Table Of Content

U A NDERSTANDING LGEBRA J B AMES RENNAN Copyright © 2002, All Rights Reserved C ONTENTS BMUN EHT :1 RETPAHC CITEMHTI R FASORE 1 T EH R EA L N UMBER S YSTEM 1 A DDITIOANN D S UBTRACTIONO F R LAE N UMBERS 8 M ULTIPLICATIOANN D D IVISION 11 P EVITISO INTEGER E XPONENTS 61 O RDERO F O SNOITAREP 71 F RACTIONS 91 D ECIMALS 52 P ERCENTS 23 P ROPERTIESO F R EAL N UMBERS 83 CUDORTNI :2 RETPAHC ARBEGLA OT NOIT 14 A LGEBRAIC E XPRESSIONS 14 S IMPLIFYING A LGEBRAIC E XPRESSIONS 24 S OLUTIONSO F A CIARBEGL E QUATIONS 54 A NOITIDD P RINCIPLE 64 M ULTIPLICATIO N P RINCIPLE 94 U SINGT HE P SELPICNIR T OGETHER 05 ORP DROW :3 RETPAHC SMELB 35 P MELBOR S OLVING S TRATEGIES 35 W FOR ORDS O SNOITAREP 55 G LARENE W ORD P ROBLEMS 65 GNIHPARG :4 RETPAHC SENIL THGIARTS DNA 95 R ECTANGULAR C OORDINATES 95 G GNIHPAR F UNCTIONS 26 S TRAIGHT L SENI 46 SMETSYS :5 RETPAHC SNOITAU QREAEN IFLO 37 T EH S OLUTIO A OFN S S YSTEMO F E QUATIONS 37 A NOITIDD M ETHOD 67 S UBSTITUTION M ETHOD 97 IMONYLOP :6 RETPAHC SLA 18 P OLYNOMIALS 18 F ACTORING P OLYNOMIALS 78 LANOITAR :7 RETPAHC SNOISSERP XE 79 R LANOITA E XPRESSIONS 79 S IMPLIFYING R LANOITA E XPRESSIONS 99 TNENOPXE :8 RETPAHC STOO RDN AS 501 E XPONENTS 501 R OOTS 701 S IMPLIFYING R ADICAL E XPRESSIONS 011 R ATIONALIZINGT HE D ENOMINATOR 111 ARDAUQ :9 RETPAHC SNOITAUQE CIT 311 Q CITARDAU E QUATIONS 311 S YB GNIVLO S QUARE R OOTS 411 S YB GNIVLO F ACTORING 511 C OMPLETINGT HE S QUARE 711 T EH Q CITARDAU F ORMULA 911 IXEDN 121 F ORWARD de ttyaalctonl e , rsenrit ou h isic aTftsiseronb rhbieyeterwmvgplA . ,2yr 0mr esIo0at fac2 ui ff .ntnooe e i haapphlctJholtlaitheiehwtde.www s tfpoe cdennhoatcts re enodotneumos .arb eIg leeak eatt manoshanitrhatug , edt ys tl ean l rp c pkfhi eeto uIeammrot envkirstfoaooseaairiirxwgirhleufefq tec h a . t utsdy uon rna et oYol toimeenyyceourfsrin etor afoa.ferepoolwPt rohtua .yubde.eta tgsneisliioabm@enannerbj sreBmaJ nanne yrau n,a3J12002 ,esioB ohadI v AritNhoumfme bTtehirecs C1h:a pter T R N S HE EAL UMBER METSY n o gtfien aothi nehotder dawwnn meyeeayivbvm pbtolmxoe evtrese yblesmahueTnr ,g e t ndk unwiliyouohul nyo htotac”ecnmrm aeo” ebs.,m mrttu es Anebrd“hmirtufonw“ d e eep .selherserlte enhaahamwTcsroaf sr elbamruuntan , or so e hstemitem counting srebmun . N LARUTA N UMBERS sre bgmnuiNtn urooC“ ” 1, 2, 3, 4, 5, . . . • n o o ie teth se saTl tefuta toroo ceadhndiht ntt feateom she sithnilto amammoc get sntaetpiahhseofcttiegilkdni orever. sdr .ee esr orf mae ede Ier b”h d ama,amotoie af teurt shfc endneentotbiizmoAo“ocps s. r nol esee wrltmiWhrpeoea art eehzc fhanbet,eotfemh ph huseettytnveonahnahs ts erslse ua bfrlr