The Realnumber System

Equations and variables are the essential ingredients of a mathematical model. But since the values that an economic variable takes are usually numerical, a few words should be said about the number system. Here, we shall deal only with so-called "real numbers."

Whole numbers such as 1,2, 3. ... are called positive integers', these are the numbers most frequently used in counting. Their negative counterparts — 1, — 2, — 3. ... are called negative integers; these can be employed, for example, to indicate subzero temperatures (in degrees). The number 0 (zero), on the other hand, is neither positive nor negative, and is in that sense unique. Let us lump all the positive and negative integers and the number zero into a single category, referring to them collectively as the set of all integers.

Integers, of course, do not exhaust all the possible numbers, for we have fractions, such as ij. 4. and which — if placed on a ruler—would fall between the integers. Also, we have negative fractions, such as — I and — ?. Together, these make up the set of all fractions.

The common property of all fractional numbers is that each is expressible as a ratio of two integers; thus fractions qualify for the designation rational numbers (in this usage, rational means ratio-nal). But integers are also rational, because any integer n can be considered as the ratio n/\. The set of all integers and the set of all fractions together form the set of all rational numbers.

Once the notion of rational numbers is used, however, there naturally arises the concept of irrational numbers—numbers that cannot be expressed as ratios of a pair of integers. One example is the number /2 = 1.4142 which is a nonrepeating, nonterminating decimal. Another is the special constant 1t = 3.1415... (representing the ratio of the circumference of any circle to its diameter), which is again a nonrepeating, nonterminating decimal, as is characteristic of all irrational numbers.

Each irrational number, if placed on a ruler, would fall between two rational numbers, so that, just as the fractions fill in the gaps between the integers on a ruler, the irrational numbers fill in the gaps between rational numbers. The result of this iilling-in process is a continuum of numbers, all of which are so-called "real numbers." This continuum constitutes the set of all real numbers, which is often denoted by the symbol R. When the set R is displayed on a straight line (an extended ruler), we refer to the line as the real line.

In Fig. 2.1 are listed (in the order discussed) all the number sets, arranged in relationship to one another. If we read from bottom to top, however, we find in effect a classificatory scheme in which the set of real numbers is broken down into its component and subcomponent number sets. This figure therefore is a summary of the structure of the real-number system.

Fractions y y

Rational numbers

Irrational numbers y v

Real numbers

Figure 2.1

Real numbers are all we need for the first 14 chapters of this book, but they are not the only numbers used in mathematics. In fact, the reason for the term "real" is that there are also "imaginary" numbers, which have to do with the square roots of negative numbers. That concept will be discussed later, in Chap.

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