Chapter 1 The Construction of Real and Complex Numbers Thus the System of Real Numbers—the definition of irrationals and the extension of the four species to the new numbers—is established. The method has the advantage of simplicity in detail. It is well for the student, after a first study of the method of Dedekind, to work it through in detail. He will then return to the former method with increased power and greater zest. The method of regular sequences is a middle-of- the-road method. It is an easy way to reach the mountain top. The traveler buys his ticket and takes the funicular. Many people prefer this mode of travel. But some like a stiff climb over rocks and across streams, and such an ascent has its advantages if the heart is good and the muscles are strong. – William Fogg Osgood Functions of Real Variables Challenge Problems: I Problem 1.1. How many congruent regular tetrahedra with edge length 1 can be packed inside a sphere of radius 1 if each tetrahedron has a vertex at the center of the sphere? A lattice point in n-dimensional Euclidean space is a point all of whose coordinates are integers. A lattice polygon in R2 is a polygon all of whose vertices are lattice points. Similarly, a lattice polyhedron in Rn is a polyhe- dron all of whose vertices are lattice points. Problem 1.2. Find all possible areas of lattice squares in R2. (Hint: The first answer you get is probably not the final answer we are looking for.) Problem 1.3. Find all possible volumes of lattice cubes in R3. Problem 1.4. Find all possible volumes of lattice hypercubes in Rn for n 3. 1

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