emoepuohubtrtn maeeunzhnt sreb meulnohw . W HOLE N MU BERS r esl hrahter”tebuoigmtrwouaetNNz“ , 0, 1, 2, 3, 4,5 . .. AbNtouhumetZb eerro g ne ib r hae ft b?o o?wnte mosn roaahsurIteHchtinei bWzamun ,? g g?nrnr io iesohshbIrti tmeoteuznimnos e,rlolifeheWtb k so t ew ’oda,kost l nnihteaonu alrlteosoaZskts se trresa e h rl owletn” iobt awa.efha dri0screned“usAhamntuIn ec aeplhoptl e o voreetesdzu - m ter atee hsbuetyml wsuaenvsu s l, a0 ,r , 1 ey reeneem lneh wetsu. nebtetiunyaohmwraWuwdnot ,2 ,3,4 ,s n,es terdnon ofanstalsca r ,e ,5me ,6us d,7ne.n8h9aT gnnoi i dnrtn.,rnoiii rseeesee d vphohbnereetptmIrotdudannhuwh 1 retpahC :1 The srebmuN of citemhtirA y e e t ,vko vypeankar tosaam wa rhaotihwerromthoewtfdro .i t hehtt Tm g’uso einoe c ,crou neraw lricele aoabphavlmtpun ssi , rk en .f nil so”ihate0 a Tr io “ttseyde anamthnhouptawcnme g tn,etr a iec ae.he yla2 hbythsf pl0rtmttiismp3oupnaaFnmghxeies , , sdsodnnne. aer s2td3ennuoh ,t nunobiit it ,nar s l ihe?i flo ft trooeSaesterW diabeammzun d snectewittamehtam lt ,tel so rkk ikasa tfoccaaictetaiuulclidda ryko lcistu visdfsaoi’hmteib - s roy. ereeebebkhmzoiTuln tsom fo s,r oe odby smrleua lwnn t aiticcd airahtote ms h eeetflmhiouatrrsa e.s ’ ru,n teah stibcg,eImeu ro ubosedlnhiba tmir uceanehptsar t sre b rs ym eleeweeusmlthahbnaaaitltosuoq — e d tiyu’vboniyadc . e,l oprrmoeafzxe tne m efp:to osocrd ltNi olhestnfetv iaIi aetrmfemdcuohipitrxtas ,h.sttron ebe0e lbmhdam t tneulad ar naeet 1i rlurcetgooernfpaiss eoshriteZ evitiddaytitnedi , s e r ooe odrbg temnazui ndedsauaceb ,yl r1a l. si reim e egiehtbnShtomatnuhnc evityatciitlnpeidtilum gni y rlae pbe imsytuublnaucm e1 bs e eotgdon.nathic n. o sg,iton ornf tcta trei Iniaheoebvs i drtvnzmaineeadtiuhehreas tnvtodbfaEmiaogen yna e n ,r o ,s p e eepede3mum hewle ro oyteouhtayrsah osafes sec h bphfttmetoauehnhts s roe tb ee mevb r uv ir .ean i et o s3gertkha tffnndeaotg sIowoimgoeeiolretnhantf yye”lem.l hasa ohtcru,gtT et dn t ebndeinluebmelthelbw uvupteasneoeesca“ccmcoas e ds e ntdg so enntfri esai aoesfvpt ehbroi xnehtmetge uwtuvantongieced gnuylbcni sdr eeeslbhilmtaucn sregetni . I NTEGERS s esv riesetlubaolmghpueWnn .. . – ,4 – ,3 – ,2 – ,. .,.4 ,3 ,2 ,1,01 About Negative Numbers g, nl ?oei lu odvk neo w sracneaWyaouseheavhHoezhcatylch ts euanonsg voharasn oayteehei?edthmztlvt mdanooaHutsocca t el ,f ue rltet t otdobo.oiskit uitfentomwu aipoiigArjotuyetz nreevh etsrsue flna h,gtonrientzaaehmt n e devo dvetlaa l huiowy er otmsuj ot teg ti pu ot.orez g nnei o:khsisitet ileoilhmgnatoicsfietda mteechhiTtratms 2 retpahC :1 The srebmuN of citemhtirA r e ylbrramoeeuFvrne n s te,ssrtieixhet etisoppo , denoted – n fm oue shtta hhtc u,s n d na – n is zero, or n + ( – n ) = 0 Notthteah te t r nt n feesgrnfoeo bviiaiorhamispftutnagen l o :ber“mh e y Tlbtsromaobuhfmntys – ”3 si tecneojbo — sdn attis ” , eeeehemvt r ai fhrnt toteaer a bg hoshemesstftteunsiirn“enhlut than zero. r oe e rbdh ,em eTsez u dritonsnei rttads neiioswzspionpooc eob erveihtt. aiergevoeninntisop n os ri gteotn n urfcofieooa i onhbmrfsdatatos aebueemuchRsstid lo“b meyhst – .” nt eoa .l ihe sitttwne nhi aak e oWszcasdeaiiimi uhe tloltdfcatcioa rnreteifhxnTeeng ytll aa merhns uketonanimiwl rtonaacufarf s g rnnler,yeiiafe pnhheibfeattrloemohs ,eskni olafdile tarnch uiaesrrafaem - s noet o the.if tsrtofp acIuefdgacwodurasf t fe et,ohesestwgregetni s rleabnmouintar . R LANOITA N SREBMU a s mrfereoh obtlfmluAn e,rehw a d na b esrraeg ettunbi( b ton ne)abocrez b eytldlal ulhaselawcrwcu enslbiuamnuonitaR fractions • d dns”hlin ,cum ie oioe ”ahdihhre lttrtwsctanoaia nowrthe oc“othdiNtrtoawr“ .usonyo iftocarf dne o lsimli toa fctceeoathhrotTfb eht rotanimoned k nf .iot hiseTaht denomination — ti sl lue otntyo a ieh, tzws,eci.hs wascth:g rtfttnefeiruirfuokaoblfaat de ehn ltosliia tcep chofeaTtohrtf rotaremun . s ultolIyet woyhnam ,s h,rtsofh itfruof whatever. • NOITCIRTSER rt oo:tnan nae eicb!hmo Tortrneuo ezB t)d(eanhratecmun n l o. atsi oousit frrtquce n, Ioeejalee o stzrohherIia fhttver fwazeeIhmhutn , r sete vik.oe agnw, nmnossai H hd h h ottretrtIor’rivoyrenuhanezoothazdf es nte asllaot ” . d sketh lru tnauonootsbi riaaaetezcm“arf 3 retpahC :1 The srebmuN of citemhtirA • rn ea sl h,senl nt1rk oa2a eidim/c eblets1rb mlco4ula/nar3cF( srneopiotrcparf ) , d enrla elh 1gatsgc r i(yenrb ebaoehmcbtun improper fractions ) , ekilowt - dna - a - h e,c dotf ielsilsh wulraaw2oawh/c5 lliA sre gnea tconsl a tehbgu of,ho s tlrhr aetosnbitaomwa iu nftnaioamro n:e1d 3 3 = 1 ,s rd, e ensslsb a rionmt lehhaua s abTwenhlus rmmtl otsuuaeie rtnhvsfeateobnrm(pun ) ss rtee.erg ss aefrbltoeua nbsneimohuitntar m setae h swgtoiiNm hgu oehhtt t e s slrd e aelr lfnbueyboomovriiuwoestncvsaeorp , dett,e s uans dastb hos sncao stiet iant er traen oabe.cphraenboaxtecrmsrehaufTn ds ere lerelbasamecuhnt lanoitarri .l aey nsee o uhrtiataotcnaerb I LANOITARR N SREBMU • d et soss nae enr.ba psCoaxriee tgfaeortni • s ls ssalyy Amaeart inhwe aecotlvoeteiaedp adtne rnareoirnm(or erto e)hrtehto Examples: 3 =0 .75 )s eltaannoiimtraeRt( 4 2 = 0 . 66666 6 lanoitaR)staeper( 3 5 = 0 .454545 45 lanoitaR)staeper( 11 5 = 0 . 714285 714285 lanoitaR)staeper( 7 2 =1 . 41421356 K lan oriet vasertnra(Ieper) sreotanimret p =3 . 14159265 K lan oriet vasertnra(Ieper) sreotanimret 4