223-Appendix1

223-Appendix1 - hof51918_app_641_670 10/17/05 3:28 PM Page...

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SECTION A1 A Brief Review of Algebra There are many techniques from elementary algebra that are needed in calculus. This appendix contains a review of such topics, and we begin by examining numbering systems. An integer is a “whole number,” either positive or negative. For example, 1, 2, 875, ± 15, ± 83, and 0 are integers, while , 8.71, and are not. A rational number is a number that can be expressed as the quotient of two integers, where b ± 0. For example, and are rational numbers, as are Every integer is a rational number since it can be expressed as itself divided by 1. When expressed in decimal form, rational numbers are either terminating or in±nitely repeating decimals. For example, A number that cannot be expressed as the quotient of two integers is called an irrational number. For example, are irrational numbers. The rational numbers and irrational numbers form the real numbers and can be visualized geometrically as points on a number line as illustrated in Figure A.1. FIGURE A.1 The number line. If a and b are real numbers and a is to the right of b on the number line, we say that a is greater than b and write a . b. If a is to the left of b , we say that a is less than b and write a , b (Figure A.2). For example, 5 ² 2 ± 12 ³ 0 and ± 8.2 ³± 2.4 FIGURE A.2 Inequalities. a b a > b b a a < b Inequalities –5 –4 –3 –2 –1 0 5 4 3 2 1 π –2.5 3 2 2 ± 2 ² 1.41421356 and ´ ² 3.14159265 5 8 µ 0.625 1 3 µ 0.33 . . . and 13 11 µ 1.181818 . . . ± 6 1 2 µ ± 13 2 and 0.25 µ 25 100 µ 1 4 ± 4 7 8 5 , 2 3 , a b ± 2 2 3 The Real Numbers 642 APPENDIX A Algebra Review A-2 hof51918_app_641_670 10/17/05 3:28 PM Page 642
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A-3 SECTION A1 A Brief Review of Algebra 643 Moreover, as you can see by noting that The symbol ± stands for greater than or equal to, and the symbol ² stands for less than or equal to. Thus, for example, ³ 3 ±³ 4 ³ 3 3 ³ 4 ²³ 3 and ³ 4 4 A set of real numbers that can be represented on the number line by a line segment is called an interval. Inequalities can be used to describe intervals. For example, the interval a ² x ´ b consists of all real numbers x that are between a and b , including a but excluding b . This interval is shown in Figure A.3. The numbers a and b are known as the endpoints of the interval. The square bracket at a indicates that a is included in the interval, while the rounded bracket at b indicates that b is excluded. Intervals may be ±nite or in±nite in extent and may or may not contain either endpoint. The possibilities (including customary notation and terminology) are illus- trated in Figure A.4. Intervals 6 7 µ 48 56 and 7 8 µ 49 56 6 7 ´ 7 8 a b x FIGURE A.3 The interval a ² x ´ b . a b b x x x x Closed interval: a x b Open interval: a < x < b Half-open interval a x < b Infinite interval x a Infinite interval x > a Infinite interval x b Infinite interval x < b Half-open interval a < x b a a b ab b x x x x a a b EXAMPLE A1.1 Use inequalities to describe these intervals.
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This note was uploaded on 01/21/2012 for the course MA 22300 taught by Professor Staff during the Spring '08 term at Purdue.

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223-Appendix1 - hof51918_app_641_670 10/17/05 3:28 PM Page...

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