How to find out the square root by long division method

If the square root of the number N is  x .  Then x *x will the number N. So in this post we will see how to find out the square root of a number in long division method.

Step-1

Consider the Fig1-a  we are finding out the square root of 225.  First we have to group the 225 as group of 2 numbers from right to left as shown in fig. If the leftmost digit is single we can leave as it is. Now consider each group from left to right. So first we consider the 2. Then find the  perfect root number who squire is just below or same to the first group of number (2). So in this case 1*1=1 and 2*2=4.  As 4 is greater than 2 we can’t consider 2 as denominator and quotient . So we will take 1 as denominator and quotient. And the product of denominator and quotient(1*1)=1 will be write below the first group of the number (2). And  subtract the product from (2) and write down the remainder in the second step.

Step-2

In the second step 1 is the current  numerator. Bring down the second group of number (25). The new numerator is 125. In order to get  first part of the denominator, add the last step quotient with last denominator. (1+1=2). Now put a new number in the right side of the denominator and multiply it with that new number. The result should be just less than or same as the numerator. In this case 25*5 =125. So  new denominator is 25 and quotient is 5. this product will subtract from numerator and the remainder will write in the third step. In our case the remainder is zero. So the current quotient is 15 and that is the square root of the number 225.

Notes

If we are finding the root of a big number we have to repeat the steps. as shown in the fig1-c. In each step  first part of denominator can be obtain by adding the last quotient and last denominator.

If we are finding out the square root of a number which not have a perfect root. Then the remainder will not become zero, after all the group of nuber bring down. in this case we can put decimal point in the quotient. And bring down group of zeros (2 zero) to the new steps. this can be repeat until how much decimal point we require. This one we can see in the fig2-a. 

 

Square root of a number with decimal point

Now we can see how to find out the square root of a number with decimal point. We can follow the above step itself. But we have to group the number after the decimal point from left to write.  And also we have to ad one zero if the rightmost digit if it is single. Please place the decimal point in the quotient, just before bring down the first decimal number group. also we can see the example in the fig2-b.

If any one need more help on this topic please contact me through the forum.