Sergei Yakovenko's blog: on Math and Teaching

Sunday, November 19, 2017

Lecture 3, Nov 14, 2017

Real numbers as solutions to infinite systems of equalities

In the past we already extended our number system by adding “missing” elements which are assumed to satisfy certain equations, based only on knowing what these equations are. It turns out that we may extend the set of rational numbers \mathbb Q to a much larger set of real numbers \mathbb R by adding solutions to (infinite numbers) of inequalities. As before, the properties of these new numbers could be derived only from the properties of inequalities between the rational numbers.

On one leg, the idea can be explained as follows. Since for any two rational numbers r,s\in\mathbb Q one and only one relation out of three is possible, r < s, r=s or r > s, we can uniquely define any, say, positive rational unknown number x by looking at the two sets, L=\{l\in\mathbb Q: 0\le l\le x\} and R=\{r\in\mathbb Q: x\le r\}. (You don’t have to be too smart at this moment: x is the only element in the intersection L\cap R 馃槈

However, sometimes the analogous construction leads to problems. For instance, if L=\{l\in\mathbb Q: l\ge 0, l^2\le 2\} and R=\{r\in\mathbb Q: r^2\ge 2\}, then L\cap R=\varnothing, since the square root of two is not a rational number, but L\cup R=\mathbb Q_+, i.e., for any positive rational number we can say whether is smaller or larger the missing number \sqrt 2. This allows to derive all properties of \sqrt 2, including its approximation with any number of digits.

Proceeding this way, we introduce (positive) real numbers by indication, what is their relative position to all rational numbers. This allows to describe the real numbers completely.

The details can be found here.

A didactic digression

Some of you complained about insufficient number of problems that are discussed during the tutorials. Everybody knows that problems and questions for self-control are the most important elements of study mathematics, especially in comparison with other disciplines. The rationale behind is the assumption that a student who understands the subject, should be able to answer these questions immediately or after some reflection. Composing such problems is an easy thing: you any mathematical argument you can stop for a second and ask yourself: “why I can do as explained?” or “under what conditions are my actions justified?”. In the lecture notes (see the link above) tens of such problems are explicitly formulated. Similar problems will await you on the exam.

However, remember one simple thing. If you already know how to solve a problem, this is not a problem but rather a job. Unless you solve these problems yourselves, there is no sense in memorizing their solutions: knowing solution of one such problem won’t help you with solving another problem unless you really understand what’s going on. There are no “typical problems”: each one of them is of its own sort, though, of course, some problems can be solved by similar methods.

A practical advice: you should not expect that all problems that appear on the exam will be discussed at length at the tutorials. There are no ready recipes to memorize. Only to understand honestly. Believe me, this is easier than memorize by heart endless formulas and algorithms.

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Sunday, December 4, 2011

Lectures 4-7, Nov. 15, 22, 29 and Dec. 6

Construction of real numbers

The idea of extending the number system from the set of rational numbers \mathbb Q by adjoining roots of polynomial equations is very interesting, but faces obvious difficulties: we need to treat all possible polynomial equations, and this still give us no guarantees whatsoever that transcendental equations (trigonometric, exponential etc). will be solvable when we expect them to be.

The alternative is to extend the set of rationals by adding “solutions to systems of inequalities”. In order for such a system to represent a unique “new” number, the equations need to be consistent (compatible between themselves) and possess some uniqueness property.

These two requirements can be implemented by consideration of the so called Dedekind cuts, which can be informally considered as sets of rational “approximations” (lower and upper) for the missing number.

In the lectures we pursue this strategy and explain how the cuts can be compared, how arithmetic operations on the cuts can be defined and why the addition of all possible cuts results in a “complete” number system.

The detailed exposition, as before, is downloadable as a pdf file. Please take a time to signal (in the comments to this post or by any other way) about all errors, inevitable in the first draft.

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