Basic Quantitative Thinking Skills


 Jocelyn Gaines
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1 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 Bsic Qutittive Thikig Skills 1.1 Qutittive Thikig i Evirometl Sciece Like it or ot, qutittive thikig forms the bsis of most techicl discussio of evirometl issues. Regultors express emissios stdrds i qutittive terms d justify them o the bsis of elborte models of risks to hum helth. Fisheries scietists rely o mthemticl models to determie popultio levels (d sustible hrvest levels) for fish popultios. Egieers use equtios tht describe spects of wter flow i urb eviromets to desig stormwter mgemet structures. Coservtio biologists use popultio dymic models to guide mgemet strtegies for edgered species. Sciece d ssocited qutittive methods form domit mode of discourse, both i evirometl sciece d i my other fields of moder life. Like it or ot, fmilirity with the covetios d pricipls of qutittive thikig is essetil for prticiptig i lrge prt of tody s discussios of evirometl issues. If you wt to be full prticipt i the evirometl policy debtes i the 1 st cetury, you will eed, t miimum, to be ble to uderstd d criticize the qutittive rgumets mrshled i defese of oe or other policy proposls. Most fields of moder sciece rely o qutittive resoig i oe form or other, d evirometl sciece is o exceptio. But evirometl sciece is iterdiscipliry field, i which scietists tried i vriety of disciplies tke prt. The covetios for use of qutittive thikig d, perhps more importtly, the covetios for how qutittive results re commuicted, vry from disciplie to disciplie. Ecologists, for exmple, re frequetly well tried i multivrite sttisticl methods, while hydrologists my hve little sttisticl triig, but hve thorough groudig i mthemticl models of wter movemet. Evirometl chemists, egieers, tmospheric physicists d so o ech brig their ow prticulr pproch to qutittive thikig to their evirometl work. Wht the, forms the core of qutittive thikig skills for evirometl scietists? While we suspect tht o two evirometl scietists would completely gree o this questio, we thik fudmetl foudtio i qutittive resoig icludes the followig: 1. Fmilirity with commo covetios of qutittive presettio i the scieces, such s use of the metric system, uderstdig of scietific ottio d comfort with the cocept of sigifict figures.. Fcility with bsic skills of umericl mipultio s used i the scieces icludig fcility with uit coversios d dimesiol lysis, d fmilirity d eve degree of comfort with use of expoets d logs. 3. Uderstdig of the use of models to elucidte the logicl implictios of theory, d to express those cosequeces i wy tht permit testig of theoreticl ides. 4. Uderstdig of sttistics to the extet of pprecitig the role of ucertity i the scieces, grsps the reltioship mog differet sources
2 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 of ucertity (such s evirometl heterogeeity d mesuremet error), d c thik itelligetly bout how to desig study to collect relible evirometl dt. My dditiol qutittive skills re used i evirometl sciece, but beyod reltively smll core, the prticulr skills vry from disciplie to disciplie. Ecologists might go o to get extesive triig i sttisticl methods, while egieers would be more likely to study sttics d stregth of mterils. Chemists would work with thermodymics d kietics of chemicl rectios, while hydrologists would study models of fluid flow through porous medi. This chpter is iteded to help icrese your qutittive litercy, specificlly i the cotext of evirometl scieces. It is divided ito five sectios: 1. This itroductio,. A review of fudmetl skills of qutittive thikig i the scieces 3. A discussio of the role of mthemticl models i sciece 4. A discussio of prticulr clss of mthemticl models clled stock d flow models tht re widely used i sciece, d 5. A itroductio to sttisticl pricipls I keepig with the overll gols of this textbook, the mteril we preset here should set you up to improve your bility to thik criticlly bout evirometl issues Tools d Tricks: Fudmetl Skills for Qutittive Thikig i the Scieces The skills we review i sectio 1. of this chpter form the bsic buildig blocks of scietific computtio. Becuse of the widely disprte bckgrouds of studets tkig evirometl sciece clsses, it is likely tht for some of you these skills will lredy be secod ture. For others they will be vguely remembered detils from sciece course tke bck i high school. But for few of you, they my etirely ew. Eve if you re fmilir with the mi cocepts covered i this sectio, review of the mteril is worthwhile, if oly becuse it c provide you with perspective o the wy tht the qutittive commuictio i the scieces hve evolved to be richly itercoected web of ides. For exmple, oe cot relly uderstd sigifict digits without first uderstdig scietific ottio, which i tur requires tht you uderstd expoets d logrithms. This presets both chllege to those of you who re ew to qutittive commuictio you will fid tht the pieces of the system oly mke cler sese i the cotext of uderstdig the whole d lso opportuity. If you persevere, d work to uderstd the tips d tricks we discuss i this sectio, you will fid tht the pricipls of this type of resoig will become esier to remember s your uderstdig becomes more complete. Exmples of the pplictio of this thikig.1 Expoets d Logrithms A cler uderstdig of logrithms d expoets (tilogrithms) is ecessry to chieve y fluecy with qutittive resoig, d rel comfort with the cocepts tht uderlie them is of gret help i developig your bck of the evelope thikig skills.
