chapter  2
26 Pages

The Real Numbers

All modern mathematicians appreciate the vast importance, to mathematics and to a huge range of mathematically based crafts and sciences, of the differential and integral calculus. And anyone who has studied a course in real analysis should have some sense of not only a rigorous foundation for the calculus, but also why such a foundation is necessary. The ramifications of the calculus for science and mathematics were so extensive and exciting that the need to remedy these and other deficiencies only began to become pressing towards end of 18th century. Although Richard Dedekind dreamt up his construction in 1858, he only published it in 1872, when he received a paper of Georg Cantor's with an alternative construction. Cantor's construction exploits Cauchy sequences of rationals. A person may have encountered the idea of a Cauchy sequence in real analysis. It is a sequence for which the person might not have been told, a limit, but whose members become arbitrarily close to each other.