Let α,β be real numbers 0≤α<1,0≤β<1. They define at the plane (y,z)∈R² the inhomogeneous linear form L_α,β(y,z)=-β+αy+z. We propose the algorithm of an expansion of this linear form into the 'inhomogeneous continued fraction' L_α,β~[0;b₁,b₂,...]mod[0;a₁,a₂,...]. Inhomogeneous continued fraction generalize the classic regular continued fraction: for β=0 every b_n=0 and we get the continued fraction expansion of the number α: L_α,0~[0]mod[0;a₁,a₂,...]. Some properties of inhomogeneous continued fractions are proved.

continued fractions, inhomogeneous Diophantine approximations, Euclid algorithm

Mathematical modelling in actual problems of science and technics