Purdue MA 22300 - A Brief Review of Algebra

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SECTION A1 A Brief Review of AlgebraThere are many techniques from elementary algebra that are needed in calculus. Thisappendix contains a review of such topics, and we begin by examining numberingsystems.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 twointegers, where b  0. For example, and are rational numbers, as areEvery 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 infinitelyrepeating decimals. For example,A number that cannot be expressed as the quotient of two integers is called anirrational number. For example,are irrational numbers.The rational numbers and irrational numbers form the real numbers and can bevisualized geometrically as points on a number line as illustrated in Figure A.1.FIGURE A.1The number line.If a and b are real numbers and a is to the right of b on the number line, we say thata is greater than b and write a . b. If a is to the left of b, we say that a is lessthan b and write a , b (Figure A.2). For example,5  2 12 0 and 8.2 2.4FIGURE A.2 Inequalities.aba > bbaa < bInequalities–5–4 –3 –2 –1054321π–2.5–√3222  1.41421356 and   3.1415926558 0.625 13 0.33 . . . and 1311 1.181818 . . .612132 and 0.25 25100144785,23,ab223The Real Numbers642 ❘ APPENDIX A ❘ Algebra Review ❘ A-2hof51918_app_641_670 10/17/05 3:28 PM Page 642A-3 ❘ SECTION A1 ❘ A Brief Review of Algebra ❘ 643Moreover,as you can see by noting thatThe symbol  stands for greater than or equal to, and the symbol  stands forless than or equal to. Thus, for example,3 4 3 3 4 3 and 4 4A set of real numbers that can be represented on the number line by a line segmentis called an interval. Inequalities can be used to describe intervals. For example, theinterval a  x b consists of all real numbers x that are between a and b, includinga but excluding b. This interval is shown in Figure A.3. The numbers a and b areknown as the endpoints of the interval. The square bracket at a indicates that a isincluded in the interval, while the rounded bracket at b indicates that b is excluded.Intervals may be finite or infinite in extent and may or may not contain eitherendpoint. The possibilities (including customary notation and terminology) are illus-trated in Figure A.4.Intervals674856 and 78495667 78abxFIGURE A.3 The intervala  x b.abbxxxxClosed interval: a ≤ x ≤ bOpen interval: a < x < bHalf-open intervala ≤ x < bInfinite intervalx ≥ aInfinite intervalx > aInfinite intervalx ≤ bInfinite intervalx < bHalf-open intervala < x ≤ baababbxxxxaabEXAMPLE A1.1Use inequalities to describe these intervals.FIGURE A.4 Intervals of real numbers.hof51918_app_641_670 10/17/05 3:28 PM Page 643644 ❘ APPENDIX A ❘ Algebra Review ❘ A-4xxx33–2–2(a) (b)(c)Solutiona.x  3 b. x 2 c. 2 x  3EXAMPLE A1.2Represent each of these intervals as a line segment on a number line.a. x 1 b. 1  x  2 c. x  2Solutiona.b.c.The absolute value of a real number x, denoted by x , is the distance from x to 0 ona number line. Since distance is always nonnegative, it follows that x  0. Forexample, 4  4 4  4 0  0 5  9  4  3  3 Here is a general formula for absolute value.33Absolute Value2x12x1xAbsolute Value■For any real number x, the absolute value of x isxxx if x  0if x 0 More generally, the distance between any two numbers a and b is the absolute valueof their difference taken in either order, as illustrated in Figure A.5.EXAMPLE A1.3Find the distance on the number line between 2 and 3.SolutionThe distance between two numbers is the absolute value of their difference. Hence,Distance  2  3  5  5The situation is illustrated in Figure A.6.| a – b |orabx| b – a |FIGURE A.5 The distancebetween a and b  a  b .hof51918_app_641_670 10/17/05 3:28 PM Page 644FIGURE A.6 Distance between 2 and 3.The geometric interpretation of absolute value as distance can be used to simplify cer-tain algebraic inequalities involving absolute values. Here is an example.EXAMPLE A1.4Find the interval consisting of all real numbers x such that x  1  3.SolutionIn geometric terms, the numbers x for which x  1  3 are those whose distancefrom 1 is less than or equal to 3. As illustrated in Figure A.7, these are the numbersthat satisfy 2  x  4.FIGURE A.7 The interval on which x  1  3 is 2  x  4.To find this interval algebraically, without relying on the geometry, rewrite theinequality x  1  3 as3  x  1  3and add 1 to each term to get3  1  x  1  1  3  1or2  x  4These rules define the expression axfor a  0 and all rational values of x.Exponential Notation–5–6–4 –3 –2 –1056432133xAbsolute Valuesand Intervals–5–4 –3 –2 –1054321A-5 ❘ SECTION A1 ❘ A Brief Review of Algebra ❘ 645Definition of axfor x $ 0■Integer powers: If n is a positive integer, thenan a  a  awhere the product a  a  a contains n factors.Fractional powers: If n and m are positive integers, thenwhere denotes the positive mth root.Negative powers: axZero power: a0 11axm anm (ma)nmanhof51918_app_641_670 10/17/05 3:28 PM Page 645EXAMPLE A1.5Evaluate these expressions (without using your calculator).a. 912b. 2723c. 813d. e. 50Solutiona.9123b. 272332 9 c. 813d.  10032103 1,000e. 50 1Exponents obey these useful laws.Laws of Exponents(100)31100321813138123729  93(27)2(327)29110032646 ❘ APPENDIX A ❘ Algebra Review ❘ A-6Laws of Exponents■For a real number a, we haveThe product law: aras arsThe quotient law:  arsif a  0The power law: (ar)s arsarasThe laws of exponents are illustrated in Examples A1.6 and A1.7.EXAMPLE A1.6Evaluate these expressions (without using a calculator).a. (22)3b. c. 274(814)Solutiona.(22)3 26b. 32 9c. 274(814)  274(23)14


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Purdue MA 22300 - A Brief Review of Algebra

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