124CHAPTER9 Irrational and Real Numbers
We started out with the natural numbers (or naturals), and took their negatives to get the integers.
Then we divided integers by each other to come up with rational numbers (or rationals). In this
chapter, we’ll study the irrational numbers (or irrationals) and real numbers (or reals), and com-
pile a full set of rules for working with real variables.The Number Hierarchy
Mathematicians have known for centuries that there are plenty of irrational numbers that can’t be
expressed as ratios of integers. Let’s see how they behave compared with the rational numbers.Rational-number “density”
Suppose we assign rational numbers to points on a horizontal line so the distance of any point
from the origin is directly proportional to its absolute value. If a point is on the left-hand side
of the point representing 0, then that point corresponds to a negative number; if it’s on the
right-hand side, it corresponds to a positive number.
If we take any two points a and b on the line that correspond to rational numbers, then the
point midway between them corresponds to the rational number (a+b)/2. (Do you recognize this
as the formula for the average, or arithmetic mean, of two numbers?) We can keep cutting an inter-
val in half, and if the end points are both rational numbers, then the midpoint is another rational
number. Figure 9-1 shows an example of this, starting with the interval between 1 and 2.
It is tempting to suppose that the points on a rational-number line are “infinitely dense.”
The point midway between any two rational-number points always corresponds to another
rational number. But do the rational numbers account for all of the points along a true geo-
metric line? The answer is “No. They don’t even come close!”Irrational numbers
As we have seen, an irrational number can’t be expressed as the ratio of two integers. Examples
of irrational numbers include:Copyright © 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use.