1.1 Real numbers 3horse is greater than 1990 and less than 2010 becomes
1990 < w < 2010.This can be read "1990 is less than w is less than 2010." In this book,
Figure 1.12 precedes Figure 1.13 because 1.12 < 1.13, and both precede
Section 1.2 because 1.13 < 1.2; the decimal system governs the number-
ing of all items except those appearing in the lists of problems at the ends
of the sections. The basic idea that the number 1.131 or 1.135 can be
assigned to an item which appears between items numbered 1.13 and 1.
is often used to bring order out of chaos and has hordes of valuable
applications. While facts of life are being considered and he still has his
full complement of readers, the author can extend his best wishes to the
canonical 20 per cent who will not complete this course. Those who
abandon their studies to work in the design department of a sport shirt
factory will be rewarded for commencement of their studies if they have
learned that items in their stocks can be identified by numbers in one
sequence and that numbers such as 416.35 and 416.351 can be assigned to
items that should be listed between 416.3 and 416.4. Numbers assigned
to items in books and factories are akin to numbers assigned to buildings
beside streets and to doors inside skyscrapers. In the best of circum-
stances, these numbers are assigned in an informative way, and they are
noticed and used when occasions arise. Persons who study Section 5.
will be well aware that they are just getting started when they reach 5.41,
that they are about halfway through the text of the section when they
reach 5.45, and that they have reached the problems at the end of the
section when they reach 5.49.
The inequality a < b is read "a is less than or equal to b." The
inequalities 4 < 5 and 5 5; 5 are both true, but the inequality 6 S 5 is
false.
The absolute value of a number x is denoted by Ixl. It is equal to x
itself if x > 0; it is equal to 0 if x = 0; and it is equal to -x if x < 0.
Thus 171 = 7, 101 = 0, and 1-41 = 4. For each x we have Ixl>-- 0.
Moreover, Ixl = 0 if and only if x = 0. For each x we have either
Ixl = x or Ixl = -x, and since x2 and (-x)2 are equal it follows that
Ixl2 = x2.
With the aid of Figure 1.12, we acquire the idea that the distance
between the point with coordinate 1 and the point with coordinate 4 is 3.
The distance between the points with coordinates - 3 and 2 is seen to be
5, that is, 2 - (-3). By considering the different typical cases, we
reach the conclusion that the distancet between the two points with coordi-
t It is far from easy to formulate and use enough axioms involving the geometry of Euclid
and the set of real numbers to prove that the number lb - al is the distance between the
points having coordinates a and b. To place our ideas upon a rigorous base, we can do
what is usually done in more advanced mathematics: construct the foundations of ordinary
geometry and analysis in such a way that the number lb - al is defined to be the distance
between the two points having coordinates a and b.