3 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 There re reltively few rules for workig with logs d tilogs, d ll of them c be deduced directly or idirectly from the bsic defiitio of expoetitio..1.1 Expoets We will derive bsic reltioships for expoetitio first, the explore relted properties of logrithms. We iclude this mteril here o the pricipl tht if you c derive importt properties of expoets from first pricipls, you will ot hve to simply memorize them. For y umber, rised to the th power is simply equl to multiplied times itself times. Eq 1.1: Oe c redily deduce the most importt properties of expoets from this defiitio. I the followig exmples, the followig defiitios pply. (The subscripts here re used merely to cout the umber of terms beig multiplied. The i re equl to for ll vlues of i.). Eq 1. m = = ( 1 L 1 ) ( L ) 1 m 1 m Exmple 1: Product of two expoets m = 1 L 1 1 L m 1 Eq 1.3 m + m ( ) ( ) = m Exmple : Rtio of two expoets Eq 1.4 Exmple 3: Meig of egtive expoet Usig the result of equtio Eq 1.4, d relizig tht the logic we pplied works eqully for m> s for >m, oe c redily determie the meig of egtive expoet.
4 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 0 Eq 1.5 = = = = 0 1 Other Useful Properties of Expoets Derivtios of the followig properties use similr logic, d will be left to the studet. This list of properties expressed s equtios my look overwhelmig, but most re simple cosequeces of the defiitio of expoetitio, d should be self evidet if you uderstd the pricipls ivolved. We offer this list primrily for referece if you hve ot used expoetitio recetly m m Eq 1.6 ( ) = Eq 1.7 ( ) b = b b Eq 1.8 = = b (provided b 0) b Eq 1.9 b = b 1 = 1 b b = = b (provided,b 0) Eq Logrithms Studets frequetly hve bit more trouble workig with logrithms th they do workig with expoets. However the two re directly relted, d oe c derive the mjor properties of logrithms if oe keeps the reltioship cler. Logrithms re simply the mthemticl iverse of expoetitio. Lets put tht cocept ito forml defiitio. I the followig reltioship, is kow s the bse of the logrithm. Eq 1.11 y = log y log ( x) This implies tht = if d oly if ( y) d y = log ( y ) x = y Commo Nottiol Covetios The ottio log (x), writte without y subscript deotig the bse of the logrithm, lmost lwys deotes log bse. (i.e. ) log ( x) log( = ). The soclled turl x logrithm is the logrithm with bse equl to the irrtiol umber e Nturl logs tur up i vriety of mthemticl cotexts d they re covetiolly deoted s x) log ( x) l( =. We will follow these covetios i this book. e
5 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 Exmple 1: The Logrithm of the product of two umbers Wht is the log of the product of two umbers? The swer is surprisigly simple, d is cetrl to the importce of logrithms i my res of sciece. Eq 1.1 Other Useful Properties of Logrithms (Proofs left to the reder). Eq 1.13 log ( 1) = 0 for ll > 0 Eq 1.14 log ( ) = 1 for ll > 0 x y Eq 1.15 log = log ( x) log ( y) y Eq 1.16 log ( x ) = y log ( x) Eq 1.17 ( ) ( ) ( b) log c log b =, for y coveiet bse, c. log (This lst reltioship is ofte clled the bse chge formul) c Geerl Commet Much of the vlue of logs is direct result of properties Eq 1.15 d Eq These two properties permit my clcultios tht ivolve multiplictio d divisio to be replced with the simpler opertios of dditio d subtrctio. These simple properties of logs uderlie the simplicity of the slide