Sergei Yakovenko's blog: on Math and Teaching

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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2 Comments »

  1. As we discovered with the students during the last tutorial, in Problem 5, the following situation is possible:
    L=\{q\in\mathbb{Q}: q<0\}, R=\{0\}. The pair (L,R) is a cut, L\cap R=\emptyset. So the correct formulation seems to be: "at least one of the sets L,R is infinite.

    Comment by dmitrybat — Monday, December 5, 2011 @ 3:28 | Reply

  2. Problem 23 should read:
    “If \alpha_1 \triangleright \alpha_2 and \beta_1 \triangleright \beta_2, then \alpha_1+\beta_1 \triangleright \alpha_2+\beta_2.”

    Comment by dmitrybat — Sunday, December 11, 2011 @ 12:06 | Reply


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