Given a function \(f(x)\), what number \(x\) can I plug in so that \(f(x)=0\)? Are there any values for \(x\) which will work?

This is a surprisingly important question in many areas of math, and was the problem Newton was trying to solve when he came up with a way to approximate solutions (if there are any) for differentiable functions. That is Newton came up with a way to find values of \(x\) so that \(f(x)\approx 0\). The very first formula at the top describes how to construct a sequence which will get closer and closer to a solution. We first make a random guess, call it \(z_0\). Then we construct the next term in the sequence using the formula up top. In other words \(z_1 = z_0 - \frac{p(z_0)}{p'(z_0)}\), \(z_2 = z_1 - \frac{p(z_1)}{p'(z_1)}\), and so on. For most choices of \(z_0\), this will converge to a solution quite rapidly. But for others it make take longer, or not at all.

The image above was generated using Newton's method. What you are seeing is a small portion of the complex plane. First, we take each point and apply Newton's method. If Newton's method works, once a point gets pretty close to a solution, it stays near the solution. So the points are colored by how many times we needed to apply Newton's in order to get that point "pretty close". So the brighter the pixel, the longer it took for that number to get close to a solution. In some cases they never get close to any solution. These are the very bright regions. Surprisingly, applying Newton's method to any differentiable function seems to produce a fractal. You can read more about Newton's method on the wikipedia page, and about Newton fractals.