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1 Numrs n surs Aritmti is t stuy o numrs n oprtions on tm. Tis sort ptr rviws wol numrs, intrs, rtionl numrs n rl numrs, wit prtiulr ttntion to t ritmti o surs n tir pproximtions. Most o tis mtril will milir rom rlir yrs. Diitl Rsours r vill or tis ptr in t Intrtiv Txtook n Onlin Tin Suit. S t Ovrviw t t ront o t txtook or tils. Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

2 A Wol numrs, intrs n rtionls A Wol numrs, intrs n rtionls Our is out numrs ris rom t two quit istint sours: T wol numrs, t intrs n t rtionl numrs r vlop rom ountin. T rl numrs r vlop rom omtry n t numr lin. Tis stion vry rily rviws wol numrs, intrs n rtionl numrs, wit prtiulr ttntion to prnts n rurrin imls. T wol numrs Countin is t irst oprtion in ritmti. Countin tins su s popl in room rquirs zro (i t room is mpty) n tn t sussiv numrs,,,..., nrtin ll t wol numrs: 0,,,,,, 6, T numr zro is t irst numr on tis list, ut tr is no lst numr, us vry numr is ollow y notr numr, istint rom ll prvious numrs. T list is tror ll ininit, wi mns tt it nvr iniss. T symol is nrlly us or t st o wol numrs. A non-zro wol numr n tor, in on n only on wy, into t prout o prim numrs, wr prim numr is wol numr rtr tn wos only ivisors r itsl n. T prims orm squn wos istintiv pttrn s onus vry mtmtiin sin Grk tims:,,, 7,,, 7, 9,, 9,, 7,,, 7,, 9, 6, 67, 7, T wol numrs rtr tn n not prim r ll omposit numrs:, 6, 8, 9, 0,,,, 6, 8, 0,,,,, 6, 7, 8, 0,, T wol numrs 0 n r spil ss, in nitr prim nor omposit. THE SET OF WHOLE NUMBERS T wol numrs r 0,,,,,, 6, Evry wol numr xpt 0 n is itr prim or omposit, n vry omposit numr n tor into prims in on n only on wy. Wn wol numrs r or multipli, t rsult is wol numr. T intrs Any two wol numrs n or multipli, n t rsult is notr wol numr. Sutrtion, owvr, rquirs t ntiv intrs s wll:, 6,,,,, so tt lultions su s 7 = n omplt. T symol numrs) is onvntionlly us or t st o intrs. THE SET INTEGERS T intrs r,,,,,, 0,,,,,, Wn intrs r, sutrt or multipli, t rsult is n intr. (rom Grmn Zln mnin Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

3 Cptr Numrs n surs A T rtionl numrs A prolm su s, Divi 7 ks into qul prts, ls nturlly to rtions, wr t wol is rtur or rokn into pis. Tus w v t systm o rtionl numrs, wi r numrs tt n writtn s t rtio o two intrs. Hr r som xmpls o rtionl numrs writtn s sinl rtions: = 7 = 0 =.7 = 7 = 00 T symol or quotint is onvntionlly us or t st o rtionl numrs. Oprtions on t rtionl numrs Aition, multiplition, sutrtion n ivision (xpt y 0) n ll rri out witin t rtionl numrs. Rtionl numrs r simplii y iviin top n ottom y tir HCF (ist ommon tor). For xmpl, n v HCF 7, so: = 7 7 = Rtionl numrs r n sutrt usin t LCM (lowst ommon multipl) o tir nomintors. For xmpl, 6 n 8 v LCM, so: = + = = = Frtions r multipli y multiplyin t numrtors n multiplyin t nomintors, tr irst nllin out ny ommon tors. To ivi y rtion, multiply y its riprol. 0 9 = 7 = 6 THE SET OF RATIONAL NUMBERS Diml nottion trmintin or rurrin imls 8 = 8 = 6 T rtionl numrs r t numrs tt n writtn s rtions, wr n r intrs n 0. Evry intr n writtn s rtion, n so is rtionl numr. Wn rtionl numrs r, sutrt, multipli n ivi (ut not y zro), t rsult is rtionl numr. Diml nottion xtns pl vlu to ntiv powrs o 0. For xmpl:.6 = Su numr n writtn s rtion 6, n so is rtionl numr. 000 Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

