If we have two real numbers $u,v\ge 0$ with the continued fractions $$u=[0,a_1,a_2,a_3,\cdots]$$ and $$v=[0,b_1,b_2,b_3,\cdots]$$ can we calculate the continued fractions of $u+v,u-v,uv,\frac{u}{v}$ only using the continued fraction representations of $u$ and $v$ ?
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2I am quite certain that we can, but I highly doubt it's going to be pretty. – Arthur Jan 14 '17 at 15:41
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I have lost the proof, but my teacher showed us that the representation of a number as continued fraction is not continuous in some useful sense. So I agree with @Arthur. – ajotatxe Jan 14 '17 at 15:46