Guest Post: Vedic Mathematics – Squaring Numbers 1

When any number is multiplied by itself , the result obtained is known as the square of that number.  In Vedic Mathematics following methods are used for calculation of squares : Ekadhika Method,  Yavadunam Method, Duplex Method, and  Anurupyena Method.  I have written about Ekadhika Method in my blog, but here, we will discuss Yavadunam method of squaring. Other methods will be discussed in the next posts.

Yavadunam Method

This method is used for squaring the numbers which are near some base. The method can be extended to other numbers which are not near base by using sub-base. The sub-formula “Yavadunam Tavdunikritya Vargamcha Yojayet” actually means – whatever the extent of the deficiency of a number from base , lessen it to the same extent and set up the square of the deficiency.

It can also be applied to excess and in this case , this excess is to be added to same extent followed by setting up the square of excess. Hence in general this mean – whatever the deviation , increase the number by that deviation and suffix the square of the deviation.

Thus squaring of numbers near the base involves two steps:

First, divide the answer into two parts, the LHS and the RHS part. The LHS part of answer is number plus deviation. The deviation may be positive or negative depending on whether the number is more or less than the base respectively.

Second , the RHS part of the answer being the square of the deviation and contains the same number of digits as the number of zeroes in the base. The excess digits, if any are to be carried over to LHS and deficit digits if any , are to be filled up by zeroes to the left of RHS part of the answer.

Example 1 : Find 96².

Solution :   96 is near hundred , so our base is 100 (RHS will have two digits)

Deviation    =  96 – 100 = -4

96² = LHS   /   RHS

LHS  = [(96+  (-4)] = 92

RHS = (-4)² = 16

So  , 96² = 9216

Example 2 : Find 114²

Solution :   114 is near hundred , so our base is 100 (RHS will have two digits)

Deviation    = 114 – 100 = 14

114² = LHS   /   RHS

RHS = (14)² = 196,as RHS can have only two digits(96),1 will be carried to the LHS

LHS  = [114 +14] +1 = 129

So  , 114² = 12996

If the number is not near any base , then we choose a suitable sub-base and find the deviation from the sub-base. The RHS part of the answer is the square of the deviation from the sub-base.

LHS part is to be modified proportionately. It is found by increasing the number by deviation from sub-base and then multiplying it by the ratio (R ) of sub-base to base.

Example 3 : Find 33²

Solution  :   Here we take Base (B) = 10

Sub-base (SB) = 30  ,

So , Ratio (R ) = SB ¸ B = 3

Deviation = 33 – 30 = 3

Now , 33² = LHS  /  RHS

RHS = 3² = 9  (will have one digit as base have one zero)

LHS  =  3 x (33 + 3) = 108

\ 33² = 1089

Example 4  : Find 47²

Solution (A)  : B = 10 , SB = 50 , R = 5

Deviation = 47 – 50 = -3

Now , 47² = LHS  /  RHS

RHS = -3² = 9

LHS  =  5 x [47 + (-3)] = 220

\ 47² = 2209

\ 47² = 2209

Solution (B) : B = 100 , SB = 50 , R = 1/2

Deviation = 47 – 50 = -3

Now , 47² = LHS  /  RHS

RHS = (-3)² = 09   (Base has two zeroes)

LHS  =  1/2 x [47 + (-3)] = 22

Example 4  : Find 606²

Solution : B = 100 , SB = 600 , R = 6  , Deviation = 606 – 600 = 6

606² = LHS / RHS  , RHS = 6² = 36 , LHS = 6 x (606 + 6) = 6 x 612= 3672

\ 606² = 367236

***

Sanjay Guilati (author) has been teaching computer and mathematics in Bhilai , state Chattisgarh in India for 15 years. Currently he is a  teacher in a senior secondary school and he is also involved in teacher training.  His online work can be found in Mathematics Academy.