I tried to learn about continued fractions of rational and irational numbers, but when it comes to complex numbers i cannot find anything resonable about that. Is there a way to write complex numbers as a finite or infinite continued fraction and what is the algorithm for this ?
A complex number $a+bi$ as a continued fraction.
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1What kind of numbers would you use as coefficients? For real numbers, you're usually limited to integer coefficients, and that makes continued fractions unique for each number, but ultimately limited to real numbers. What alternative do you propose? – 2017-01-14
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0@Arthur, the real part should easily be written as a continued fraction, but i have no idea what should i do with $sqrt(-1)$ and its multiples. – 2017-01-14
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1That's why I'm asking you! Of course you could make continued fractions with complex numbers, but if you don't _limit_ your space of possible coefficients, then it won't be very interesting. If you allow every complex number to be a coefficient, then for any $z\in \Bbb C$, we have $z = z + \frac{0}{\ddots}$. – 2017-01-14
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0so the coefficients should be real or complex ? – 2017-01-14
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1Here's an article about using Gaussian integers as the coefficients: http://www.math.tamu.edu/~dhensley/SanAntonioShort.pdf – 2017-01-14