# Sergei Yakovenko's blog: on Math and Teaching

## 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 $L\cap R$ 😉

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.

## Crash course on linear algebra and multivariate calculus

Real numbers as complete ordered field. Finite dimensional linear spaces over $\mathbb R$. Linear maps. Linear functionals, the dual space. Linear operators (self-maps of linear space), invertibility via determinant. Affine maps, affine spaces.

Polynomial nonlinear maps and functions, re-expansion as a tool to construct linear (affine) approximation. Differential. Differentiability of maps, smoothness of functions.

Inverse function theorem.

Vector fields, parameterized curves, differential equations.

The first set of notes is available here here.

# Real numbers

There are certain situations when the rational numbers are apparently not sufficient: for instance, the function $f(x)=x^2-2$ is negative at $x=0$, positive at $x=2$ but does not take the intermediate value zero: $\forall x\in\mathbb Q\ f(x)\ne 0$. Another situation concerns the possibility to define the notions of supremum and infimum for infinite sets: the set $A=\{x\in\mathbb Q: x^2<2\}$ is bounded from two sides, but among its upper bounds $B=\{b\in\mathbb Q:\ \forall a\in A\ a\leqslant b\}$ there is no minimal one.

The idea is to adjoin to $\mathbb Q$ solutions of infinitely many inequalities.

For any rational number $a\in\mathbb Q$ one can associate two subsets $L,R\subset\mathbb Q$ as follows: $L=\{l\in \mathbb Q: l\le a\}$ and $R=\{r\in\mathbb Q: a\le r\}$. Then the number $a$ is the unique solution to the infinite system of inequalities of the form $l\le x\le r$ for different choices of $l\in L,\ r\in R$. This system has the following two features:

1. it is self-consistent (non-contradictory): any lower bound $l$ is no greater than any upper bound $r$, i.e., $L\le R$, and
2. it is maximal: together the two sets give $\mathbb Q=L\cup R$, and none of the sets can be enlarged without violating the first condition.

Definition.
A (Dedekind) cut is any pair of subsets $L,R\subseteq\mathbb Q$ satisfying the two conditions above.

If a rational number $a\in\mathbb Q$ satisfies all the inqualities $l\le a,\ a\le r$ for all $l\in L,\ r\in R$, then we call it a root (or a solution) of the cut. Every rational number is the solution to some cut $\alpha=(L,R)$ as above, and this happens if and only if $L\cap R=\{a\}$. Yet not all cuts have rational solutions (give an example!).

We can associate cuts without rational solutions with “missing” numbers which we want to adjoin to $\mathbb Q$. For this purpose we have to show how cuts can be ordered (in a way compatible with the order on $\mathbb Q$) and how arithmetic operations can be performed on cuts.

## Order on cuts

Let $\alpha=(L,R),\ \beta=(L',R')$ be two different cuts. We declare that $\alpha\triangleleft\beta$, if $L\cap R'\ne\varnothing$, i.e., if there is a rational number $a\in\mathbb Q$ that is at the same time an upper bound for the cut $\alpha$ and a lower bound for the cut $\beta$. If both cuts have rational solutions, this number would be squeezed between these solutions. In the similar way we define the opposite order $\alpha\triangleright\beta$ if and only if $L'\cap R\ne\varnothing$.

To see that this definition is indeed a complete order, we need to check that for any two cuts $\alpha,\beta$ one and only one of the three possibilities holds: $\alpha\triangleleft\beta,\ \alpha\triangleright\beta$ or $\alpha=\beta$ (meaning that $L=L',R=R'$). This is a routine check: if the first two possibilities are excluded, then $L\cap R'=L'\cap R=\varnothing$, and therefore $(L\cup L', R\cup R')$ is a self-consistent cut. But because of the maximality condition, this means that $L\cup L'=L=L'$ and $R\cup R'=R=R'$, that is, $\alpha=\beta$.

