We assume à priori that we are not using a calculator.
If we require accuracy of nearest tenth or hundredth, we need to know the size of the number we're working with. It would not be reasonable to estimate the square-root of 284029 to the hundredth place without a calculator. On the other hand, with one, we can get the exact value and estimation is not required.
The following method should work comfortably well with numbers below hundred, all in the head. We will take the example of estimating the square-root of 82 to the hundredth, without a calculator (nor pen and paper), using Newton's method.
Newton's method to find square-roots depends on an initial estimate, which is not a problem if we know the multiplication table.
We know that 9*9=81, and 10*10=100, so the square-root of 82 should lie between 9 and 10, but close to 9. We will use the initial estimate, x0=9.
A better estimate is given by x1=x0-(x0^2-82)/(2x0) .................(1) =9-(81-82)/(18) =9+1/18 =9.0555...
So to two decimal places, it will be 9.06. In fact, the exact value is 9.0554, and our estimate was accurate to 3 places of decimal, or almost 4.
If we set x0=9.055 and apply to (1) again, we will get a still better estimate, but the squaring and division will need some help, pen and paper or a calculator.
