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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