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\ 06-Jan-2025 Bharat Arora Proof-7 Reader’s Sign _______________________ Date __________

Learning by Doing

objective: To find the square of a number using an interesting method (diagonal
method).

Materials required: Paper sheet, Pen/Pencil, Ruler
Procedure:

Let us find the square of 145.

• Count the number of digits in the given number. Let there be n digits
in the number. So draw a squared grid with n rows and n columns on
a paper sheet, creating n sub-squares.
2
Here, in the number 145, n = 3. So draw a 3 × 3 grid (see figure alongside).

1 4 5
• Draw a diagonal of each sub-square and write the digits of the number 1
to be squared along the left vertical side and top horizontal side of the 4
grid (see alongside).
5

• Multiply each digit on the left side of the square with each digit at the 1 4 5
top of the columns one by one. Write the ones digit of the product below 1 0 0 0
1 4 5
the diagonal and tens digit above the diagonal in the corresponding 4 0 1 2
sub-squares as shown. 0 4 2 6 2 0
5
5 0 5
• Finally, add diagonally. Carry over if required in addition as follow:

1 4 5
0 0 0
1
1 4 5
0 1 2
4
4 2 6 2 0
0
0 + 1 + 0 = 1 + 1 = 2 0 + 2 + 0 = 2 5 0 5
5
0 + 4 + 1 + 4 + 0 + 2 = 1 1
5 + 2 + 6 + 2 + 5 = 2 0

0 2 1 0 2 5

Thus, (145) = 21025
2

147 Squares and Square Roots