Method 9

Square of four digit numbers
(from the same process of duplex of method 7 )

Example 3 : - Find the square of 3242 .

Answer : -

Step 1 : -

Write according to the format
number of digits = 4 so number of zeros = 4 - 1 = 3

0003242

Step 2 : -

Now take the duplex of 2 = 2² = 4

Obtained digit = 4

Step 3 : -

Now take the duplex of 42 = 2.(4 * 2) = 16

Retain 6 and 1 is for carry .

Obtained digit = 64

Step 3 : -

Now take the duplex of 242 = 2.(2 * 2) + 4² = 24

Add carry from the previous step
i.e.
24 + 1 = 25

Retain 5 and 2 is for carry .

Obtained digit = 564

Step 4 : -

Now take the duplex of 3242 = 2.(3 * 2) + 2.(2 * 4) = 12 + 16
= 28

Add carry from previous step
i.e.
28 + 2 = 30

Retain 0 and 3 is for carry to the next step

Obtained digit = 0564

Step 5 : -

Now take the duplex of 03242 = 2.(0 * 2) + 2.(3 * 4) + 2² = 28

Add carry from the previous step
i.e.
28 + 3 = 31

Retain 1 and 3 is for carry .

Obtained digit = 10564

Step 6 : -

Now take the duplex of 003242 = 2.(0 * 2) + 2.(0 * 4) + 2.(3 * 2) = 12

Add carry
i.e.
12 + 3 = 15

Retain 5 and 1 is for carry

Obtained digit = 510564

Step 7 : -

Now take the duplex of 0003242
= 2.(0 * 2) + 2.(0 * 4) + 2.(0 * 2) + 3² = 9

Add carry
i.e.
9 + 1 = 10

Place it to the left

10510564

Hence the square of the 3242 is 10510564 .

Algebraic Proof

Try the proof yourself by seeing the proofs of method 7 and 8 .

Exercises :

Find the squares of the following numbers :

1.) 5532
2.) 2286
3.) 8732
4.) 5467