Definition of the numbers 1, i, and the square root of 2

In order to make precise sense out of the concepts we study in mathematical analysis, we must first come to terms with what the "real numbers" are. Everything in mathematical analysis is based on these numbers, and their very definition and existence is quite deep. We will, in fact, not attempt to demonstrate (prove) the existence of the real numbers in the body of this text, but will content ourselves with a careful delineation of their properties, referring the interested reader to an appendix for the existence and uniqueness proofs.

Although people may always have had an intuitive idea of what these real numbers were, it was not until the nineteenth century that mathematically precise definitions were given. The history of how mathematicians came to realize the necessity for such precision in their definitions is fascinating from a philosophical point of view as much as from a mathematical one. However, we will not pursue the philosophical aspects of the subject in this book, but will be content to concentrate our attention just on the mathematical facts. These precise definitions are quite complicated, but the powerful possibilities within mathematical analysis rely heavily on this precision, so we must pursue them. Toward our primary goals, we will in this chapter give definitions of the symbols (numbers) $-1,i,$ and $\sqrt{2}.$

The main points of this chapter are the following:

1. The notions of least upper bound ( supremum ) and greatest lower bound ( infimum ) of a set of numbers,
2. The definition of the real numbers $R,$
3. the formula for the sum of a geometric progression ( [link] ),
4. the Binomial Theorem ( [link] ), and
5. the triangle inequality for complex numbers ( [link] ).