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.
翻译:在本文中,我们引入了一个新的概念, 即$\ expslon$- arithmetics, 用于任何固定维度的真实矢量。 基本的想法是使用理性值的矢量( 所谓的理性矢量) 来接近 $\ epslon$ 范围内同一维度的真实矢量。 对于固定维度的理性矢量, $m, 它们可以形成一个面积, 即 $\ momxxxb* (\ alpha)$( pha) 的合理字段的第一级扩展 $\ mathbf $( pha), $\ alpha$ 的最小多元度为$ $\ mathbf $( mathbf) $( $ $) 。 然后, 实际矢量的计算, 如添加、 减量、 倍增、 倍增、 和分割, 可以通过在 $\\ exslon 范围内使用其近度的合理矢量的合理矢量矢量的数值来定义 。