Newtons’ Method

Squaring is a lossy process. That is, when a number is squared the root is lost. The search for roots is further complicated  because many whole numbers have irrational roots.

Even intuition is misleading, take a guess that the square root of 4 might be half (and you’d be right), but geometric growth would indicate that this is not the case as the root rapidly becomes a smaller and smaller portion of the square.

But Newton. He provides a method to make a guess and increase it by some small delta. His method, given a guess z looks like this:

z_1 = z_0 - \frac{z^2 -x}{2z}

Turning around and feeding z back into the expression iteratively yields an answer which tends closer and closer to the root.

Mind boggling basic calculus.