Squaring - How to find the Square of numbers mentally without calculator - Math Tricks

SQUARE

In this post i will give top tricks to square numbers within seconds without calculator or pen-paper. Square of any number is the product of number with itself. It works on the same principle of multiplication, same tricks of multiplication are applicable here but there are some additional tricks to find the squares of numbers quickly and easily. Conventional method takes lot of time and we need paper-pen too to solve questions using that method. Here we will learn some new methods and tricks to solve squaring problems.

Square of Two Digit Numbers

Formula: (AB)2 = (A)2 | 2AB | (B)2
After writing in terms of given formula, keep the unit digit of the rightmost number and transfer rest of the digits to the left and add them.

Example: Find the square of 13?
Solution: Here A = 1 ; B = 3
                  ⇒ (13)2 = (1)2 | 2 x 1 x 3 | (3)2
                                 =   1    |       6      | 9
                                 = 169

Example: Find the square of 48?
Solution: Here A = 4 ; B = 8
                 ⇒ (48)2 = (4)2 | 2 x 4 x 8 | (8)2

Now keep the unit digit of the rightmost number and transfer rest of the digits to the left and add them.
                           = 16|6 4 ⃔|6 4

                           = 16 ⃔|70 |6 4
                           = 2304

 

Second Method:

Remember square of numbers up-to 25 for this method.

22 = 4                      102 = 100                    182 = 324 

32 = 9                      112 = 121                    192 = 361

42 = 16                    122 = 144                    202 = 400

52 = 25                    132 = 169                    212 = 441

62 = 36                    142 = 196                    222 = 484

72 = 49                    152 = 225                    232 = 529

82 = 64                    162 = 256                    242 = 576

92 = 81                    172 = 289                    252 = 625 

 1.  Square of numbers from 26 to 50: 

Example: Find the square of 37?

Solution:  Step 1  Find how much less the number is than 50 = 50 - 37 = 13

 Step 2  Subtract this '13' from 25 = 25 - 13 = 12

 Step 3  Write the square of '13', which is 169.

 Step 4  Ignore last two digits and add the remaining digits of this square with '12'.

        ⇒ 12 ⃔|169 = 1369

 2.  Square of numbers from 51 to 75:

Example: Find the square of 73?

Solution:  Step 1  Find how much more the number is from 50 = 73 - 50 = 23

 Step 2  Add this '23' to 25 = 25 + 23 = 48

 Step 3  Write the square of '23', which is 529.

 Step 4  Ignore last two digits and add the remaining digits of this square with '48'.

        ⇒ 48 ⃔|529 = 5329

 3.  Square of numbers from 76 to 100:

Example: Find the square of 97?

Solution:  Step 1  Find how much less the number is from 100: Deficit = 100 - 97 = 3

 Step 2  Subtract this deficit from the number = 97 - 3 = 94

 Step 3  Write the square of  this Deficit = '32' = 09

 Step 4  Ignore last two digits and add the remaining digits of this square with '94'.

        ⇒ 94 | 09 = 9409

Squaring of Three Digit Numbers

Formula: (abc)2 = a(abc + bc)| + (bc)2
Add “abc” with “bc” and multiply with “a”. Now add (bc)2 to this product after ignoring the last two digits of this (bc)2.

Example: Find the square of 113?
Solution: Here A = 1 ; B = 1 ; C = 3
         ⇒ (113)2 = [1(113 + 13)| + (13)2]
                         = 126 ⃔|169
                         = 12769

Square of Numbers near to multiples of power of 10

Example: Find the square of 67?

Solution: Nearest Base = 70

 Step 1  Find how much less the number is from 70: Deficit = 70 - 67 = 3

 Step 2  Subtract this deficit from the number = 67 - 3 = 64

 Step 3  As Base is (7 x 10) multiply this number '64' with 7  = 448

 Step 4  Write the square of  this Deficit = '32' = 9

 Step 5  Now add both the numbers

        ⇒ 448 | 9 = 4489

 SPECIAL CASES 

Numbers ending with 1

Example: Find the square of 71?
Solution:  Step 1  Write number like this: a = 70 + 1
 Step 2  Square first number (70) = 4900
 Step 3  Double this number and add one to it.
        ⇒ 2 x 70 + 1 = 141
 Step 4  Now add both the numbers.
        ⇒ 4900 + 141 = 5041
We can represent it in terms of Formula as: (a - 1)2 + 2(a -1) + 1
Find the square of 81: 802 + 2 x 80 + 1 = 6400 + 161 = 6561

Number ending with 4

Example: Find the square of 74?
Solution:  Step 1  Write number like this: a = 75 - 1
 Step 2  Square first number (75) = 5625
 Step 3  Double this number and subtract one from it.
        ⇒ 2 x 75 - 1 = 149
 Step 4  Now subtract this product from the square of first number.
        ⇒ 5625 - 149 = 5476
We can represent it in terms of Formula as: (a + 1)2 - 2(a +1) - 1

Numbers ending with 5 

Formula: (n5)2 = [n(n + 1)| 25] 

In a given number neglect “5” and multiply the remaining number “n” with its next number “n+1” and write “25” at last of this product.

Example: Find the square of 35?
Solution: Here n = 3
                  ⇒ (35)2 = [3(3 + 1)| 25]
                                 = 1225

Number ending with 6

Example: Find the square of 76?
Solution:  Step 1  Write number like this: a = 75 + 1
 Step 2  Square first number (75) = 5625
 Step 3  Double this number and add one to it.
        ⇒ 2 x 75 + 1 = 151
 Step 4  Now add this product to the square of first number.
       ⇒ 5625 + 151 = 5776
We can represent it in terms of Formula as: (a - 1)2 + 2(a -1) + 1

Number ending with 9

Example: Find the square of 79?
Solution:  Step 1  Write number like this: a = 80 - 1
 Step 2  Square first number (80) = 6400
 Step 3  Double this number and subtract one from it.
        ⇒ 2 x 80 - 1 = 159
 Step 4  Now subtract this product from the square of first number.
        ⇒ 6400 - 159 = 6241
We can represent it in terms of Formula as: (a + 1)2 - 2(a +1) - 1

Square of Numbers with Repeated Digits 

Count the number of 1's and then reverse the order. If there are four 1's, then count 1,2,3,4 and then reverse it like 3,2,1 and that's the answer 1234321.

(11)2   = 121
(111)2 = 12321
(3333)2 = (3)2 x (1111)2 = 11108889

NOTE:
 1.  Square of a number can never have 2, 3, 7 & 8 at its units place.
 2.  The difference between any perfect square and its predecessor is: 2n + 1

These are the fastest squaring tricks. After practicing above methods u will be able to find the squares of any two digit or three digit numbers within mind without the need of calculator or paper-pen. U can find squares up-to 1000 easily. Try your conventional methods with above mentioned methods and compare the result to feel the difference. Good Luck 👍