Sergei Yakovenko's blog: on Math and Teaching

Tuesday, October 31, 2017

Lecture 1, October 31, 2017

שלום כיטה א!

The main feature that distinguishes the Calculus (or Mathematical Analysis) from other branches of mathematics is the repeated use of infinite constructions and processes. Without infinity even the simplest things, like the decimal representation of the simple fraction \frac13=0.333333\dots becomes problematic.

Yet to deal with infinity and infinite constructions, we need to make precise our language, based on the notions of sets and functions (maps, applications, – all these words are synonymous).

Look at the first section of the lecture notes here.

You are most welcome to start discussions in the comments to this (or any other) post. Don’t be afraid of asking questions that may look stupid: this never harms! Write in any language (besides Hebrew/English, I hope that Ghadeer will take care of questions in Arabic, and I promise to deal with French/Spanish/Catalan/Italian/Russian/Ukrainian questions) 😉 Subscribe for updates on this site with your usual emails, to be independent from any dependence 😉

Looking forward for a mutually beneficial interaction in the new semester!


Sunday, October 27, 2013

Lectures 1 and 2 (Oct. 22 & 29, 2013)

We start our course by first carefully walking in a zoo with several surprising, wonderful or dangerous beasts, all caught in the Jungle of Infinity. To tame them, we need to wear some protective gear made of ironclad formulas, operate from the safety of well defined sets and use the tools provided by functions 😉

The lecture notes (considerably updated and revised in comparison with the previous years) can be found here. Please report typos, errors and complain about obscure instances in the comments.

Please pay attention to the problems scattered over the text: they are mostly for self-control, but if you are not sure, you may write your solutions and submit them (e.g., in the comments to this post, but also directly to Inna, if you want to keep it private).

Sunday, November 13, 2011

Lecture 2. November 8, 2011

Infinity: first accurate steps

  1. Finite and infinite subsets of \mathbb N.
  2. Admissible infinite operations: infinite unions and intersections. Quantifiers.
  3. “Small” and “large” infinite subsets of \mathbb N.
  4. One-to-one maps as the means of comparison between various infinite sets.
  5. The first “paradox”: \mathbb N\times\mathbb N\simeq\mathbb N.

Download the pdf file for the complete exposition.

Thursday, November 3, 2011

Lecture 1, Nov. 1, 2011

First Encounter with Infinity

In the first lecture we discuss the dangers that are inherently present when we transcend our finite intuition and consider infinite quantities, constructions etc.

The first step to do is to clean up our language, restricting it to the most transparent and unambiguous grammar and vocabulary. This is the language of the sets and operations on them, and logical formulas involving quantifiers, to formulate meaningful (true or false) constructions.

Read the first lecture here.

P.S. Yet another funny and instructive book by N. Ya. Vilenkin, “In Search of Infinity”, can be downloaded here. Read slowly and enjoy!

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