Here is another way to approximate squares.

Hi,

Some times we  need to an approximate range among which the square root of a number lies.
For that the conventional method is time consuming.

Here is an interesting way to find an approximate range of a square root.

For instance 4059. It does not look like a perfect square number, but let us see what is the nearest perfect square number to it.
4059
now divide 4059 as    40|59

the nearst perfect square to 40 is 36 which is 6^2

so put a six below 40 and for the 2 digits of 59 put a zero beside 6 which makes it 60

so 4059 is a number greater than 60^2 for sure now we can do 2 things.

Eiter follow approximation approach by checking 65^2 and 70^2.

Now 65^2 is 4225 and 70^2 is 4900. Ideally 4059 lies between 60^2 and 65^2 and from there we can very easily calculate the squares of 64,63,62,61(using 25-75 approach) on the fly and approximate to the extent required.

One more way is to divide 4059 with 60 which gives a number and now the nearest perfect square lies between these 2 numbers.

Hope this is useful.

Cheers!!