We have seen the applications of the Base Method in multiplication of numbers. There is a corollary of the Base Method called the ‘Yavadunam’ rule. This rule is helpful in squaring numbers.
In this chapter we will study the Yavadunam rule and its applications in squaring numbers.
RULE
Swamiji had coined the Yavadunam rule in a Sanskrit line. When translated into English, it means:
‘Whatever the extent of its deficiency, lessen it to the same extent and also set up the square of the deficiency.’
We thus see that the rule is composed of two parts. The first part says that whatever the extent of the deficiency, we must lessen it to the same extent. The second part simply says—square the deficiency.
While writing the answer we will put the first part on the LHS and the second part on the RHS.
Let us have a look at an example:
(Q) Find the square of 8.
•We take the nearest power of 10 as our base (in this case 10 itself).
•As 8 is 2 less than 10, we should decrease 8 further by 2 and write the answer so obtained, viz. 6 as the LHS of the answer.
•Next, as the rule says, we square the deficiency and put it on the RHS. The square of 2 is 4 and hence we put it on the RHS.
•The LHS is 6 and RHS is 4. The complete answer is 64. Thus, the square of 8 is 64.
Since the Yavadunam rule is a corollary of the Base Method, the method used in this rule is exactly similar to the Base Method. However, we represent it in a different manner. Look at the examples given below:
(Q) Find the square of 96.
•In this case, we take the nearest power of ten, viz. 100.
•The difference of 100 and 96 is 4 and so we further subtract 4 from 96 and make it 92.
•We square 4 and make it 16 and put it on the RHS.
•The complete answer is 9216.
(Q) Find the square of 988.
•We take the nearest power of 10, i.e. 1000 as base.
•The difference of 988 and 1000 is 12 and therefore we subtract 12 from 988 and make it 976. This becomes the left half of our answer.
•We square twelve and put it on the RHS as 144.
•Thus, the square of 988 is 976144.
(Q) Find the square of 97.
•We take the nearest power of 10 namely 100 as base.
•The difference between 100 and 97 is 3 and therefore we further remove 3 from 97 and make it 94.
•Now, we square 3 and write it as 09 (because the base has two zeros) and put it on the RHS.
•The complete answer is 9409.
We have seen how the Yavadunam rule can be used in squaring numbers that are below a certain power of ten. In the same way, we can also use the rule to square numbers that are above a certain power of ten. However, instead of decreasing the number still further by the deficit we will increase the number still further by the surplus.
(Q) Find the square of 12.
•We take the nearest power of 10 closer to 12 which is 10 itself.
•The difference between 10 and 12 is 2 and so we further add 2 to 12 and make it 14. This becomes our LHS.
•We square the surplus 2 and make it 4 and this becomes our RHS.
•Thus, the square of 12 is 144.
(Q) Find the square of 108.
•We take 100 as the base and 8 as the surplus.
•We further add the surplus 8 to the number and make it 116.
•We square the surplus and make it 64.
•The final answer is 11664.
(Q) Find the square of 14.
•We take 10 as the base and 4 as the surplus.
•We further add 4 to 14 and make it 18 (LHS).
•We square 4 and write 16 as the RHS. 142 = 18/ 6 = 19/6
(We have observed in the previous chapter that if the base is 10, the RHS can be a single-digit answer only. In this case, it is a two-digit answer, namely, 16. Hence, we carry over the extra digit 1 to LHS and add it to 18.)
(Q) Find the square of 201.
Before we use the Yavadunam rule to solve the question, let us recall how the Base Method would have been used to solve it.
Base Method:
We multiplied the actual base by 2 to get the working base.
Thus, we multiplied the LHS by 2 to get the final answer.
Using the Yavadunam rule:
•The actual base is 100 and the working base is 200 (100 * 2).
•The surplus of 201 over 200 is 1 and so we further increase it by 1 making it 202. This becomes our LHS.
•We square the surplus 1 and put it on the RHS after converting it into a two-digit number, viz. 01.
•The complete answer is 202/01.
•But since we have multiplied the actual base by 2 to get the working base, we multiply the LHS by 2 and make it 404. The RHS remains the same.
•The final answer is 404/01. More Examples:
(a) 932 = 86/49
(b) 9972 = 994/009 (c) 132 = 16/9
(d) 152 = 20/ 5 = 22/5
(e) 9602 = 920/ 600 = 921/600 (f) 99852 = 99700225
(g) 2022 = 40804
(h) 3012 = 90601
EXERCISE
Find the squares of the following numbers using the Yavadunam Rule.
PART A
(1) 7
(2) 95
(3) 986
(4) 1025
(5) 1012
PART B
(1) 85
(2) 880
(3) 910
(4) 18
(5) 1120
PART C
(1) 22
(2) 203
(3) 303
(4) 498 (Hint: Take working base as 1000/2 = 500)
(5) 225 (Hint: Take working base as 1000/4 = 250)
