Sergei Yakovenko's blog: on Math and Teaching

Tuesday, November 7, 2017

Lecture 2, November 7, 2017


The basic set theory allows us to construct a set \mathbb N=\{|,||,|||,||||,|||||,\dots\} with a function “next”, denoted by \mathrm{Succ}:\mathbb N\to\mathbb N\smallsetminus\{|\}, which is bijective. This set describes the process of counting objects and is the most basic structure. Starting from a distinguished element denoted by 1, we construct an infinite number of elements 2=\mathrm{Succ}(1),\ 3=\mathrm{Succ}(2),\ 4=\mathrm{Succ}(3) etc. There are two axioms guaranteeing that the set \mathbb N indeed coincides with what we call the set of natural numbers:

  1. \forall x\in\mathbb N\ \mathrm{Succ}(x)\ne 1
  2. Any element x\in\mathbb N is obtained by the iteration of \mathrm{Succ}: x=(\mathrm{Succ}\circ\cdots\circ\mathrm{Succ})(1).

Using this function and its partial inverse  one can define on \mathbb N the order and the operations of addition (as repeated addition of 1 which is just evaluation of \mathrm{Succ}) and multiplication (repeated addition).

However, not all equations of the form x+a=b or x\cdot a=b are solvable. One can enlarge \mathbb N by adding solutions of all such equations, obtaining the set of integer numbers \mathbb Z which is a commutative group with respect to the operation of addition, and finally the set of rational numbers \mathbb Q in which division is available by any nonzero number.

Division by zero is impossible: if we add “solution of the equation 0\cdot x=1” as a new imaginary element, then we will not be able to do some arithmetic operations on it. Still, if we are ready to pay this price, then the rational numbers can be extended by a new element so that, say, the function f(x)=1/x would be everywhere defined and continuous.

Details are available in the lecture notes here.


Wednesday, October 28, 2015

Lecture 1, Tue Oct 27, 2015

‘שלום כיתה א

Welcome to the 2015/6 season of the Rothschild–Caesaria course of Analysis for high school teachers! You are welcome to bookmark this site and check it for all kind of information relevant for the course, from room changes to new handouts, updated lecture notes etc. Below follows the brief synopsis of the first lecture.


We discussed all kinds of paradoxes and possible controversies that may appear if we allow infinite sets, infinite procedures etc. They are listed in Section 1 (pages 1-5) here.


The next subject was devoted to the numbers we use. The natural numbers \mathbb N=\{1,2,3,\dots\} can be axiomatically defined using the Peano axiom system, i.e., using the symbol | (usually written as 1) and the operation “next after x” (denoted in various sources as x^+ or \textrm{Succ}(x)). Applying this operation several times, one gets elements ||,|||,||||,|||||\dots which are usually denoted by 2,3,4,5,\dots. This construction emulates the process of counting, which is how the natural numbers appeared in the human culture. More about this here, pages 1-4.

From the “usual” natural numbers one can construct larger sets of “numbers”. This can be done in more than one way, e.g., the negative integer numbers can be introduced like here (sect. 1.2, pages 4-6).

Yet there is a more general construction which works surprisingly often. The idea is to “add solutions of equations which are not solvable in the usual sense”. For instance, the negative number -n can be introduced as the “solution” of the equation x+n+1=1 which has no solution x\in\mathbb N. Using the equation, we can derive rules of manipulation with such numbers. Once we check that they are not mutually contradicting (this is a boring but necessary step), the “extension” is done. For details see sect. 1.3 of the same Note.

This process, however, does not work always. Sometimes “ideal solutions” cannot be introduced without violating the existing rules. For instance, if we decide to add “solution” of the equation 0\cdot x=1 (kind of “infinity”) which has no solutions over \mathbb Z, then we get a contradiction: such “ideal number” cannot be added with the usual integers from \mathbb Z, see Sect. 1.4.

If we start with \mathbb N and extend it so that all linear equations of the form ax+b=c are solvable (except for the “impossible” case above), the result will be the set of all \mathbb Q of rational numbers. It is a field: addition, subtraction and multiplication is always possible in \mathbb Q, while division is possible by nonzero numbers only.

However, if we want solvability of equations of degree higher than 1, then the rational numbers again become insufficient. The equations x^2-2=0 and x^2+1=0 are not solvable in \mathbb Q, albeit for “different reasons”. Still we can adjoin either of them (or both) to \mathbb Q, see Sect. 2. In principle, we can adjoin (this would require some hard work) solutions to all polynomial equations of the form a_0 x^n+ a_1x^{n-1}+\cdots +a_{n-1} x+a_n=0 with rational coefficients a_0,\dots,a_n\in\mathbb Q. The corresponding set is called the (field of) algebraic numbers \overline{\mathbb Q}.

Still for many reasons it is insufficient. Algebra is not all 😉

Sunday, November 13, 2011

Lecture 3, Nov 10, 2011 (Thu)

Number systems

Leopold Kronecker (1823-1891) famously quipped, “God made the natural numbers; all else is the work of man”. So we start working.

  1. Construct non-positive integers by adjoining “formal solutions” to the equations x+n=m for n\ge m
  2. Embed \mathbb N into \mathbb Z, identifying the above solution with the difference m-n for m>n.
  3. Define arithmetic operations on these “new numbers” via manipulations with the corresponding equations.
  4. Prove that with the “new numbers” the addition operation is always invertible, and x+n=m is always solvable with any n,m\in\mathbb Z.
  5. The construction can be essentially reproduced to define fractions as “formal solutions” to the equations of the form qx=p with p,q\in\mathbb Z. To avoid an obvious non-uniqueness, consider only case where p,q do not vanish simultaneously.
  6. Derive the formulas for addition/subtraction and multiplication/division. Note that these formulas sometimes give the forbidden combination 0\cdot x=0.
  7. Two ways to solve the problem:
    • keep the addition/subtraction always defined, but exclude the root of 0\cdot x=1, or
    • keep the “ideal element” and have a nice picture and lots of simplification in geometry, but live with arithmetic prohibitions.
  8. The ring \mathbb Q of rationals and the “circle” of the “rational projective line” \mathbb Q P^1:

Download the pdf file for the complete exposition.

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