2 Analytic geometry in two dimensions
of real numbers or the real-number system. Except where explicit state-
ments to the contrary are made, the word number in this book always
means real number. It is assumed that we are all familiar with the idea
that numbers can be represented or approximated in decimal form. The
equality -- = 0.5 and the approximation
(1.11) it = 3.14159 26535 89793
must not frighten us. Searching questions about the possibility of
"representing" 9r and other numbers by "infinite decimals" can be post-
poned. Our decimal system was devised by Hindus and was carried to
Europe by Arabs in the twelfth century and earlier, but it took a few
centuries to convince Europeans that they should and could teach the
system to all of their children.I-4 -3 -2 -1 (^01234) X
Figure 1.
With each number x we associate a point on a line as in Figure 1.12.
The line is called the real line or the x axis, and the point associated with
0 (zero) is called the origin 0 (oh). If x is positive, say 2, the point
associated with x lies x, say 2, units to the right of the origin. If x is
negative, say -3, the point lies -x, say 3, units to the left of the origin.
This correspondence between numbers and points is one to one; that is,
to each number there corresponds exactly one point and to each point
there corresponds exactly one number. The number is called the coordi-
nate of the point. While points and numbers are entities of different
kinds, we sometimes find convenience in abbreviating our language by
using "the point x" to mean "the point having coordinate x." The part
of the x axis upon which positive numbers are plotted, or located, is called
the positive x axis.
The statement a = b is read "a equals b" or "a is equal to b." Simi-
larly, the statement "a 0 b" is read "a is not equal to b" or "a is different
from b." Thus the statements 2 = 2 and 2 ; 3 are true. The state-
ments 2 P6 2 and 2 = 3 are false.
When two numbers a and b are so related that the point corresponding
to a lies to the left of the point corresponding to b
d b x as in Figure 1.13, we say that a is less than b and
write a < b. In this case we say also that b is
Figure 1.greater than a and write b > a. For example,
2 < 6, -3 < 1, and -4 < -2. This terminology
agrees with common usage when temperatures are being compared; we
say that a temperature -3° is less, or lower, than a temperature P. Theinequality -2 < 0 means that -2 is less than 0 and that -2 isnega-
tive. The inequality 4 > 0 means that 4 is greater than 0 and that 4 is
positive. The statement that the weight w, measured in pounds, of a