4 A Wol numrs, intrs n rtionls I rtionl numr n writtn s rtion wos nomintor is powr o 0, tn it n sily writtn s trmintin iml: = = 0. n 78 = = I rtionl numr nnot writtn wit powr o 0 s its nomintor, tn rpt ivision will yil n ininit strin o iits in its iml rprsnttion. Tis strin will yl on t sm rminr rurs, ivin rurrin iml. = = 0.6. (wi s yl lnt ) 6 = = (wi s yl lnt 6) 7 7 = =.. 8. (wi s yl lnt ) Convrsly, vry rurrin iml n writtn s rtion su lultions r isuss in Yr in t ontxt o omtri sris. Prnts Mny prtil situtions involvin rtions, imls n rtios r ommonly xprss in trms o prnts. PERCENTAGES To onvrt rtion to prnt, multiply y 00 %: 0 = 0 00 % = % To onvrt prnt to rtion, rpl % y % = 00 = 0 Mny prolms r st solv y t unitry mto, illustrt low. Exmpl A tl mrk $00 s n isount y 0%. How mu os t tl now ost? A tl isount y 0% now osts $00. Wt ws t oriinl pri? SOLUTION 00% is $00 0 0% is $0 7 70% is $980 so t isount pri is $ : 70% is $00 7 0% is $ % is $000 so t oriinl pri ws $000. A Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

5 6 Cptr Numrs n surs A Exris A Not: Qustions r non-lultor qustions. Writ s rtion in lowst trms: 0% 80% 7% % Writ s iml: 60% 7% 9% 6.% Writ s prnt: Writ s prnt: Ftor into prims: Cnl rtion own to lowst trms Exprss rtion s iml y rwritin it wit nomintor 0, 00 or Exprss trmintin iml s rtion in lowst trms Exprss rtion s rurrin iml y iviin t numrtor y t nomintor Fin t lowst ommon nomintor, tn simpliy: i j FOUNDATION Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

6 A Wol numrs, intrs n rtionls 7 Fin t vlu o: 0 Fin % o $. Fin 7.% o 00 k. Inrs $6 000 y 0%. Drs ours y 0%. Exprss rtion s iml i 8 j 8 DEVELOPMENT Stv s ounil rts inrs y % tis yr to $80. Wt wr is ounil rts lst yr? Jonn riv 0% isount on pir o sos. I s pi $, wt ws t oriinl pri? Mrko spnt $ tis yr t t Estr Sow,.% inrs on lst yr. How mu i spn lst yr? Exprss rtion in lowst trms, witout usin lultor Us your lultor to in t t rurrin imls or,,, CHALLENGE 0,,. Is tr pttrn? Us your lultor to in t rurrin imls or,,,, n 6. Is tr pttrn? T numrs you otin in tis qustion my vry pnin on t lultor us. Us your lultor to xprss s iml y ntrin. Sutrt 0. rom tis, multiply t rsult y 0 8, n tn tk t riprol. Sow ritmtilly tt t inl nswr in prt is. Is t nswr on your lultor lso qul to? Wt os tis tll you out t wy rtions r stor on lultor? Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

7 8 Cptr Numrs n surs B B Rl numrs n pproximtions Tis stion introus t st o rl numrs, wi r s not on ountin, ut on omtry ty r t points on t numr lin. Ty inlu ll rtionl numrs, ut s w sll s, ty lso inlu mny mor numrs tt nnot writtn s rtions. Dlin wit rl numrs tt r not rtionl rquirs spil symols, su s n : ut wn rl numr ns to pproximt, iml is usully t st ppro, writtn to s mny iml pls s is nssry. Dimls r us routinly in mtmtis n sin or two oo rsons: Any two imls n sily ompr wit otr. Any quntity n pproximt s losly s w lik y iml. Evry msurmnt is only pproximt, no mttr ow oo t instrumnt, n rounin usin imls is usul wy o sowin ow urt it is. Rounin to rtin numr o iml pls T ruls or rounin iml r: RULES FOR ROUNDING A DECIMAL To roun iml to, sy, two iml pls, look t t tir iit. I t tir iit is 0,,, or, lv t son iit lon. I t tir iit is, 6, 7, 8 or 9, inrs t son iit y. Alwys us rtr tn = wn quntity s n roun or pproximt. For xmpl,.87.8, orrt to two iml pls. (look t 7, t tir iit).87.8, orrt to on iml pl. (look t, t son iit) Sintii nottion n rounin to rtin numr o siniint iu T vry lr n vry smll numrs ommon in stronomy n tomi pysis r sir to omprn wn ty r writtn in sintii nottion: 000 =. 0 6 (tr r our siniint iurs) = 6. 0 (tr r iv siniint iurs) T iits in t irst tor r ll t siniint iurs o t numr. It is otn mor snsil to roun quntity orrt to ivn numr o siniint iurs rtr tn to ivn numr o iml pls. To roun, sy to tr siniint iurs, look t t ourt iit. I it is, 6, 7, 8 or 9, inrs t tir iit y. Otrwis, lv t tir iit lon , orrt to tr siniint iurs , orrt to our siniint iurs. T numr n in norml nottion n still roun tis wy:.0.0, orrt to our siniint iurs. Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

