Can we add, subtract , multiply and divide numbers only using their continued fraction representation?

Question

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 $v$ ?