Seker ramat horaa '15-'16: http://www.cs.biu.ac.il/~katzmik/sekerramathoraa1516.pdf

Keisler's textbook *Elementary calculus*:
https://www.math.wisc.edu/~keisler/calc.html

Keisler's textbook in searchable format:
http://www.cs.biu.ac.il/~katzmik/keislercalcsearchable.pdf

Keisler's *Foundations of infinitesimal calculus* (companion
volume)
http://www.cs.biu.ac.il/~katzmik/keislerfoundations07.pdf

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

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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