Prof. Mikhail Katz

Infinitesimal Calculus 88-132, '17 schedule: sundays 16-18 in building 404 room 9, mondays 16-18 in building 504 room 6

Seker ramat horaa '15-'16:

Keisler's textbook Elementary calculus:
Keisler's textbook in searchable format:

Keisler's Foundations of infinitesimal calculus (companion volume)

Grade for the course is based 80% on final exam, 10% on targil grade, and 10% obligatory class participation

syllabus for the course

lecture 1, lecture 2, lecture 3, lecture 4 lecture 5, lecture 6, lecture 7, lecture 8, lecture 9, lecture 10, lecture on sequences: Lecture on inf, sup, sequences, etc.
Cauchy's definition of continuity
Cauchy's definition of continuity in BW

Final exams: Final '15 moed A, solutions for final '15 moed A, Final '15 moed B, solutions for final '15 moed B, final '16 moed A, solutions for final '16 moed A, final '16 moed B, solutions for final '16 moed B, final '16 moed C, final '17 moed A, solutions for final '17 moed A, final '17 moed B, solutions to final '17 moed B, final '18 moed A, solutions to final '18 moed A,

Old homework from '14: targil 1, targil 2, targil 3, targil 4, targil 5, targil 6, targil 7, targil 8, targil 9, targil 10, targil 11, targil 12, targil 13

Bochan from '14: bochan

Answers to some Frequently Asked Questions (FAQ).

Question 1. Does infinity belong to the hyperreals?
Answer. The infinity symbol ∞ is often added to the reals in calculus and analysis courses. The resulting number system is sometimes called the extended reals. This extended number system is of course not a field, and is not related to the hyperreals. The hyperreals form a field that contains many infinite elements, but the infinity symbol is not one of them. One can also adjoin the infinity symbol to the hyperreal line, resulting in an extended hyperreal line. I am not sure we will need this in the course but if we do it will be signaled appropriately.

Question 2. Which elements are added when one passes from the reals to the hyperreals?
Answer. What is added is all infinitesimals, all infinite numbers, but also combinations like 1+ε where ε is infinitesimal.

Question 3. What is a finite number which is not real?
Answer. An example already appeared above, namely 1+ε where ε is infinitesimal.

Question 4. In Cantorian set theory that all the students are familiar with to one extent or another, there is the notion of cardinality of a set. How is this related to the hyperreals?
Answer. Cantor developed a theory of infinite cardinalities including the fact that the cardinality of the reals is greater than the cardinality of the natural numbers, etc. This is of course a different notion of infinity than that of an infinite hyperreal. Cantor by the way was hostile to infinitesimals and at some point claimed to have proved that they are inconsistent. Another point, by the way, is that the famous philosopher Russell accepted Cantor's claim of inconsistency as fact and reproduced it in his books, including his famous "Principles of Mathematics". This kind of tidbit is strictly speaking not necessary but it might spice up class or tirgul presentation if you notice that the students are drifting off to sleep :-)

Question 5. Is there an infinitesimal number greater than epsilon?
Answer. The student probably answered himself, "2ε". Another example given in class is square root of ε.

Question 6. Why doesn't our lecturer let us use the concept of "tends to zero" when speaking of infinitely small numbers?
Answer. What I personally told them in the lecture is that the idea of a sequence (1/n) tending to zero is an excellent intuitive point of approach to infinitesimals. So they can certainly use the concept provided they understand that this is a preliminary, intuitive stage toward grasping the concept of an infinitesimal number. However, in the end a positive infinitesimal is a fixed number that's smaller than every positive real number. A student asked me in class if it is possible to think of an infinitesimal as 0.0000...1 with a lot of zeros. I told him that the answer is affirmative, provided that there are infinitely many zeros there before a final 1.

Question 7. Why can't one define finite numbers as numbers between -H and H ?
Answer. All finite numbers are indeed between -H and H if H is infinite. However, there are some infinite numbers that are also there. For example, H/2 is also between H and -H. So the property only works in one direction and cannot serve as a definition.

Question 8. They asked for an example of an infinitesimal. How does one respond?
Answer. For advanced students, such an example can be given with respect to a construction of the hypereals in terms of sequences of real numbers. Here the equivalence class of the sequence (1/n) provides such an example. This connects well with question 6 above on tending to zero. Namely, the sequence tends to zero. However, it is not the sequence itself but rather its equivalence class that defines a hyperreal infinitesimal. One can also mention that students already have natural intuitions of such numbers, when they think about what they feel is a very small discrepancy between 1 and 0.999... Over the hyperreals one can formalize such intuitions if one thinks of a number 0.999...9 with a specific infinite number of digits 9.

Question 9. Is zero an infinitesimal?
Answer. The convention following Keisler's book is to define the number zero to be infinitesimal. This may seem contrary to intuition but turns out to be convenient technically. Unlike every other infinitesimal, zero is not invertible.

Question 10. Why does Keisler all of a sudden goes back to the old approach with limits after he has defined things using standard part?
Answer. In Keisler's approach, limits themselves are defined via standard part. Therefore when limits start appearing in the course this is not to be interpreted as a throw-back to the old method, but rather as an application of the standard part approach.

Question 11. How does one define continuity of a function on an arbitrary domain?
Answer. The notion of continuity on a closed interval is defined via one-sided continuity at both endpoints. Fist we define continuity at a point, then on an open interval (in the natural way). But then, if we want to extend it to arbitrary intervals (closed, half closed) we need one-sided continuity. Thus we don't bother students with notions of continuity on a general domain where for example any function defined on Z turns out to be continuous. Dealing with general domains only confuses the students at this stage in their learning.

Question 12. What kind of weird definition of inverse function is that?
Answer. Keisler's definition of the inverse function is the following. If y=f(x) is a function then x=g(y) is its inverse provided f and g have the same graph in the (x,y) plane. Thus inverse fuctions can be defined before the composition of functions is defined. Keisler's definition in terms of the graphs is a very nice one and is different from the traditional one using composition of functions.

Question 13. Wow, I am impressed. Are there ANY mistakes in Keisler's book?
Answer. In december '15 Meny Shlossberg pointed out that there is a gap in Keisler's proof of the direct test on page 138. Here the existence of a maximum is used in the proof even though such existence is not proved until a later section. Keisler corrected it in the online edition of the book.

More on infinitesimals
Bar Ilan University wiki site for the course 89-132
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