$ε$-Arithmetics for Real Vectors

In this paper, we introduce a new concept, namely $\epsilon$-arithmetics, for real vectors of any fixed dimension. The basic idea is to use vectors of rational values (called rational vectors) to approximate vectors of real values of the same dimension within $\epsilon$ range. For rational vectors of a fixed dimension $m$, they can form a field that is an $m$th order extension $\mathbf{Q}(\alpha)$ of the rational field $\mathbf{Q}$ where $\alpha$ has its minimum polynomial of degree $m$ over $\mathbf{Q}$. Then, the arithmetics, such as addition, subtraction, multiplication, and division, of real vectors can be defined by using that of their approximated rational vectors within $\epsilon$ range. We also define complex conjugate of a real vector and then inner products for two real vectors and two real vector sequences of the same length.