打开网站“无穷小微积分”，点击“ElementaryCalculus”按钮，等待数秒钟，问题的答案就会从遥远的互联网“云端”飞到你的视屏上。我国的00后大学生就是懂网络、会英文的一代新人。
容易想象，读者不出几秒钟的时间，就会找到该书的第一章的1.4节“实数与超实数”，其中有下面的一段文字（请见本文附件）。这段文字正面回答了微积分为什么需要引入超实数无穷小，我国的中学生都能读懂这段文字。该书作者的学术水平就表现在这里！书如其人也。
感言：如果今年全国数百万大学新生都采用这本微积分参考书，该有多好啊！菲氏微积分可以休矣！
附：
1.4 REAL AND HYPER REAL NUMBERS
But for a verysmall increment of time Lit, the velocity will change very little, and theaverage velocity Liy/ Lit will be close to the velocity at time t0 . To get thevelocity v0 at time t0 , we neglect the small term Lit in the formula Dave =2to + Lit,
and we are left with the value
v0 = 2t0 .
When we plot y against t, the velocity is the same as the slope of thecurve y = t2, and the average velocity is the same as the average slope. Thetrouble with the above intuitive argument, whether stated in terms of slope orvelocity, is that it is not clear when something is to be"neglected." Nevertheless, the basic idea can be made into a usefuland mathematically sound method of finding the slope of a curve or the velocity.What is needed is a sharp distinction between numbers which are small enough tobe neglected and numbers which aren't. Actually, no real number except zero issmall enough to be neglected. To get around this difficulty, we take the boldstep of introducing a new kind of number, which is infinitely small and yet notequal to zero. A number e is said to be irifinitely small, or infinitesimal, if
-a < e <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 hypeiTealnumbers, 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 at the end of the book. In thischapter, we shall simply list the properties of the hyperreal numbers neededfor the calculus. First we shall give an intuitive picture of the hyperrealnumbers and show how they can be used to find the slope of a curve. The set ofall hyperreal numbers is denoted by R*. Every real number is a member of R*,but R* has other elements too. The infinitesimals in R* are of three kinds:positive, negative, and the real number 0. The symbols Lix, Liy, ... and theGreek letters e (epsilon) and Ci (delta) will be used for infinitesimals. If aand bare hyperreal numbers whose difference a - b is infinitesimal, we say thata is irifinitely close to b. For example, if Lix is infinitesimal then x0 + Lixis infinitely close to x0 . If e is positive infinitesimal, then - e will be anegative infinitesimal. 1/e will be an irifinite positive number, that is, itwill be greater than any real number. On the other hand, - 1/e will be an infinitenegative number, i.e., a number less than every real number. Hyperreal numberswhich are not infinite numbers are called finite numbers. Figure 1.4.3 shows adrawing of the hyperrealline. The circles represent "infinitesimalmicroscopes" which are powerful enough to show an infinitely small portionof the hyperrealline. The set R of real numbers is scattered among the finitenumbers. About each real number c is a portion of the hyperrealline composed ofthe numbers infinitely close to c (shown under an infinitesimal microscope forc = 0 and c = 100). The numbers infinitely close to 0 are the infinitesimals.In Figure 1.4.3 the finite and infinite parts of the hyperrealline wereseparated from each other by a dotted line. Another way to represent the infiniteparts of the hyperrealline is with an "infinite telescope" as inFigure 1.4.4. The field of view of an infinite telescope has the same scale asthe finite portion of the hyperreal line, while the field of view of aninfinitesimal microscope contains an infinitely small portion of the hyperreal lineblown up.
容易想象，读者不出几秒钟的时间，就会找到该书的第一章的1.4节“实数与超实数”，其中有下面的一段文字（请见本文附件）。这段文字正面回答了微积分为什么需要引入超实数无穷小，我国的中学生都能读懂这段文字。该书作者的学术水平就表现在这里！书如其人也。
感言：如果今年全国数百万大学新生都采用这本微积分参考书，该有多好啊！菲氏微积分可以休矣！
袁萌 6月16日
附：
1.4 REAL AND HYPER REAL NUMBERS
But for a verysmall increment of time Lit, the velocity will change very little, and theaverage velocity Liy/ Lit will be close to the velocity at time t0 . To get thevelocity v0 at time t0 , we neglect the small term Lit in the formula Dave =2to + Lit,
and we are left with the value
v0 = 2t0 .
When we plot y against t, the velocity is the same as the slope of thecurve y = t2, and the average velocity is the same as the average slope. Thetrouble with the above intuitive argument, whether stated in terms of slope orvelocity, is that it is not clear when something is to be"neglected." Nevertheless, the basic idea can be made into a usefuland mathematically sound method of finding the slope of a curve or the velocity.What is needed is a sharp distinction between numbers which are small enough tobe neglected and numbers which aren't. Actually, no real number except zero issmall enough to be neglected. To get around this difficulty, we take the boldstep of introducing a new kind of number, which is infinitely small and yet notequal to zero. A number e is said to be irifinitely small, or infinitesimal, if
-a < e <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 hypeiTealnumbers, 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 at the end of the book. In thischapter, we shall simply list the properties of the hyperreal numbers neededfor the calculus. First we shall give an intuitive picture of the hyperrealnumbers and show how they can be used to find the slope of a curve. The set ofall hyperreal numbers is denoted by R*. Every real number is a member of R*,but R* has other elements too. The infinitesimals in R* are of three kinds:positive, negative, and the real number 0. The symbols Lix, Liy, ... and theGreek letters e (epsilon) and Ci (delta) will be used for infinitesimals. If aand bare hyperreal numbers whose difference a - b is infinitesimal, we say thata is irifinitely close to b. For example, if Lix is infinitesimal then x0 + Lixis infinitely close to x0 . If e is positive infinitesimal, then - e will be anegative infinitesimal. 1/e will be an irifinite positive number, that is, itwill be greater than any real number. On the other hand, - 1/e will be an infinitenegative number, i.e., a number less than every real number. Hyperreal numberswhich are not infinite numbers are called finite numbers. Figure 1.4.3 shows adrawing of the hyperrealline. The circles represent "infinitesimalmicroscopes" which are powerful enough to show an infinitely small portionof the hyperrealline. The set R of real numbers is scattered among the finitenumbers. About each real number c is a portion of the hyperrealline composed ofthe numbers infinitely close to c (shown under an infinitesimal microscope forc = 0 and c = 100). The numbers infinitely close to 0 are the infinitesimals.In Figure 1.4.3 the finite and infinite parts of the hyperrealline wereseparated from each other by a dotted line. Another way to represent the infiniteparts of the hyperrealline is with an "infinite telescope" as inFigure 1.4.4. The field of view of an infinite telescope has the same scale asthe finite portion of the hyperreal line, while the field of view of aninfinitesimal microscope contains an infinitely small portion of the hyperreal lineblown up.