rule, device for rpid clcultio of multiplictio d divisio tht ws mde obsolete by ivetio of iexpesive pocket clcultors. Logs lso permit compct disply or compriso of dt tht covers very lrge rge of vlues. A vriety of mesurble qutities, from cidity of queous solutios, to oise levels, to the severity of erthqukes re covetiolly reported i vlues derived from logrithmic scle.. Scietific Nottio Perhps the sigle most commo re i which you will be fced with workig with expoets d logs is i delig with very lrge or very smll umbers i compct fshio. Scietists d egieers use vrious forms of scietific ottio to del with this situtio. Scietific ottio logiclly seprtes umericl vlue ito two prts, coefficiet d power of te. The coefficiet is geerlly umber with sigle digit to the left of the deciml plce (tht is it hs vlue x such tht ). The power of te (or expoet) is lwys iteger. The product of the coefficiet d rised to the expoet produces the origil umericl vlue. My computer progrms d clcultors express very lrge d
6 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 very smll umbers i form of scietific ottio. Ofte such umbers re writte s the coefficiet, followed by cpitl letter E, followed by the expoet. Some exmples my help clrify the cocept. Eq = ,697,000 = = 5.43 (sometimes writte s E ) 6 6 (sometimes writte s.697 E 6) (sometimeswritte s 5.43 E 6) I effect, scietific ottio cts to shift the deciml, llowig us to do much of our mth with vlues betwee 1 d, determiig the order of mgitude of the result t the ed of our clcultios. While this is of cosiderble vlue eve tody (especilly i the cotext of bck of the evelope clcultio), it ws of criticl importce i the dys before redy vilbility of computers d clcultors. Slide rules mde it reltively quick d esy to multiply d divide umbers betwee 1 d. But tht techology would hve bee of little use without simple wy to hdle multiplictio d divisio of lrger d smller umbers. Scietific ottio provided wy to trsform my clcultios ito clcultios ivolvig umbers with oly sigle digit to the left of the deciml. Tody, fcility with scietific ottio is prt of scietific litercy. Uderstdig of this ottiol covetio while perhps less criticl tody th geertio go helps to mke the metric system of uits more comprehesible, ids with rpid clcultio, d helps to structure thikig bout the order of mgitude of qutities of scietific or egieerig iterest...1 Mthemticl Opertios i Scietific Nottio Scietific ottio mkes certi clcultios more coveiet, but if you re ot used to workig i scietific ottio, it c tke little gettig used to. The rules for clcultig i scietific ottio re cosequece of the wy scietific ottio is defied. You should be ble to figure these rules out for yourself with little thought. To describe the bsic mthemticl opertios, we eed to defie the vribles d b s follows: Eq 1.19 = x b = y p q Additio (or Subtrctio) We cot directly dd the vlues of the coefficiets, x d y, becuse they re ech multiplied by differet powers of te. The coefficiet x might be multiplied by te to produce, while y might be multiplied by millios to produce b. We must write d b so tht they re both expressed s vlue multiplied by the sme power of te. Luckily, this is geerlly ot too difficult to do. Lets express the vlue of i terms tht mtch the power of te i which b hs bee expressed, mely the expoet q. Eq 1.0 = = p q q x ( ) p q q ( x ) Now we re i positio to dd d b, expressig both i terms of the sme expoet.
