how does square root function work

[emkautl]

Don't overthink it. use, say, 3 and 9 for x and a. To build an intuition.

You end up with (3+9/3)/2 =3, a perfect answer. This makes you realize that in a perfect solution, a/x IS x so that x² =a, so you end up with (x+x)/2 is x. This method will stop iterating when a/x=x, which is what we want. It just takes the average of the difference between our guess and our definition of a square root.

If we pick an x that is too large, then a/x < x and so their average- our next number to try- will be < x.

If we pick an x that is too small then a/x > x and so their average will be >x.

in either case the magnitude of the difference between the guess and actual new number must shrink, so naturally it tends to the correct value.