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Square of the number

     This article is used to find out the square of  a number in a short period of time compare to normal maths. It increase the speed of your calculations. The answer obtained by multiplying a number by itself is the square of that number.

Example :- 6×6= 36 which is the square of 6.

                        8×8= 64 which is the square of 8.

    

1. Square of numbers ending in 5 :-

Example 1-:

352 =  ?

Step 1:-Multiply the first part by the first part plus 1:

3×(3+1) =3×4 =12

Step 2:-Write the number 25 next to the result from step 1:

352 = 12_25 = 1225

Example 2-: (in 3 digits)

1252 =  ?

Step 1:-Multiply the first part by the first part plus 1:

12×(12+1) =12×13 =156

Step 2:-Write the number 25 next to the result from step 1:

1252 = 156_25 = 15625

Example 3-: (in decimal numbers)

9.52 =   ?

Step 1:-Multiply the first part by the first part plus 1:

9×(9+1) =9×10 =90

Step 2:-Write the number 25 next to the result from step 1:

9.52 = 90_25 = 90.25

Note :- Last digit 5 is the 2nd part and rest number is 1st part.

 

2.     The second method of doing squares is from this formula. Formula means by adding and subtracting. A square of one more or one less of a number (with unit digit zero) can be easily extracted by this method. In the same way, the square of a number can be known or can be easily known. Square of one less or one more of that number will also be found by this method.

Example 1 :- Help of 402=1600, We find square of 39 and 41.

392 = 402-40-39=1600-79=1521

412 = 402+40+41=1600+81=1681

Example 2 :- Help of 452=2025, We find square of 44 and 46.

442 = 452 -45-44 = 2025-89 = 1936

462 = 452+45+46 = 2025+91 = 2116

Example 3 :- Help of 4002=160000, We find square of 399 and 401.

3992 = 4002-400-399 =160000-799 = 159201

4012 = 4002+400+401 = 160000+801 = 160801

 

3.     Squaring number near base :-

When number is more than base

Example 1-:

1072 =  ?

Step 1 :- Find the difference between given number and base

Difference = 107-100=7

Step 2 :- Now add given number and difference

107+7=114

Step 3 :- Find the square of difference

72=49

Step 4 :- Write the result of step 3, next to the result from step 2:

1072 = 114_49 =11449

 

Example 2-:

1122 =  ?

Step 1 :- Find the difference between given number and base

Difference = 112-100=12

Step 2 :- Now add given number and difference

112+12=124

Step 3 :- Find the square of difference

122=144

Step 4 :- Write the result of step 3, next to the result from step 2:

1122= 124_144 =124+1_44=12544 (because base have 2 zeros)

 

Example 3-:

10112 =  ?

Step 1 :- Find the difference between given number and base

Difference = 1011-1000=11

Step 2 :- now add given number and difference

1011+11=1022

Step 3 :- Find the square of difference

112=121

Step 4 :- Write the result of step 3, next to the result from step 2:

10112= 1022_121 =1022121

When number is less than base

Example 4-:

972 =  ?

Step 1 :- Find the difference between given number and base

Difference = 100-97=3

Step 2 :- now subtract difference from given number

97-3=94

Step 3 :- Find the square of difference

32=09

Step 4 :- Write the result of step 3, next to the result from step 2:

972= 94_09 =9409

Note: 10,100,1000,10000 etc. are called base.


4.     This forth method is commonly used for squares of two-digit numbers. Before learning this, the student needs good practice of multiplication and division.

Formula : (a+b)2 = a2+2×a×b+b2

Let first digit = a , and second digit = b

Example 1 :- 492

1st part            2nd part            3rd part

   42                  2×4×9                  92

   16                   72                       81

   16                   72                      81

   16                 80(72+8)              1

   16                   80                        1

  24(16+8)           0                        1

    492 = 2401  is Answer



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