Method 8

Square of three digit number
(from the same process of duplex of method 7)

Note : - This method only looks long but after a small practise it will become so easy that you will complete the square in one line .

Example 2 : - Find the square of 215 .

Answer : -

Step 1 : -

Write according to the way
number of digits = 3 number of zeros = 3 -1 = 2

00215

Step 2 : -

Write the duplex of 5 = 5² = 25

Retain 5 and 2 is for carry .

Obtained digit = 5

Step 3 : -

Write the duplex of 15 = 2.( 1 * 5 ) = 10

Add carry of previous step
i.e.
10 + 2 = 12

Retain 2 and 1 is for carry .

Obtained digit = 25

Step 4 : -

Write the duplex of 215 = 2.(2*5) + 1² = 21

Add carry
i.e.
21 + 1 = 22

Retain 2 and 2 is for carry .

Obtained digit = 225

Step 5 : -

Write the duplex of 0215 = 2.(0 * 5) + 2.(2 * 1) = 4

Add carry
i.e.
4 + 2 = 6

Obtained digit = 6225

Step 6 : -

Write the duplex of 00215 = 2.(0 * 5) + 2.(0 * 1) + 2² = 4

Place it to the left

= 46225

Hence the square of 215 is 46225 .

Algebraic Proof : -

Any three digit nuumber ' abc ' can be represented as

(100.a + 10.b + c) .

The square of above number is

(100.a + 10.b + c)²

= (100.a)² + (10.b)² + c² + 2.(100.a).(10b) + 2.(10.b).c + 2.(100.a).c

= 10000.a² + 100.b² + c² + 2.a.b.1000 + 2.b.c.10 + 2.a.c.100
(arranging in increasing powers of 10)

= 10⁴.a² + 10³.2.a.b +10².(2.a.c + b²) + 10.2.b.c + c²

In the above

c² is represented by Step 2 .

2.b.c is represented by Step 3.

(2.a.c. + b²) is represented by Step 4 .

2.a.b. is represented by Step 5 . &

a² is represented by Step 6 .

Exercises : -

Find the square of the following numbers :

1.) 332
2.) 442
3.) 112
4.) 887
5.) 264