Squares of numbers -2

Let us look into a more general method of squaring. We saw in the previous post how to square a number ending with 5 , now we will explore a more general method.

Normally, for competitive examinations squares till 25 or 30 are supposed to be remembered. Using the following method we can compute that mentally.

Consider, a base number (say, b) which is the last digit of the number obtained when we multiply the unit's digit of the number (whose square we are supposed to find out) by 2.

For example, consider 37, i.e we need to find square(37). Base number b for 7(since 7 is the unit's digit of 37) = b(7) = last digit( 7*2) = last digit(14) =4.
Hence, b(1)=2, b(2) =4, b(3)=6, b(4)=8, b(6)=2, b(7)=4, b(8)=6, b(9)=8.

Step 1 Square the unit's digit of the number whose square is to be calculated, i.e 7 * 7 = 49. Now, 9 will be the last digit of the our final answer and 4 will be carried over.

Step 2 Now, multiply the number remaining after the unit's digit of the original number is removed(say, X) with base number (b) and add to it the carry over from the previous step. i.e. X * b(7) + carry over = 3*4 + 4=16. Now, 6 will be our ten's digit and 1 will be carry over. So, our final answer will end with 69.

Step 3 Since, the unit's digit of our original number (37) is 7 (which is greater than 5) we have to multiply X with (X+1) , i.e X * (X+1) and add the carry over from the previous step. If the unit's digit is lesser than 5, then we need to multiply X with itself i.e X * X . In our example, 3*(one more than it) + carry over from previous step = 3*4 + 1 =13. So, our final answer will be 1369. Let us take one more example. Consider, 98

Step 1 8*8 = 64. last digit is 4 and carry over is 6.

Step 2 9*base number for 8 + carry over= 9*6 + 6 = 60. So, second last digit of our final answer is 0 and 6 is the carry over.

Step 3 Since 8 is greater than 5, therefore 9*one more than it + carry over = 9*10 + 6 = 96.
Final answer = 9604.

Consider one more example of say, 94.

Step 1 4*4 = 16. last digit =6, carry over =1.

Step 2 9 * base number of 4 + carry over = 9*8 +1 = 73. So, our final answer will end with 36. Carry over = 7.

Step 3 Since, unit's digit of the original number i.e 94 is 4 (less than 5) so we need to multiply 9 with itself and add the carry over from the previous step,
i.e 9*9 + 7 = 81 +7 = 88.

Final answer = 8836.

Note, we can use this method for 3 digit numbers also. Instead of considering only the ten's place digit in Step 2 and 3, we need to consider the hundred's place number also. i.e suppose we need to find the square of 124.

Step 1 4*4 = 16. Last digit =6 , carry over =1.

Step 2 12(number remaining after the unit's digit of the original number is removed) * base number for 4 + carry over = 12 * 8 + 1= 97. So, final answer will end with 76. Carry over = 9.

Step 3 Now, since 4(unit's digit of 124) is less than 5 so we need to multiply X with itself. i.e. 12*12 + carry over = 144+9=153.

So, final answer = 15376.