For example, to multiplybywe multiply 4 by 5 andbyas follows:To divide these numbers, we proceed likewise:410105101245101010120.81010120.810281032010101220102221023(41010)(51012)(45)(10101012)1012,101051012,41010(a1>n)mam>n(a1>n)nan>naa1>n,a1>21a1a:a1>2a1>n(23)32333(a#b)nan#bn(32)33233632333113323335(an)manmanamanman#amamna01A-2APPENDIX 2Mathematics Review

APPENDIX 2Mathematics ReviewA-3When performing additions or subtractions of numbers in scientific notation, we mustbe careful to begin by expressing the numbers with the same power of 10. For exam-ple, the sum ofandisA 2.4AlgebraAn equation is a mathematical statement that tells us that one quantity or a combinationof quantities is equal to another quantity or combination. We often have to solve forone of the quantities in the equation in terms of the other quantities. For instance, wemay have to solve the equationforxin terms ofaandb. Hereaandbare numerical constants or mathematical expres-sions which are regarded as known, andxis regarded as unknown.The rules of algebra instruct us how to manipulate equations and accomplish theirsolution. The three most important rules are:1. Any equation remains valid if equal terms are added or subtracted from its leftside and its right side.This rule is useful for solving the equationWe simply subtractafrom bothsides of this equation and findthat is,To see how this works in a concrete numerical example, consider the equationSubtracting 7 from both sides, we obtainorNote that given an equation of the formwe may want to solve forain terms ofxandb, ifxis already known from some other information butais a math-ematical quantity that is not yet known. If so, we must subtractxfrom both sides ofthe equation, and we obtainMost equations in physics contain several mathematical quantities which sometimesplay the role of known quantities, sometimes the role of unknown quantities, depend-ing on circumstances. Correspondingly, we will sometimes want to solve the equationfor one quantity (such asx), sometimes for another (such asa).2. Any equation remains valid if the left and the right sides are multiplied or dividedby the same factor.This rule is useful for solvingaxbabxxab,x2x57x75xbaxaabaxab.xab1.510931081.51090.31091.810931081.5109

A-4APPENDIX 2Mathematics ReviewWe simply divide both sides bya, which yieldsorOften it will be necessary to combine both of the above rules. For instance, to solvethe equationwe begin by subtracting 10 from both sides, obtainingorand then we divide both sides by 2, with the resultor3.Any equation remains valid if both sides are raised to the same power.This rule permits us to solve the equationRaising both sides to the powerwe findorAs a final example, let us consider the equation(as established in Chapter 2, this equation describes the vertical position of a particlethat starts at a heightand falls for a timet; but the meaning of the equation need notconcern us here). Suppose that we want to solve fortin terms of the other quantities inthe equation. This will require the use of all our rules of algebra. First, subtractxfromboth sides and then addto both sides. This leads toand then toNext, multiply both sides by 2 and divide both sides byg; this yieldst22g(x0x)12gt2x0x012gt2x0x12gt2x0x12gt2x0xb1>3(x3)1>3b1>313,x3bx3x622x62x16102x1016xbaaxaba

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