## Arithmetic operations on cuts

If $\alpha=(L,R),\ \beta=(L',R')$ are two cuts which have rational solutions $a,b$, then these solutions satisfy inequalities $L\le a\le R,\ L'\le b\le R'$ (check that you understand the meaning of this inequality between sets and numbers ;-)!) Adding these inequalities together means that $c=a+b$ satisfies the infinite system of inequalities $L+L'\le c\le R+R'$, where $L+L'$ stands for the so called Minkowski sum $L+L'=\{l+l':\ l\in L,\ l'\in L'\}$ (the same for $R+R'$). This allows to define the summation on cuts.

Definition.
The sum of two cuts $\alpha=(L,R),\beta=(L',R')$ is the cut $\gamma=(L+L',R+R')$ with the Minkowski sum in the right hand side.

To define the difference, we first define the cut $-\alpha$ as follows, $-\alpha=(-R,-L)$, where (of course!) $-L=\{-l: l\in L\},\ -R=\{-r: r\in R\}$. Note that the upper and lower bounds exchanged their roles, since multiplication by $-1$ changes the direction (sense) of the inequalities. Then we can safely define $\alpha-\beta$ as $\alpha + (-\beta)$. Again, one has to check that this definition is well-behaving and all arithmetic properties are preserved.

To define multiplication, one has to exercise additional care and start with multiplication between positive cuts $\alpha,\beta\triangleright 0$ (do it yourselves!) and then extend it for negative cuts and the zero cut. After introducing this definition, one has to make a lot of trivial checks:

1. that for cuts having rational solutions, we get precisely what we expected, that is, the new operation agrees with the old one on the rational numbers,
2. that they have the same algebraic properties (associativity, distributivity, commutativity etc) as we had for the rational numbers,
3. that they agree with the order that we introduced earlier exactly as this was the case with the rational numbers,
4. … … …. …. …

Of course, nobody ever wrote the formal proofs of these endless properties! (Life is short and one should not waste it for nothing). Yet every mathematician can certainly provide a formal proof for any of them, and nobody of countless students who passed through this ordeal ever voiced any concern about validity of these endless nanotheorems. So wouldn’t we.

## Achievement of the stated goals

Once we constructed the extension of the rational numbers by all cuts and denote the result $\mathbb R$ and call it the set of real numbers, one has to verify that all the problems we started with, were actually resolved. There is a number of theorems about the real numbers that look dull and self-evident unless we know that a heavy price had to be paid for that. Namely, we can guarantee that:

1. Any subset $A\subset\mathbb R$ which admits at least one upper bound, admits the minimal upper bound called $\sup A=\sup_{a\in A}a$ (and, of course, the analogous statement holds for $\inf A$).
2. If $\varnothing\ne I_k=[a_k,b_k]\subseteq\mathbb R$ is a family of nested nonempty closed intervals, $I_1\supseteq I_2\supseteq I_3\supseteq\cdots$, then the intersection $I_\infty=\bigcap_{k=1}^\infty I_k$ is also nonempty.
3. Any function $f:[a,b]\to\mathbb R$ continuous on the closed segment $[a,b]$, takes any intermediate value between $f(a)$ and $f(b)$.

For more detailed exposition, read the lecture notes here.

## 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.

## Existence of limits and completeness of the real numbers system

• Monotonicity and its implications.
• Nested intervals and their common point
• Boundedness as another property stable by finite alterations
• Converging subsequense of  a bounded sequence
• But why we are so sure that there are no gaps on the real line? And what is a real line?

Construction of the number system: from natural numbers toward scary numbers

• Completion by algebraic operations: from $\mathbb N$ to $\mathbb Q$ via $\mathbb Z$. Everything you need to solve linear equations
• Problems  with quadratic equations: irrationalities and negative discriminants. An idea of algebraic number.
• Problems with transition to limit: the ubiquitous $\pi$ and much, much more
• Infinite decimal fractions: completion by “adding limits of monotone sequences”.
• Operations with real numbers: ordered field. Completeness “axiom”.

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