7 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 Eq 1.1 Agi, exmple my help. Eq ( ) ( ) = = 8647 Multiplictio Eq 1.3 b = ( x y) p+ q Divisio Eq Orders of Mgitude Scietists ofte tlk bout the order of mgitude of vlue. The order of mgitude refers to the vlue of the expoet of umericl vlue expressed i scietific ottio. But it is eve more commo for scietists to spek of two umbers s differig by certi umber of orders of mgitude. Two vlues tht differ by order of mgitude differ by fctor of te ( 1 ). Vlues tht differ by three orders of mgitude differ by fctor of thousd ( 3 ). This termiology is used i settigs i which it does ot mke sese to py too much ttetio to the exct vlue of two umbers. For exmple, the mss of mrble (which is somewhere uder 0 g) d the mss of lrge utomobile (which is bout 1 metric to, or 00 kg, or 1,000,000 g) differ by pproximtely six orders of mgitude. Give these huge discrepcies i mss, for most prcticl problems, it relly would ot be too importt to kow the exct weight of the mrble. The termiology is used s sort of shorthd tht gives scietists quick referece for the pproximte reltive mgitude of differet vlues..3 Sigifict Digits Numbers c express qutities with rbitrry precisio (you c lwys just keep ddig digits to the right of the deciml plce). Ufortutely, we c mesure y relvlued qutity, d my itegervlued qutities with oly limited precisio, so it is esy to express vlues with more precisio th they deserve. Somehow, scietists must commuicte to oe other the ctul precisio with which umbers hve bee mesured or estimted. The most complete wy of doig tht is to report ech vlue with qutittive estimte of the ucertity of tht umber (such s its stdrd error), but i my cotexts tht is both lbor itesive d uecessry. The covetio of reportig umbers with pproprite umber of sigifict digits evolved s secod, less lborious (lbeit less precise) wy of keepig trck of d commuictig the precisio of umbers. The ide behid the covetio of sigifict digits is simple. If there is substtil ucertity i the vlue you report t certi order of mgitude (e.g. tes, hudreds, or thousds), do t report y digits t lesser order of mgitude. The digits tht cout s sigifict re those tht would pper i the coefficiet if the umber were writte i
8 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 scietific ottio, with the proviso tht zeros c be dded to the right of the deciml to idicte icresed precisio. Similr rules pply to umbers lrger th oe, but becuse zeros re ofte used to idicte plce (d thus powers of te) the usge c be mbiguous. Eq Sigifict Digits i Mesured Qutities The rules for determiig how my sigifict digits to use i your ow work re esiest to uderstd i the cotext of tkig mesuremets. As geerl rule, oe should record mesuremets to the first digit tht you must estimte, give the mesurig techology beig employed. I other words, you record ll the digits bout which you re resobly certi, d just the first digit bout which you hve some ucertity. Applyig this stdrd frequetly clls for little judgmet. Most commercilly vilble meter sticks (the metric equivlet of yrdstick) hve divisios tht oe c iterpret to the erest millimeter. Oe my be ble to estimte mesuremets t slightly fier resolutio by iterpoltig betwee the divisios. But mufcturig tolerces for meter sticks re ofte poor, d divisio mrks o poorly mde oes re iccurtely prited. Moreover, meter sticks re ofte used i field sciece to mke quick mesuremets of vrible or imprecisely defied properties such s wter depth or height of vegettio. Thus depedig o the ccurcy with which the meter stick ws produced d the type of mesuremet beig tke it might be pproprite to record mesuremets to the erest cetimeter (s for mesurig the depth of wter i rpidly flowig strem) or erest millimeter (legth of plt stem or lef). Fier resolutio (while superficilly possible by iterpoltig betwee the divisios o the scle) will seldom be pproprite..3. Sigifict Digits i Clculted Qutities My qutities i sciece re ot mesured directly, but clculted from other mesured vlues, d here the rules for determiig the umber of sigifict digits to report i your work re more complex. For exmple, specific grvity is clculted by combiig mesuremet of volume (which hs certi ccurcy) with mesuremet of mss (lso of limited precisio). The rules for determiig the umber of sigifict digits i clculted vlues differ for dditio or subtrctio d multiplictio d divisio. Additio (d Subtrctio) The result c be o more ccurte (i vlue of the lest sigifict digit, ot umber of digits) th the lest ccurte of the vlues dded together. Thus the result of the dditio of two vlues is oly s ccurte s the less ccurte of the two umbers beig dded together. To give you exmple, it oly mkes sese to dd weight of few grms weight expressed i metric toes (00 kg, or 1 x 6 g) if the weight i toes ws mesured to the erest grm. If it ws mesured oly to the erest kilogrm, it would mkes little sese to dd just few grms to tht mout, sice the mesuremet of the lrger object s mss could esily be off by severl hudred grms.
