实数与超实数位于同一条几何直线上，无穷小ε闪亮登场
实数系统R与超实数系统R*位于同一条几何直线L上，有点儿难于想象，匪夷所思。J.Keisler发明显微镜与望远镜来观察几何直线的“微观结构”，发现无穷小ε就在原点附近。
几何直线的“微观结构”很神奇，让00后大学生大开眼界，增长知识。00后大学生学习微积分赢在学习的起跑线上，指的就是这一点。
00后上大学学习微积分的目的不限于提高公式计算的能力，主要的目的是：学会慎密的逻辑思维，日后不做糊涂。请见本文附件。
袁萌 7月11日
附：
A number ε is said to be irifinitely small, or infinitesimal,if
-a < ε< a
for every positive real number a. Then the only real number that isinfinitesimal is zero. We shall use a new number system called the hyperrealnumbers, which contains all the real numbers and also has infinitesimals thatare not zero. Just as the real numbers can be constructed from the rationalnumbers, the hyperreal numbers can be constructed from the real numbers. Thisconstruction is sketched in the Epilogue （结束语）at the end of thebook. In this chapter, we shall simply list the properties of the hyperrealnumbers needed for the calculus. First we shall give an intuitive picture of thehyperreal numbers and show how they can be used to find the slope of a curve.The set of all hyperreal numbers is denoted by R*. Every real number is amember of R*, but R* has other elements too. The infinitesimals in R* are ofthree kinds: positive, negative, and the real number 0. The symbols Δx, Δy, ... and theGreek letters ε(epsilon) and δ(delta) will beused for infinitesimals. If a and b are hyperreal numbers whose difference a -b is infinitesimal, we say that a is irifinitely close to b. For example, ifLix is infinitesimal then x0 + Lix is infinitely close to x0 . If ε is positive infinitesimal, then – ε will be a negative infinitesimal. 1/ε will be an irifinite……（以下省略）