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Methods of Squaring

As we have seen in the case of multiplication, there are several methods for squaring also. Let us see the methods one by one.

Method 1: Base 10 Method

Understand it by taking few examples:
  • •Let us find out the square of 9. Since 9 is 1 less than 10, decrease it still further to 8. This is the left side of our answer.
  • •On the right hand side put the square of the deficiency that is 12. Hence, the answer is 81.
  • •Similarly, 82 = 64, 72 = 49.
  • •For numbers above 10, instead of looking at the deficit we look at the surplus.
     
    For example,
    112 = (11 + 1); 10 + 12 = 121
    122 = (12 + 2); 10 + 22 = 144
    142 = (14 + 4); 10 + 42 = 18 ; 10 + 16 = 196
    and so on.
This is based on the identities (a + b) (a  b) = a2  b2 and (a + b)2 = a2 + 2ab + b2.
 
We can be use this method to find the squares of any number, but after a certain stage, this method loses its efficiency.

Method 2: Base 50n Method here, (n is any natural number)

This method is nothing but the application of (a + b)2 a2 + 2ab + b2.
 
This can be seen in the following example:
 
Example-1
Find the square of 62.
Solution
Because this number is close to 50, we will assume 50 as the base.
(62)2 = (50 + 12)2 = (50)2 + 2 × 50 × 12 + (12)2 = 2500 + 1200 + 144
 
To make it self explanatory a special method of writring is used.
(62)2 = [100’s in (Base)]2 + Surplus | Surplus2
= 25 + 12 | 144 = 38 | 44 [Number before the bar on its left hand side is number of hundreds and on its right hand side are last two digits of the number.]
(68)2 = 25 + 18 | 324 = 46 | 24
(76)2 = 25 + 26 | 676 = 57 | 76
(42)2 = 25 − 8 | 64 = 17 | 24 [(a − b)2 = a2 − 2ab + b2]
 
 
Example-2
Find the square of 112.
Solution
Since this number is closer to 100, we will take 100 as the base.
(112)2 = (100 + 12)2 = (100)2 + 2 × 100 × 12 + (12)2 = 10000 + 2 × 1200 + 144
(112)2 = [100’s in (Base)]2 + 2 × Surplus | Surplus2
= 100 + 2 × 12 | 122 = 125 | 44
 
Alternatively, we can multiply it directly using base value method.
 
Had this been 162, we would have multiplied 3 in surplus before adding it into [100’s in (Base)]2 because assumed base here is 150.
(162)2 = [100’s in (Base)]2 + 3 × Surplus | Surplus2
= 225 + 3 × 12 | 122 = 262 | 44
 

Method 3 : 10n Method

This method is applied when the number is close to 10n.
With base as 10n, find the surplus or deficit (×Again answer can be arrived at in two parts
(B + 2x) |x2
The right-hand part will consist of n digits. Add leading zeros or carry forward the extra to satisfy this condition.
1082 = (100 + 2 × 8) | 82 = 116 | 64 = 11664
1022 = (100 + 12 × 2) | 22 = 104 | 04 = 10404
932 = (100 – 2 × 7) | (7)2 = 86 | 49  8649
10062 = (1000 + 2 × 6) | 62 = 10|12 | 036 = 1012036
The right-hand part will consist of 2 digits. Add leading zeros or carry forward the extra to satisfy this condition.
632 = (25 + 13) | 132 = 38 | 169 = 3969
382 = (25 – 12) + (12)2 = 13 | 144 = 1444
 
Square Mirrors
142 + 872 = 412 + 782
152 + 752 = 512 + 572
172 + 842 = 712 + 482
262 + 972 = 622 + 792
272 + 962 = 722 + 692
 
 
Some Special Cases
  1. Numbers ending with 5
If a number is in the form of n5, the square of it is n (n+1) | 25
Example: 452 = 4 × 5 | 25 = 2025
1352 = 13 × 14 | 25 = 18225
This is nothing but the application of the multiplication method using the sum of units digits.
We can use this method to find out the squares fractions like Description: 2184.png…also.
Process: Multiply the integral portion by the next higher integer and addDescription: 2178.png.
For example, Description: 2172.png





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