Friday, January 7, 2011

A Maths Trick - Where Am I Going Wrong

Book: Figuring: The Joy of Numbers - Shakuntala Devi
Pg 75 - Squares and Square Roots

If the number from which the square root is to be extracted is higher than 40000 you use a slightly different method.
For example, to extract the square root of 537289.
First break it up into groups of two figures, starting from the right:
                                       53 72 89
To find the hundreds figure of the square root go to the first memorised table.
53 stands between the squares of 7 and 8. the lower of these numbers is our first figure, 7.
To find the tens figure we must find the difference between 53 and the square of 7, that is between 53 and 49.
                                   53 - 49 = 4

We now put 4 before the left-hand digit of the second group of figures in the number from which we are extracting the square. this gives 47.
We divide this number by twice the figure already arrived as the first in the square root - 7 - plus 1.
This gives us 15
                      47 / 15 = 3, with remainder 2
The quotient thus obtained is the tens digit of the square root we are extracting, so the first two digits of the root are 73.

We know that the units digit must be 7 or 3, for the number ends in a 9. As the quotient obtained, 3, was greater than the remainder, 2, we take the smaller of the two possible figures and arrive at the complete square root of 733.

When the quotient is smaller than the remainder take the larger of the two possible figures as the units digit.
Now let us try for the square of 666 i.e., 443556
Step 1: Group 44 35 56
Step 2: 44 lies between 36(6 square) and 49(7 square). So we take 6.
Step 3: Subtract 6 square from 44
                  44 - 36 = 8
Step 4: Put 8 before 35 to get 83.
Step 5: Divide 83 by twice the figure already arrived (6) plus 1
So, 83 / 13 = 6, with remainder 5

As the last digit of 443556 is a 6, one's digit of square root must be either 4 or 6.
Step 6: As quotient is greater than remainder, we take the smaller of 4 & 6, i.e., 4 which gives us 664.
But square root of 443556 is 666 and not 664.

Please share your thoughts as comments or email me at ajay184f[AT]GMAIL.COM


SzaLaci said...

That is because 666 is cursed :-)

Anonymous said...

Hi Ajay,

Are you sure this method is correct every time. I have tried with 885481. It is square of 941 but by this method answer is 939. I don't think this method is trustworthy.


Ajay Balamurugadas said...


I have stopped using that book. Though I learnt many things for the first time, I have found more than one instance where the example itself is mentioned incorrectly. I do not recommend this book. :|

Thanks for your time.