9 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 Eq = 5. 4 Additio is oly s ccurte s the less ccurte umber beig dded together. The more ccurte umber 1.7 i this cse is rouded to mtch the less ccurte oe before crryig out the dditio. Multiplictio d Divisio The results of both multiplictio d divisio hve the sme umber of sigifict digits s the LESS ACCURATE of the umbers you strted with. Eq = 13 = = = (two sigifict digits) (oe sigifict digit) Ofte, these rules will pper couter ituitive, especilly whe it mes reportig result with fewer sigifict figures. Eq Complex Clcultios I complex clcultios, icludig my sttisticl clcultios, if you re usig computer or pocket clcultor, you should crry out your computtios with ll vilble digits, d roud the result to the pproprite umber of sigifict digits oly t the ed of your clcultios. Roudig of itermedite results c itroduce sigifict umericl errors..4 The Metric System d SI Uits The system of mesuremets commoly clled the metric system is the stdrd set of uits for scietific ctivity worldwide. The moder icrtio of this iterlly cosistet system of scietific uits is the Itertiol System of Uits (or System Itertioledes Uités, ledig to the bbrevitio SI). SI uits ow domite itertiol commerce d idustry, d re the domit uits for egieerig prctice, t lest outside of the Uited Sttes. Becomig comfortble with SI uits will tke you log wy towrds beig ble to be full prticipt i techicl discussio o evirometl issues. For Americs who hve grow up usig Imperil uits (feet, gllos, pouds), lerig to work with metric uits is very much like lerig to use foreig lguge. As you re first workig with these uits, you my fid yourself metlly trsltig from metric to imperil uits d bck gi. As you gi fluecy i use of metric uits, you will fid it esier to thik i terms of the metric uits directly. If you go o to use metric mesuremets o regulr bsis, you my eve fid yourself wtig to buy bout _ kg (isted of bout 1 poud) of eggplt for dier.
10 Bohle d Austi Text PRELIMINARY DRAFT 1/6/ System Itertiol Uits SI uits re defied by itertiol greemets goig bck to Chges i the SI system eeded to reflect moder techologies d chges i scietific uderstdig re mde uder the uspices of itertiol covetio. All SI uits re ll bsed o seve bsic uits. These uits re ssumed to be idepedet of oe other i the sese tht oe could (t lest i pricipl) redefie the uit of y oe qutity without ffectig the other six. The seve bsic qutities d the SI uits used to mesure them re give i the followig tble: SI Bse Uits Bse qutity Nme Symbol legth meter m mss kilogrm kg time secod s electric curret mpere A thermodymic temperture Kelvi K mout of substce mole mol lumious itesity cdel cd Derived Uits All other SI uits re defied s combitios of the seve bsic uits. Volume hs dimesios of legth cubed, so the most turl uit of mesuremet of volume is the meter cubed. Velocity hs dimesios of distce per uit time, d thus is most directly mesured i meters per secod. Some combitios of the seve bsic uits re used so frequetly tht, s mtter of coveiece, they hve bee give mes. Exmples iclude the coulomb ( uit of electricl chrge, equl to A s), The joule ( uit of eergy, equl to kg m s  ), d the Volt ( mesure of electricl potetil, equl to m kg s 3 A 1 ). Usig Prefixes to Rescle Bsic d Derived Uits My bsic d derived uits tur out to be uwieldy i certi situtios, eve i everydy prctice. For exmple, it becomes tedious to write trvel distces, such s the distce from Portld, Mie to Wshigto, D.C. i terms of meters. We would ted to write the distce betwee the two cities s 890 kilometers, ot s 890,000 meters. The situtio gets eve more uwieldy for stroomers, mesurig iterstellr distces, or tomic physicists, estimtig distces withi toms. The SI system of uits iclude itertiolly greed upo list of prefixes tht permit people to scle both bsic d derived uits for greter coveiece for prticulr purposes. The prefixes re give i the followig tble. Note tht the vlues o the left of the tble re used to build ew uits tht re lrger th the bsic SI uits, while the prefixes o the right re used to build uits tht re smller tht the bsic SI uits. The symbols used for the prefixes tht correspod to fctors greter th fctor of 00 ( 3 ) re ll cpitlized. Tht s esy to remember big letters idicte big uits. We c see from the tble o the ext pge tht the fmilir uit of distce, the kilometer, correspods to 3 meters, or 00 meters, while the millimeter correspods to 3 meters. Wvelegths of light re ofte mesured i ometers, which correspod to 9 meters, or oe billioth of meter. Thus we could express the distce betwee Portld, Mie d Wshigto, D.C. i y of the followig wys (ote tht ll vlues re expressed to oly two sigifict digits). Of course, some of these re much more coveiet th others which is the whole poit of usig the prefixes.
