Today, almost everywhere, students submit evaluations of their professors as part of routine feedback. No special “quid pro quo” is expected beyond the grade you receive for your exams — and those grades remain confidential. Still, I would like to complement those grades with a public statement of my own.
I truly loved teaching your class, and each of you deserves a special thank-you for the role you played in the process.
- Maya: I tried to convince you that you didn’t need to copy everything from the screen into your notebook, since you would receive the lecture notes afterward. You convinced me instead that writing by hand helps you remember the key points more firmly. Perhaps one day I’ll be curious enough to take a look at your notes.
- Lee: Frankly, I used your eyes the way we use LED indicators on the dashboard of smart devices. When they blinked blue/green/marine, I knew I could move forward with confidence. When the color shifted to yellowish, I understood it was time to slow down and rephrase the last sentence. Thankfully, I never saw the red light flashing!
- Itay: I watched you carefully storing each new piece of information, making sure the overall structure would stand firm and serve as a solid ladder for your future students to climb to the very top.
- Rafi: Your enthusiasm for mathematical challenges at every level is truly amazing. If I watched you with the sound turned off, I could easily imagine a Beitar Jerusalem fan excitedly describing yesterday’s beautiful game to his friends. I envy your students — your passion seems highly contagious.
- Shahar: Your calm appearance cannot hide the artist inside who genuinely admires the beauty of mathematics. It is very difficult to transmit the sense of beauty from one person to another, but I hope in your case I succeeded. We will return to this theme in your final project on mathematical foundations of musical harmony.
- Doron: You remind me of a seasoned boatswain (bosun / רַב־חוֹבֵל) who double-checks every single piece of equipment to make sure it is in perfect working order. With such a solid, responsible attitude toward both studying and teaching, your boat will conquer the highest seas!
- Nadav: You were the ideal smoke detector complementing Lee’s LED panel. Always calm, yet closely following everything that happened in the class. Unlike a standard smoke detector that shrieks at the first sign of trouble, you quietly raised a question — most often signaling that I had just made a mistake or admitted an inaccuracy.
by adding solutions of all possible infinite systems of two-sided inequalities 
, where 
is a partition of all rational numbers into two non-empty sets so that each one of the above inequalities defines a nonempty set (eventually, a point). This is possible since the rationals 
. 
“seals” all gaps between rational numbers, guaranteening that 
e.a. are in it. The arithmetic operations between real numbers are extending the corresponding rational operations performed on the respective inequalities, and the order 
representing the geometric 2-plane, 3-space e.a., do not carry the linear order, so we have to replace it by a close notion of a distance between points of these powers. The distance is a symmetric nonnegative function of two arguments that satisfies the triangle inequality. 
of the given radius and center. This paves the way to introducing the first notions of the topology, those of open and closed sets. By definition, a set 
is open, if together with each point 
it contains a round open ball centered in 
. 
. The informal idea of a continuity of 
at 
of radius 
, no matter how small (but positive!) 
is, around the image 
one can find a small open round ball 
of radius 
around 
(of course, wherever defined). This notion appears first as a local one (continuity in a point), but can be globalized (continuity on the entire domain). 
: the notation 
means that a set denoted by 
is an element of another set denoted by 
. We say that the two sets are equal if and only if they consist of the same elements. There is a unique set 
which has no elements at all. 
: we say that a set 
and write 
, if any element 
is also an element of 
. By definition, 
and intersection 
of any two sets. It is important to stress that there is no problem with defining infinite unions and intersections! 
which may or may not include the number 
(this is a convention). The set of Peano axioms which starts, say, with two elements 
and for any element 
introduces a unique element 
, the successor of 
), allows us to define the set of natural numbers 
“inductively” (imitating the human process of counting) and immediately leads us to the conclusion that 
and 
are sometimes solvable with respect to 
in the natural numbers. Yet we can extend the set of naturals 
of integer numbers, closed by subtraction, and the set 
is always solvable if 
(such a set is called a field with the four arithmetic operations) and can be equipped with the complete order 
. 
expressing the length of the diagonal of a unit square. In the same way the circumference 
of the unit circle is also not a rational number. To cope with this, we need to extend the rational numbers even more, this time adding to them solutions of all (infinite) systems of two-sided inequalities of the form 
with 
. On this way we construct the set of real numbers 
and variance 
the normalized sum 
converges to the so called normal distribution with the density function 
.
we introduced two very important numeric characteristics, expectation and variance.
is (in the simplest setting) the weighted sum of values which the variable 
takes with weights equal to the corresponding probabilities; it is a finite number or can be 
. If the expectation is zero, this means that 
describes the spread of random variable around its expected value 
) equal to the expectation of the square 
. Small variance means that 
its preimage 
must be an admissible (random) event. 
. If necessary, we add indication of the random variable and write 
. 
Sergei Yakovenko's blog: on Math and Teaching 
