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# Real Numbers and the Real Line This section reviews real numbers, inequalities, intervals and absolute values. ## Real Numbers Much of calculus is based on properties of the real number system. Real numbers are numbers that can be expressed as decimals, such as - -3/4 = -0.75000... - 1/3 =...

# Real Numbers and the Real Line This section reviews real numbers, inequalities, intervals and absolute values. ## Real Numbers Much of calculus is based on properties of the real number system. Real numbers are numbers that can be expressed as decimals, such as - -3/4 = -0.75000... - 1/3 = 0.33333... - √2 = 1.4142... The dots in each case indicate that the sequence of decimal digits goes on forever. Every conceivable decimal expansion represents a real number, although some numbers have two representations. For instance, the infinite decimals.999... and 1.000... represent the same real number 1. A similar statement holds for any number with an infinite tail of 9s. The real numbers can be represented geometrically as points on a number line called the real line. The symbol R denotes either the real number system or, equivalently, the real line. The properties of the real number system fall into three categories: algebraic properties, order properties, and completeness: - 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. You can never divide by 0. ## Order Properties The order properties of real numbers are given in Appendix 4. The following useful rules can be derived from them, where the symbol ⇒ means "implies." ### Rules for Inequalities If a, b and c are real numbers, then: 1. a < b ⇒ a + c < b + c 2. a < b ⇒ a − c < b − c 3. a < b and c > 0 ⇒ ac < bc 4. a < b and c < 0 ⇒ bc < ac > Special case: a < b ⇒ −b < −a 5. a > 0 ⇒ 1/a > 0 6. If a and b are both positive or both negative, then a < b ⇒ 1/a < 1/b Notice the rules for multiplying an inequality by a number. Multiplying by a positive number preserves the inequality; multiplying by a negative number reverses the inequality. Also, reciprocation reverses the inequality for numbers of the same sign. For example, 2 < 5 but −2 > −5 and 1/2 > 1/5. The completeness property of the real number system is deeper and harder to define precisely. However, the property is essential to the idea of a limit (Chapter 2). 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. Many theorems of calculus would fail if the real number system were not complete. 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. ## Special Subsets of Real Numbers We distinguish three special subsets of real numbers. 1. The natural numbers, namely 1, 2, 3, 4,... 2. The integers, namely 0, ±1, ±2, ±3,... 3. 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 ≠ 0*. Examples are 1/3, -9/9 = -9, 4/200 and 57/1. The rational numbers are precisely the real numbers with decimal expansions that are...