How to approximate square root of a non perfect square number?

Answer 1

There is a method that looks similar to long division, but is more complicated.

However, a simple method is to bracket multiples of 100.

Suppose you want the square root of 37.

In terms of perfect squares 62 = 36 < 37 < 49 = 72

so that 6 < sqrt(37) < 7

Now look at 3700 and consider perfect squares of the integers between 10*6 = 60 and 10*7 = 70.

You will find that 602 = 3600 < 3700 < 3721 = 612

Next look at 370000 and the perfect squares of integers in the range 10*60 = 600 and 10*61 = 610

This gives 6082 = 369664 < 370000 < 370881 = 6092

The following step gives

This gives 60822 = 36990724 < 37000000 < 37002889 = 60832

So, to 2 decimal places, the answer is 6.08.

Or, on the grounds that the second inequality is much closer, you could say that the answer to 3 dp is 6.083. Note however, that that arguement does not always work.

The above method can be continued indefinitely - provided you can calculate the bracketing squares. And that can be done manually well beyond the range of an ordinary calculator. Or even standard spreadsheets.

Methods, such as Newton Raphson, are not much help because they require you to calculate the square root as part of the process.