‘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 can be axiomatically defined using the Peano axiom system, i.e., using the symbol | (usually written as 1) and the operation “next after ” (denoted in various sources as or ). Applying this operation several times, one gets elements which are usually denoted by . 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 can be introduced as the “solution” of the equation which has no solution . 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 (kind of “infinity”) which has no solutions over , then we get a contradiction: such “ideal number” cannot be added with the usual integers from , see Sect. 1.4.
If we start with and extend it so that all linear equations of the form are solvable (except for the “impossible” case above), the result will be the set of all of rational numbers. It is a field: addition, subtraction and multiplication is always possible in , 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 and are not solvable in , albeit for “different reasons”. Still we can adjoin either of them (or both) to , see Sect. 2. In principle, we can adjoin (this would require some hard work) solutions to all polynomial equations of the form with rational coefficients . The corresponding set is called the (field of) algebraic numbers .
Still for many reasons it is insufficient. Algebra is not all 😉