## 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 to a much larger set of real numbers 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 one and only one relation out of three is possible, < , or > , we can uniquely define any, say, positive rational unknown number by looking at the two sets, and . (You don’t have to be too smart at this moment: is the only element in the intersection 馃槈

However, sometimes the analogous construction leads to problems. For instance, if and , then , since the square root of two is not a rational number, but , i.e., for any positive rational number we can say whether is smaller or larger the missing number . This allows to derive all properties of , 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.