Tricky Methods to find the square of a number quickly ( Vedic math )

What's square of a number?
When we multiply a number with itself the resultant number is called square of that number.

Finding square by adding consecutive odd number:-

1² = 1

2²=1+3=4

3²=1+3+5=9

4²=1+3+5+7=16

5²=1+3+5+7+9=25

....... So on

Finding Square by using a Base:-

Base:- Bases are generally the powers of 10.

i.e 10,100,1000,10000,........ etc

Case_I:(if the number is less then The Base)

Exp:(1)

let the number is 96

   Here the base is 100 

   Here the number is less then from base by 04

  Then ,

             96²= 96-04 | (04)²

                        = 9216
So the answers is 9216

Exp:(2)

Let the number is 992

 Here the base is 1000

 Here the number is less then from base by 008

Then ,

          992²= 992-008|(008)²

                          = 984064
So the answers is 984064

Exp(3):

Let the number is 9975

  Here the base is 10000

  Here the number is less then base by 0025

Then,

         9975²=9975-0025/(0025)²

                             =99500625
So the answers is 99500625

Case_II:(If the number is greater then the Base)

Exp:(1)

Let the number is 17

  Here the base is 10

  And the number is greater than the base by 07

Then,

         17² = 17+7|(7)²

                 = 24|49

                 =289 
( here the base is the 1st power of 10 so 4 is carried towards the left and added with 24)

So the answers is 289

Exp:(2)

Let the number is 115

  Here the base is 100

  And the number is greater then from base by 15

Then, 

          115²= 115+15|(015)²

                          =130225
So the answers is 130225

Exp(3):

Let The number is 1025

   Here the base is 1000

   And the number is greater than base by 25

Then,

         1025²= 1025+25|(025)²

                          = 1050625
So the answers is 1050625

If the unit digit of the number is "5"

Let MN be the number where N is 5

Then ,

            MN² =M*(M+1)|5²

Exp(1):

Let the number is 45

 45²= 4*5|(5)²

          =2025
So the answers is 2025

Exp(2):

Let the number be 95

   95²= 9*10| 5²

            =9025
So the answers is 9025

Exp(3):

Let the number be 125

 125²= 12*13|5²

           =15625
So the answers is 15625



Dwanda yoga / Duplex combination

The Dwandwa Yoga or 'Duplex Combination' can be used for general purpose squaring.
To proceed further we need to know the Dwandwa of certain numbers.

D( a ) = a²
D( ab ) = 2ab
D( abc ) = 2ac + b²
D( abcd ) = 2ad + 2bc
D( abcde ) = 2ae + 2bd + c²
D( abcdef ) = 2af + 2be + 2cd      and so on....

As we can see above, D of any number is the sum of square of the middle number and two times the product of the other pairs.
Square of a number is given by

( ab )² = D( a ) | D( ab ) | D( b )
( abc )² = D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
( abcd )² = D(a) | D(ab) | D(abc) | D(abcd) | D(bcd) | D(bc) | D (c)

Exp(1):
Let the number is 23
Then,

      (23)² = (ab)² = D(a) | D(ab) | D(b)
                               = 4     |    12    |    9
Since Dwanada must have only one digit, we carry over '1' of '12' to LHS.
Therefore it becomes     4 |₁ 2| 9
So the  answer is 529

Exp(2):
Let the number is 527
Then,

    (527)² = ( abc )²
           = D( a ) | D( ab ) | D(abc) | D( bc ) | D( c )
           =   25   |     20    |     74   |    28    |   49
           =   25   |₂     0    |₇     4   |₂    8    |₄   9
          =   277729
So the answer is 277729

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