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 minimal 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 product and convolution of two real vectors and two real vector sequences (signals) of finite length. With these newly defined concepts for real vectors, linear processing, such as filtering, ARMA modeling, and least squares fitting, can be implemented to real vector-valued signals, which will broaden the existing linear processing to scalar-valued signals.

翻译：在本文中,我们引入了一个新的概念, 即$\ epsilon$- arthmetics, 用于任何固定维度的真正矢量。 基本的想法是使用理性值的矢量( 所谓的理性矢量) 来接近在$\ epsilon 范围内同一维度的真正矢量的矢量。 对于固定维度的理性矢量的理性矢量, $m, 它们可以形成一个字段, 即 $\ mathbf ⁇ (\ alpha), 美元, 美元, 任何固定维度的真正矢量的最小多元度为$$ 。 基本的想法是使用合理值的矢量的矢量( 所谓的理性矢量) 来接近在 $\ epsluslonon 范围内的同一维度的真正矢量的矢量的矢量( $\ mathbf) 。 然后, 算算算算算算,,, 例如添加、 减、 倍增、 倍增、 和 等现有 直线性 信标的 等, 等 将安装到 直径 等 的 等 直径 的 等 的 等 直压 等 的 的 。