1.1 Real numbers 5x = y; but the equation 0.2 = 0.3 does not imply that 2 = 3. We must
always be suspicious of results obtained by division unless we know that
the divisor is not 0.
In order to pass literacy tests and to converse with our fellow men, it is
necessary to know that the numbers
(1.17) , -4, -3, -2, -1, 0, 1, 2, 3, 4,
are called integers. The numbers 1, 2, 3, - are the positive integers.
If m and n are integers and n 0 0, the solution of the equation nx = m is
written in the form m/n and is called a rational (ratio-nal) number. Each
integer m is a rational number because it is the solution of the equation
1x = m and is m/1. There is no greatest integer, because to each integer
n there corresponds the greater integer n + 1. Likewise, there is no
least integer, but 1 is the least positive integer. If e (epsilon) and a are
positive numbers, then there is a positive integer n for which ne > a; this
is the Archimedes property of numbers. Another basic fact which is
easier to comprehend than to prove is that if x is a number, then there is
an integer n for which n <- x < n + 1.
As we near the end of this introductory section, we call attention to
some additional terminology which is more important than beautiful
and to which we shall slowly become accustomed as we proceed. When
a < b, the set of points having coordinates x for which a <- x <- b is called
the closed interval of points (or numbers) from a to b. The points a and b
are end points of the closed interval, and they belong to (or are points of)
the closed interval. The set of points (or numbers) for which a < x < b
is called the open interval from a to b. The points a and b are still called
end points of the interval, but they do not belong to the open interval.
In each case, the number b - a is called the length of the interval. Thus
the length of an interval is the distance between its end points. When
a < b, the relationsb-a+b_b-a>0 a+b-a_b-a>
2 2 2 2imply that a < (a + b)/2 < b and that the point having the coordinate
(a + b)/2 lies between and is equidistant from the points having coordi-
nates a and b. This point is called the mid-point of the interval (open or
closed) having its end points at a and b. If b < a, the above inequalities
are reversed, but (a + b)/2 is still midway between a and b.
The following problems promote understanding of statements made by
use of inequality and absolute-value signs. For example, the inequality
148 < x < 152 says that the number x (which might be the weight of a
man) is greater than 148 and less than 152. This means that x differs
from 150 by a number with absolute value less than 2 and hence that