11 Bohle d Austi Text PRELIMINARY DRAFT 1/6/005 Eq 1.8 Prefixes for Lrge Uits Prefixes for Smll Uits Fctor Nme Symbol Fctor Nme Symbol 4 yott Y 1 deci d 1 zett Z  ceti c 18 ex E 3 milli m 15 pet P 6 micro µ 1 ter T 9 o 9 gig G 1 pico p 6 meg M 15 femto f 3 kilo k 18 tto hecto h 1 zepto z 1 dek d 4 yocto y Oe c use prefixes with y of the med bsic d derived uits of the SI. The oly exceptio (d it is firly selfevidet oe) is for uits of mss. Oe should me uits of mss with referece ot to the bsic uit of mss, the kilogrm, but with referece to the grm, eve though the grm is ot formlly oe of the seve bsic uits of the SI. Thus i discussios of the globl crbo cycle, oe would spek i terms petgrms of crbo, ot terkilogrms. Additiol Uits ot Formlly Prt of SI My uits tht re used everydy i evirometl sciece re ot formlly prt of the SI system, but re curretly ccepted by itertiol uthorities for use with the SI uits. These iclude such uits s the liter (00 cm 3, or 1 dm 3, or 3 m 3 ), the metric to (sometimes writte toe, equl to 00 kg, or 1 Mg), the miute (60 secods), d the hectre (the re of squre 0m o side, 000 m, or 0.01 km ). For Further Iformtio The U.S. Ntiol Istitute of Stdrds d Techology mitis web site tht provides dditiol detils o SI uits t the URL: Dimesiol Alysis Dimesiol lysis is profoudly useful techique for resoig from wht you kow to wht you eed to kow. It c lso be used to geerte isights uvilble y other wy. It plys especilly importt role i certi fields. For exmple, i fluid dymics, dimesioless coefficiets (Reyolds umber, Froude umber, etc.) cpture importt spects of fluid flows. The bsic pricipl of dimesiol lysis is simple. If you hve equtio reltig two qutities, the dimesios (legth, mss, time, etc.) of the qutities o the two sides of the equtio must be similr.
12 Bohle d Austi Text PRELIMINARY DRAFT 1/6/ Do t Memorize Formuls Ler Pricipls Studets begiig to ler qutittive subjects ofte fid themselves memorizig formuls. But memoriztio hs severe disdvtge you ted to forget wht you memorize rther quickly. You my get through the fil exm, but you re ulikely to hve effective workig kowledge of the mteril yer or two fter you fiish the course. If you ler uderlig pricipls d few simple pproches for resoig from wht you DO kow to wht you eed to kow, you will reti workig kowledge quite bit loger. Furthermore, usig these strtegies for lerig will help you better uderstd the mteril d provide quick wy for you to check or supplemet your memory durig exms. Becuse of the iterdiscipliry ture of evirometl sciece, it is likely tht my of you will be clled upo t some time i your creer whether s scietists, policymkers, or iformed citizes to reso bout somethig you hve ot studied i yers. Thus it is importt for you to reti or be ble to regeerte your workig kowledge. I this cotext, dimesiol lysis is extremely vluble tool for beig ble to geerte or regeerte workig uderstdig..6 Uit Coversios
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA
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