Introduction

Now that we have forged our basic tools, we shall demonstrate more of the magic of mathematics. As an artist evokes, out of a few lines on a canvas, scenes of great beauty, and as a musician conjures up exquisite melodies from combinations of a few notes, so the mathematician with a few penetrating logical steps portrays results of simple elegance. Often, like the product of the magician's wand, these results seem quite mysterious, despite their simplicity, until their origin is perceived.

In this chapter, we shall employ the basic results derived in the previous chapters to obtain some of the most famous inequalities in the field of mathematical analysis. These inequalities are the everyday working tools of the specialist in this branch of mathematics.

In Chapter 5, we shall then show how these new relationships may be used to solve a number of interesting problems that, at first sight, seem far removed from algebra and inequalities. The applications are continued in Chapter 6, where we discuss and extend the notion of distance.

This, indeed, is one of the fascinations of mathematics—that simple ideas applied one after the other, in the proper order, yield results that never could have been envisaged at the outset.