Continued Fractions
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, using the quadratic equation. Nice huh! Maybe that’s why I only hear of continued fractions, and not continued radicals… because they are equivalent. Of course that only proves a simple case, where the radical/fraction is periodic. What if it isn’t; then can you find a way to convert a given continued radical into a continued fraction? If you can, then it would make perfect sense not to have a theory of continued radicals, otherwise…
Another interesting question is, what is 
? From the equation above, I would think -1, but matlab disagrees. When I use it to approximate the sequence, the numbers fluctuate between 0.5+.866j and 0.5-.866j. Unfortunately, I forgot so much that I can’t remember how to plot with MatLab, so I can’t see what’s going on with the sequence of approximations.