Let us consider a number 0.04783, which is not a perfect square, to find the square root of 0.04783, we should do a special long division, where we pair, the numbers in two, starting from decimal point in either direction. Here, however, we have chosen a number which has 0 to the left of decimal point.
To the right of decimal the pairs formed are 0. color(red)(04)color(blue)(78)color(red)(30). We write the number in long form of division as shown below.
color(white)(xx)ul(0. color(white)(x)2color(white)(xx)1color(white)(xx)8color(white)(xx)7color(white)(xx)0)
ul2|0. color(red)(04)color(white)(x)color(blue)(78)color(white)(x)color(red)(30)color(white)(x)bar(00)color(white)(.)bar(00)color(white)(.)bar(00)color(white)(.)bar(00)
color(white)(xxxx)ul(04)color(white)(X)darr
color(red)(4)1|color(white)(X)00color(white)(x)78
color(white)(xxxx)ul(00color(white)(x)41)
color(white)(x)color(red)(42)8|color(white)(xx)37color(white)(.)30
color(white)(xxxxxx.)ul(34color(white)(.)24)
color(white)(xx)color(red)(436)7|color(white)(.)3color(white)(x)06color(white)(.)00
color(white)(xxxxxXx)ul(3color(white)(.)05color(white)(.)69)
color(white)(xx)color(red)(4374)0|color(white)(x)color(white)(.)00color(white)(.)31color(white)(.)00
Here we have first pair is 04 and the number whose square is just equal or less than it is 02, so we get 2 and write its square 04 below 04. The difference is 00 and now we bring down next two digits 78.
As a divisor we first write double of 2 i.e. 4 and then find a number x so that 4x (here x stands for single digit in units place) multiplied by x is just less than the number, here 0078. We find for x=1, we have 41xx1=41 and get the difference as 37.
Now as we have a remainder of 37, we bring 30 making the number 3730. Observe that we had brought as divisor 2xx2=4, but this time we have 21, whose double is 42, and so we make the divisor as 42x and identify an x so that 42x multiplied by x is just less than 3730. This number is just 8, as making it 9 will make the product 429xx9=3861>3760. With 8, we get 428xx8=3424 and remainder is 306.
Next, we do not have anything, but for accuracy we can bring 00. The divisor will now be 218xx2=436 and identify an x so that 436x multiplied by x is just less than 30600. The number we get is 7.
We continue to do similarly and find remainder is just 31, while we will have 2187xx2=4374 i.e. 4374x and this x ought to be zero.
It is evident that we have reached desired accuracy and sqrt0.04783~~0.2187
Another example is [here.](https://socratic.org/questions/how-do-you-find-the-square-root-of-4783)
