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Alternative Methods of Squaring Numbers

Column Method :

In this section let us learn how to find the square of two or three digit numbers without actual multiplication. This method is also called as column method. We make use of the identity

(a + b)2 = a2 + 2ab + b2
Consider the example: 742
To square the number 74 we make three columns for a2, 2ab and b2.

Step I
Take a as 7 and b as 4. Then


Step II
Underline the units digit of column III and add the tens digit to the column II.

Step III
Similarly underline the units digit of column II and add the tens digit to the column I.

Step IV
Now underline the digits in the column I.

Writing all the underlined digits together in an order, we obtain the required square.
742 = 5476.



 Find the square of 32.


 a as 3 and b as 2.

 322 = 1024


Diagonal Method :-
In the case of more number of digits we have another method called as diagonal method. Consider a number 27 and 531 for squaring.

Step I
Form a square and divide it into sub-squares based on number of digits, row and column wise. Then draw the diagonals of the smaller squares.

Step II
Multiply each digit on the left of the square with each digit on top of the column one–by–one.

Step III
If the product has a single digit number, write it below the diagonal. If it has two digit numbers, write the tens digit above the diagonal and units digit below the diagonal.


Step IV

Add all the digits diagonally starting from the lowest diagonal.

Step V
Underline all the units’ digit and carry over the other digit to the next diagonal above.

Pattern method for a perfect square ending in 25.
Study the following patterns.

   2 5  
x 2 5
6 2 5

In 625, the ones and tens place is occupied by
25(= 5 x 5), i.e. the product of the digits in the ones place. 

The hundreds place is occupied by 6( = 2 x 3) 
=( number in tens place ) x (successor of number in ten place) 

   7 5
X 7 5

In 5625 the once and tens place is occupied by 25(= 5 x 5).
The thousands and hundreds place
is occupied by 56( = 7 x 8).

Squares of all numbers ending in 5 can be found out by the above method: 
(i) The number in the tens place is multiplied by its successor and the product written down. 
(ii) The product 5 x 5(=25) is then written down after that.

Pythagorean triplets :-

Three natural numbers a, b and c are called a Pythagorean triplet if a2 + b2 = c2.
For any number n > 1,  (2n, n2 - 1, n2 + 1) is a Pythagorean triplet. 

Thus, for n = 3, (6, 8, 10) is a Pythagorean triplet since  
102 = 100 = 62 + 82 

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