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Square of a Number Using Vedic Mathematics

Let’s see how we can find square of a number faster using Vedic Mathematics (Nikhilam method)

Example 1: Find square of $12$

Step 1

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=12-10=2$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=12+2=14$

Step 2

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=2^2=4$

Therefore, answer $=144$

Example 2: Find square of $13$

Step 1

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=13-10=3$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=13+3=16$

Step 2

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=3^2=9$

Therefore, answer $=169$

Example 3: Find square of $14$

Step 1

$10$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=14-10=4$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=14+4=18$

Step 2

Our base $10$ has a single zero. Therefore, right side of the answer has a single digit and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=4^2=16.$ But right side of the answer can have only a single digit because our base $10$ has a only single zero. Hence, from the obtained number $16,$ we will take right side as $6$ and $1$ is taken as a carry which will be added to our left side. Hence left side becomes $18+1=19$

Therefore, answer $=196$

Example 4: Find square of $106$

Step 1

$100$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=106-100=6$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=106+6=112$

Step 2

Our base $100$ has two zeros. Therefore, right side of the answer has two digits and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=6^2=36$

Therefore, answer $=11236$

Example 5: Find square of $112$

Step 1

$100$ is the nearest power of $10$ which can be taken as our base. The deviation to our base $=112-100=12$ (To find the deviation, just remove the leftmost digit "$1$" and you will get it quickly)

To find the deviation, just remove the leftmost digit

Left side of the answer is the sum of the number and deviation. Hence, left side of the answer $=112+12=124$

Step 2

Our base $100$ has two zeros. Therefore, right side of the answer has two digits and that can be obtained by taking the square of the deviation.

Hence, right side of the answer $=12^2=144.$ But right side of the answer can have only two digits because our base $100$ has only two zeros. Hence, from the obtained number $144,$ we will take right side as $44$ and $1$ is taken as a carry which will be added to our left side. Hence left side becomes $124+1=125$

Therefore, answer $=12544$

Video Tutorial

Note

This method is extremely useful for competitive examinations and if practiced well, square of a number can be determined within seconds using the same.

Comments(77)

profileDebdut Mukherjee
2014-01-19 15:36:44 
What about the square of 312, 487, 1117, etc.? Will this technique work? 
like 0 dislike 0 reply
profilepk
2013-11-24 09:16:01 
Is it possible to find out square of any 4 or more digit number using this method
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profileJay
2013-11-24 16:09:50 
Yes, but this method is easy when the number is close to 10,100,1000, etc
like 0 dislike 0
profilesams
2013-11-03 23:33:43 
how to go with the above method to find the square root of 41
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profileHarish Kumar K V
2013-11-08 16:32:34 
41 - 10 = 31 which is RHS(deviation) and LHS is (deviation + number ) 31 + 41 = 72
so square root is 72_

now again find the sqaure root for 31 in the same method
31 - 10 = 21 which is RHS(deviation) and LHS is (deviation + number ) 21 + 31 = 52
so square root is 52_

similarly 21 - 10 = 11 which is RHS(deviation) and LHS is (deviation + number ) 21 + 11 = 32
so square root is 32_

Now sqaure of 11 is 121 but since it is base of 10 take only 1 put in dash(321) and add 12 to 2 of the LHS which is 32
That gives you 2 + 12 = 14 where 1 is carry and gets added to 3 and become 4, so finally square root is 441

Now from here go in the reverse direction
2+44 = 46 with 4 carry added to 5 becomes 96 and square root as 961 for 31

take the last number from 961 which is 1 and put it in 72_ and add 96 to 2 of 72 which gives 96+2 = 98 and carry as 9 which to be added to 7 and 7+9 = 16 combining 1681 is the square of 41


like 0 dislike 0
profileSenthilkumar
2014-03-05 14:05:45 
    41
    41 *
-------
    41
164
-------
1681 Ans

isn't that simple?

like 0 dislike 0
profilejisstpalelil
2014-04-15 07:31:51 
U r right senthilkumar, this trick is not much easy 4 such no.
like 0 dislike 0
profilesweetdreams
2013-09-25 05:03:59 
36^2=(3*10)+6

to find rhs: 6^2=36
base has only 1 zero, therefore 3 is a carry

to find lhs: 36+6=126+carry=126+3=129
 therefore

result is 1296
like 0 dislike 0 reply
profileAkshay
2013-09-01 13:16:32 
I can give alternate method to find of the square of any number

let say 26

RHS= 6^2=36 then right 6 carry 3

middle term 2(2*6)=24+carry i.e. 27 so number is 76 carry 2

LHS= 2^2=4+carry i.e. 4+2=6

Answer =676
like 0 dislike 0 reply
profilesourav
2013-08-05 20:30:35 
how to do 36^2
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