(b) eventually repeating (ending with a block of digits that repeats over and over), for example 23 = 2.090909 Á = 2.09 11 The rational numbers are precisely the real numbers with decimal expansions that are either (a) terminating (ending in an infinite string of zeros), for example, 3 = 0.75000 Á = 0.75 4 The natural numbers, namely 1, 2, 3, 4, Á The integers, namely 0, 1, 2, 3, Á The rational numbers, namely the numbers that can be expressed in the form of a fraction m>n, where m and n are integers and n Z 0. We distinguish three special subsets of real numbers.
The topic is best saved for a more advanced course, but Appendix 4 hints about what is involved and how the real numbers are constructed. Many theorems of calculus would fail if the real number system were not complete. Roughly speaking, it says that there are enough real numbers to “complete” the real number line, in the sense that there are no “holes” or “gaps” in it. However, the property is essential to the idea of a limit (Chapter 2). The completeness property of the real number system is deeper and harder to define precisely. Also, reciprocation reverses the inequality for numbers of the same sign. Multiplying by a positive number preserves the inequality multiplying by a negative number reverses the inequality. Notice the rules for multiplying an inequality by a number. If a and b are both positive or both negative, then a 6 b Q Rules for Inequalities If a, b, and c are real numbers, then:Ī 6 b Q a + c 6 b + c a 6 b Q a - c 6 b - c a 6 b and c 7 0 Q ac 6 bc a 6 b and c 6 0 Q bc 6 ac Special case: a 6 b Q -b 6 -a 1 a 7 0 Q a 7 0 The following useful rules can be derived from them, where the symbol Q means “implies.” The order properties of real numbers are given in Appendix 4. You can never divide by 0.ġ Copyright © 2005 Pearson Education, Inc., publishing as Pearson Addison-Wesley The algebraic properties say that the real numbers can be added, subtracted, multiplied, and divided (except by 0) to produce more real numbers under the usual rules of arithmetic. The properties of the real number system fall into three categories: algebraic properties, order properties, and completeness. The symbol denotes either the real number system or, equivalently, the real line. The real numbers can be represented geometrically as points on a number line called the real line.
A similar statement holds for any number with an infinite tail of 9’s. 999 Á and 1.000 Á represent the same real number 1. Every conceivable decimal expansion represents a real number, although some numbers have two representations.
Real numbers are numbers that can be expressed as decimals, such as -Ģ2 = 1.4142 Á The dots Á in each case indicate that the sequence of decimal digits goes on forever. Real Numbers Much of calculus is based on properties of the real number system. Real Numbers and the Real Line This section reviews real numbers, inequalities, intervals, and absolute values. We also discuss the use of graphing calculators and computer graphing software. The topics include the real number system, Cartesian coordinates in the plane, straight lines, parabolas, circles, functions, and trigonometry. PRELIMINARIES OVERVIEW This chapter reviews the basic ideas you need to start calculus.
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