Squares of Any Number

Case 1: Below a base number.

Base number can be 10, 100, 1000 or higher.

997^2 = ?

1000 - 997 = 3

997^2 = 997 - 3 | 3^2

997^2 = 994 | 009  ••

997^2 = 994009

•• Number of zeros of base number indicates number of digits for the last group of the answer. That is why 3^2 = 009 .

Other examples:

96^2 = ?

100 - 96 = 4

96^2 = 96 - 4 | 4^2

96^2 = 92 | 16

96^2 = 9216

9987^2 = ?

10000 - 9987 = 13

9987^2 = 9987 - 13 | 13^2

9987^2 = 9974 | 0169 ••

9987^2 = 99740169

Case 2: Above the base number.

13^2 = ?

13 - 10 = 3

13^2 = 13 + 3 | 3^2 •••

13^2 = 16 | 9

13^2 = 169

••• If a number is below the base, subtract. If a number is above the base, add. That is why 13 + 3 and not 13 - 3.

Other examples:

108^2 = ?

108 - 100 = 8

108^2 = 108 + 8 | 8^2

108^2 = 116 | 64

108^2 = 11664

1012^2 = ?

1012 - 1000 = 12

1012^2 = 1012 + 12 | 12^2

1012^2 = 1024 | 144

1012^2 = 1024144

Case 3: Near a multiple of base number.

204^2 = ?

Base number is 2*100. Follow the same procedure with case 1 and case 2, but this time, first group of the answer will be multiplied by base factor (2).

204^2 = ?

204 - 200 = 4

204^2 = 2*(204+4) | 4^2

204^2 = 416 | 16

204^2 = 41616

Other examples:

196^2 = ?

200 - 196 = 4

196^2 = 2*(196 - 4) | 4^2

196^2 = 2*192 | 16

196^2 = 384 | 16

196^2 = 38416

6080^2 = ?

6080 - 6000 = 80

6080^2 = 6*(6080+80) | 80^2

6080^2 = 6*6160 | 6400

6080^2 = 36960 + 6 | 400 ••••

6080^2 = 36966400

•••• Number of zeros of base number indicates number of digits for the last group of the answer. Carry over the excess digits to the first group ( 36960+6).

#Matamata