The first and most elementary construction of real numbers -- by Karl Weierstraß (Q6169838)

This survey is devoted to an unknown concept of real numbers. This concept was formulated by Karl Weierstraß. One can note the following: ``In the Mathematics Library of Goethe University Frankfurt, a hitherto unknown manuscript of 171 pages was discovered by the librarian Boram Schröter in summer 2016. It contains the beginning of Weierstraß' lecture from winter 1880/81. These notes (from Emil Strauss (1859--1892)) give a painstaking record of Weierstraß' discourse, enabling the historian of mathematics, for the first time, to disclose Weierstraß' true concept of real numbers. Weierstraß created an all-embracing concept of number. He started with the natural numbers \(1, 2, 3, \dots\), and \(0\), enlarged them to the fractions and the rational numbers and then added the irrational numbers. Lastly, he completed his monoid (only zero needs to be added) of fractions and irrational numbers to a group, which of course is a ring with 1; and moreover he gave procedures of division for fractions as well as for the irrational numbers and the general real numbers (the last two, however, are false). In the end he obtained a huge world of numbers, wherein each number (except the fractional numbers and the general real numbers) is unique.'' The paper contains the main and additional descriptions. The main consideration consists of explanations of Weierstraß' ideas and are related to such items: \begin{itemize} \item[--] ``Natural numbers and quantities -- the philosophical foundations''. \item[--] ``Fractions -- the start of arithmetics''. \item[--] ``Irrational numbers -- the start of analysis''. Here such Weierstraß' notions as finite and infinite irrational numbers are noted. \item[--] ``General real numbers -- completing the basis of calculus''. \end{itemize} An additional part of the consideration contains a historical overview, certain views of some mathematicians, and the section ``Retrospect: where Weierstraß succeeded and where he did not succeed \dots.''.