8 B Rl numrs n pproximtions 9 Unortuntly, tis n miuous. For xmpl, wn w s numr su s 00, w o not know wtr it s n roun to, or siniint iurs. Tt n only onvy y nin to sintii nottion n writin,. 0 or.0 0 or Tr r numrs tt r not rtionl At irst ln, it woul sm rsonl to liv tt ll t numrs on t numr lin r rtionl, us t rtionl numrs r lrly spr s inly s w lik lon t wol numr lin. Btwn 0 n tr r 9 rtionl numrs wit nomintor 0: Btwn 0 n tr r 99 rtionl numrs wit nomintor 00: Most points on t numr lin, owvr, rprsnt numrs tt nnot writtn s rtions, n r ll irrtionl numrs. Som o t most importnt numrs in tis ours r irrtionl, su s n, n t numr tt will introu in Cptr 9. T squr root o is irrtionl T numr is prtiulrly importnt, us y Pytors torm, is t ionl o unit squr. Hr is proo y ontrition tt is n irrtionl numr rr tis proo s xtnsion. Suppos tt wr rtionl numr. Tn oul writtn s rtion in lowst trms. Tt is, =, wr n v no ommon tor. n w know tt > us is not wol numr. Squrin, =, wr > us >. Bus is in lowst trms, is lso in lowst trms, wi is impossil, us =, ut >. Tis is ontrition, so nnot rtionl numr. T Grk mtmtiins wr rtly troul y t xistn o irrtionl numrs. Tir onrns n still sn in morn Enlis, wr t wor irrtionl mns ot not rtion n not rsonl. Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

9 0 Cptr Numrs n surs B T rl numrs n t numr lin T wol numrs, t intrs, n t rtionl numrs r s on ountin. T xistn o irrtionl numrs, owvr, mns tt tis ppro to ritmti is inqut, n mor nrl i o numr is n. W v to turn wy rom ountin n mk us o omtry. 6 DEFINITION OF THE SET OF REAL NUMBERS T rl numrs r in to ll t points on t numr lin. All rtionl numrs r rl, ut rl numrs su s n r irrtionl. At tis point, omtry rpls ountin s t sis o ritmti. 0 An irrtionl rl numr nnot writtn s rtion, or s trmintin or rurrin iml. In tis ours, su s numr is usully spii in xt orm, su s x = or x =, or s iml pproximtion orrt to rtin numr o siniint iurs, su s x. or x.6. Vry osionlly, rtionl pproximtion is usul or tritionl, su s. 7 T rl numrs r otn rrr to s t ontinuum, us t rtionls, spit in ns, r sttr lon t numr lin lik spks o ust, ut o not join up. For xmpl, t rtionl multipls o, wi r ll irrtionl, r just s ns on t numr lin s t rtionl numrs. It is only t rl lin itsl wi is ompltly join up, to t ontinuous lin o omtry rtr tn llin prt into n ininitu o isrt points. Opn n los intrvls Any onnt prt o t rl numr lin is ll n intrvl. An intrvl su s x is ll los intrvl us it ontins its npoints. x An intrvl su s < x < is ll n opn intrvl us it os not ontin its npoints. An intrvl su s x < is nitr opn nor los (t wor l-los is otn us). x x In irms, n npoint is rprsnt y los irl i it is ontin in t intrvl, n y n opn irl i it is not ontin in t intrvl. T tr intrvls ov r oun us ty v two npoints, wi oun t intrvl. An unoun intrvl in ontrst must itr opn or los, n t irtion tt ontinus to or to is rprsnt y n rrow. T unoun intrvl x is los intrvl us it ontins its x npoint. T unoun intrvl x < is n opn intrvl us it os not ontin x its npoint. Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

