Method 7

Finding the square of a number

This method is based on the sutra "Dwadanda Yoga" which means a 'Duplex combination process ' . For learning this method we have to learn one small extra thing which is known as Duplex .

Duplex

For a single digit ' a ' number the duplex is ' a² '

For a two digit number ' ab ' the duplex is 2.(a * b)

For a three digit number ' abc ' the duplex is 2.(a * c) + b²

For a four digit number ' abcd ' the duplex is 2.(a * d) + 2.(b * c)

For a five digit number ' abcde ' the duplex is 2.(a * e) + 2.(b * d) + c²

And so on .........................

The concept is that take the two numbers equidistant from the beginning and from the end , then multiply these two numbers and then multiply the result by 2 And then add the result in the previously result obtained by the same process . Carry on in the same way until middle digit is reached in case of odd digit number only . Then add the square of this middle digit to the previous result to obtain the duplex .

Now explaining the method with the help of an example

Example 1 : - Find the square of 64 .

Answer : -

Step 1 : -

First write the number and to the left of it write as many zeros equal to one less than the number of digits in the number .
In number of digits in the number = 2
So number of zeros equal to = 1

Write in this way
064


Step 2 : -

First find the duplex of 4 = 4² = 16

Take 6 as the left most digit of the equation and 1 is for carry

Step 3 : -

Then find the duplex of 64 = 2.(6 * 4) = 48

Add carry to it
i.e.
48 + 1 = 49

Retain 9 as next digit from left and 4 is for carry

So obtained digit is 96

Step 4 : -

Find the duplex of 064 = 2.(0 * 4) + 6² = 36

Add carry to it
i.e.
36 + 4 = 40

Place it to the left of digits obtained till now

4096

Hence the square of 64 is 4096 .

Note : - This method only looks long . After small practise you will become so perfect that you will even calculate the square without even writing anything . In the beginning you can also practise one line implementation with paper

Algebraic Proof

Any number of the form ' ab ' can be represented as

( 10.a + b )
Square of the number is

( 10.a + b )² = (10.a)² + 2.(10.a).b + b²
= a².100 + 2.a.b.10 + b²

which shows that the square goes in the same format as above .

This is because in step 1 we do b² . Then we do 2.(a * b )
and then we do a² and all are placed on their respective places .i.e. first step
gives unit's place ,second step gives ten's place and third step gives hundred's
place .

Exercises : -

Find the square of the folowing numbers : -

1.) 54
2.) 22
3.) 34
4.) 92
5.) 45