of zeros in the base.
Step B is now at hand. The left hand side and the right hand side, sometimes referred to as the LHS and RHS, are where we will get the answer to the multiplication question in this stage. According to Step B, the right-hand side of the solution should have as many digits as there are zeros in the base.
In this example the base 100 has two zeros. Hence, the RHS will be filled in by a two-digit number.
Let us make provisions for the same:
100
97 – 3
99 – 1
– –
We have separated the LHS and the RHS with a straight line. The RHS will have as many digits as the number of zeros in the base and so we have put empty blanks in the RHS of equal number.
Step C: Multiply the differences in RHS.
The third step (step C) says to multiply the differences and write the answer in the right-hand side.
In example A we multiply the differences, viz. -3 by -1 and get the answer as 3. However, the RHS has to be filled by a two-digit number. Hence, we convert the answer 3 into 03 and put it on the RHS.
100
97 – 3
99 – 1
0 3
STEP D: Put the cross answer in the LHS.
Now we come to the last step. At this stage we already have the right-hand part of the answer. If you are giving any competitive exam and the right-hand part of the answer uniquely matches with one of the given alternatives, you can straight-away tick that alternative as the correct answer. However, the multiplication in our case is still not complete. We still have to get the left hand side of the answer.
Step D says to put the cross answer in the left hand side.
In this example, the cross answer can be obtained by doing (97 – 1) or (99 – 3). In either case the answer will be 96. This 96 we will put on the LHS. But we already had 03 on the RHS.
Hence, the complete answer is 9603.
It’s not lengthy as it appears. I emphasized and explained every single step because I was describing the process for the first time. It appears long as a result. The instances we solve next will show that this is not the case in reality. Let’s solve few more examples:
1. 14 X 15
Base: 10
14 + 4
X 15 + 5
(21 ) (0)
The number of digits on the RHS is more than the number of zeros in the base. In this case, we have carried over as we do in normal multiplication.
Case 2: When the base is not a power of 10. 51 x 32 = ? (Important)
All the while we were taking only powers of 10, namely 10, 100, 1000, etc. as bases, but in this section we will take numbers like 40, 50, 600, etc. as bases.
In the problems that will follow, we will have two bases — an actual base and a working base. The actual base will be the normal power of ten.
The working base will be obtained by dividing or multiplying the actual base by a suitable number. Hence, our actual bases will be numbers like 10,100, 1000, etc. and our working bases will be numbers like 50 (obtained by dividing 100 by 2) or 30 (obtained by multiplying 10 by 3), 250 (obtained by dividing 1000 by 4).
Actual Bases: 10, 100, 1000, etc.
Working bases: 40, 60, 500, 250, etc.
Base: 100
Working base: 100 / 2 = 50 ( instead of division, we can use multiplication, 10 x 5 = 50, in order to get the working base, see Case 3 explained below.)
51 + 1
X 32 – 18
(33) (-18)
= 161/2 (-18)
= 16 (50-18)
= 1632 is the answer.
In this case the actual base is 100 (therefore RHS will be a two-digit answer). Now, since both the numbers are close to 50 we have taken 50 as the working base and all other calculations are done with respect to 50.
Since 50 (the working base) is obtained by dividing 100 (the actual base) by 2, we divide the LHS 33 by 2 and get 161/2 as the answer on LHS. The RHS we get by subtracting RHS from 50. The complete answer is 1632.
Case 3: Another case when the base is not a power of 10. 58 x 42 = ? (Important)
Base: 10
Working base: 10 x 6 = 60
58 – 2
42 – 18
40 (36)
= 240 (36)
= 2436 is the answer.
In this case the actual base is 10 (therefore RHS will be a one-digit answer). Now, since both the numbers are close to 60 we have taken 60 as the working base. Since, 60 is obtained by multiplying 10 by 6 so, we multiply the LHS 40 by 6 and get the answer 240. The RHS should be in one digit so, add 3 (carried over) to LHS to get the complete answer..
The complete answer is 2436.
3. Multiplying a number above the base with a number below the base.
Base: 100
Working base: 100/2 = 50
58 + 8
X 42 – 8
50 (-64)
= (50/2) * 100 (-64)
At this point, we have the LHS and the RHS. Now, we multiply LHS with the base and subtract the RHS to get the final answer.
= 2500 – 64
= 2436
Case 4: Multiplying numbers with different bases.
Suppose we want to multiply 877 by 90. In this case the first number is closer to the base 1000 and the second
number is closer to the base 100. Then, how do we solve the problem.
Multiply 85 by 995
Here, the number 85 is close to the base 100 and the number 995 is close to the base 1000. We will
multiply 85 with 10 and make both the bases equal thus facilitating the calculation. Since, we have
multiplied 85 by 10 we will divide the final answer by 10 to get the accurate answer.
Base: 1000
850 – 150
995 – 5
845 (750)
= 845750 divided by 10 gives 84575.
- We multiply 85 by 10 and make it 850. Now, both the numbers are close to 1000 which we will take as our base.
- The difference is -150 and -5 which gives a product of 750.
- The cross answer is 845 which we will put on the LHS.
- Thus, the complete answer is 845750. But, since we have multiplied 85 by 10 and made it 850 we have to divide the final answer by 10 to get the effect of 85 again. When 845750 is divided by 10 we get 84575.
Thus, the product of 85 into 9995 is 84575.