Newton's method is used to find the root(s) of an equation. It is useful when the roots
of the equation are not integers. To use Newton's method, a starting number is chosen, and
when it is entered into a recursive function, it eventually converges towards a root. For
example, the equation x^3 - 1 = 0 has only one real root: 1. Any real number that you
choose as a starting number will converge to 1. However, when you take into account
imaginary numbers as well, it gets much more interesting. x^3 - 1 = 0 has two complex
roots as well. A natural question to ask would be which root will your chosen
starting number converge to. Usually, your starting number will converge towards the
nearest root. However, this is not the case near the borders where a starting number
is nearly as close to one root as it is to another. The image below is essentially a map
of starting points and which root they converge to with the equation x^3 - 1 = 0. Each
point on the image represents a starting point which is coloured a different colour
depending on which root it converged to. The boundary is fractal. Also, although it
may seem paradoxical, between any two colours in the boundary, there exists the third.
Have no idea what I'm talking about? This might help. Explore Newton's method online
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