10 B Rl numrs n pproximtions For tos wo njoy strit initions, t rl lin itsl is n unoun intrvl tt is ot opn n los, n sinl point is ll nrt intrvl. 7 INTERVALS An intrvl is onnt prt o t numr lin. A los intrvl su s x ontins its npoints. An opn intrvl su s < x < os not ontin its npoints. An intrvl su s x < is nitr opn nor los. A oun intrvl s two npoints, wi oun t intrvl. An unoun intrvl su s x ontinus to or to (or ot). An ltrntiv nottion or intrvls will introu in Yr. Exris B Clssiy ts rl numrs s rtionl or irrtionl. Exprss tos tt r rtionl in t orm in lowst trms, wr n r intrs k 0.. l 7 Writ numr orrt to on iml pl Writ numr orrt to two siniint iurs Us lultor to in numr orrt to tr iml pls i % j 0. m n. o Us lultor to in numr orrt to tr siniint iurs To ow mny siniint iurs is o ts numrs urt? FOUNDATION Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

11 Cptr Numrs n surs B 7 Clssiy intrvl s opn or los or nitr (tt is, l-los). i 0 x 7 ii x > iii x 7 iv < x v x < vi < x < 0 vii x 6 viii x < Clssiy intrvl in prt s oun or unoun. 8 Writ intrvl in symols, tn skt it on sprt numr lin. T rl numrs rtr tn n lss tn. T rl numrs rtr tn or qul to n lss tn or qul to 0. T rl numrs lss tn 7. T rl numrs lss tn or qul to 6. DEVELOPMENT 9 T rl numrs lss tn or qul to 6.Us lultor to vlut xprssion orrt to tr iml pls Us Pytors torm to in t lnt o t unknown si in trinl, n stt wtr it is rtionl or irrtionl Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

12 B Rl numrs n pproximtions Clult, orrt to our siniint iurs: j k (.0) ( ) ( ) ( 6 7) + ( ) i (87. 0 ) ( ) l.6.9 Us lultor to nswr qustions n. Writ nswr in sintii nottion. CHALLENGE T sp o lit is pproximtly m/s. How mny mtrs r tr in lit-yr (wi is t istn tt lit trvls in yr)? Assum tt tr r 6 ys in yr n writ your nswr in mtrs, orrt to tr siniint iurs. T nrst lr lxy is Anrom, wi is stimt to lit-yrs wy. How r is tt in mtrs, orrt to two siniint iurs? T tim sin t Bi Bn is stimt to.8 illion yrs. How lon is tt in sons, orrt to tr siniint iurs? How r woul lit v trvll sin t Bi Bn? Giv your nswr in mtrs, orrt to two siniint iurs. T mss o proton is k n t mss o n ltron is k. Clult, orrt to our siniint iurs, t rtio o t mss o proton to t mss o n ltron. How mny protons, orrt to on siniint iur, r tr in k? Prov tt is irrtionl. (Apt t ivn proo tt is irrtionl.) Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

13 Cptr Numrs n surs C C Surs n tir ritmti Numrs su s n our onstntly in tis ours us ty our in t solutions o qurti qutions. T lst tr stions o tis ptr rviw vrious mtos o lin wit tm. Squr roots n positiv squr roots T squr o ny rl numr is positiv, xpt tt 0 = 0. Hn ntiv numr nnot v squr root, n t only squr root o 0 is 0 itsl. A positiv numr, owvr, s two squr roots, wi r t opposits o otr. For xmpl, t squr roots o 9 r n. Not tt t wll-known symol x os not mn t squr root o x. It is in to mn t positiv squr root o x (or zro, i x = 0). 8 DEFINITION OF THE SYMBOL x For x > 0, x mns t positiv squr root o x. For x = 0, 0 = 0. For x < 0, x is unin. For xmpl, =, vn tou s two squr roots, n. T symol or t ntiv squr root o is. Cu roots Cu roots r lss omplit. Evry numr s xtly on u root, so t symol x simply mns t u root o x. For xmpl: 8 = n 8 = n 0 = 0 Wt is sur? T wor sur is otn us to rr to ny xprssion involvin squr or ir root. Mor prisly, owvr, surs o not inlu xprssions su s 9 n 8, wi n simplii to rtionl numrs. 9 SURDS An xprssion n x, wr x is rtionl numr n n is n intr, is ll sur i it is not itsl rtionl numr. T wor sur is rlt to sur surs r irrtionl. Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

14 C Surs n tir ritmti Simpliyin xprssions involvin surs Hr r som lws rom rlir yrs or simpliyin xprssions involvin squr roots. T irst pir rstt t inition o t squr root, n t son pir r sily provn y squrin. 0 LAWS CONCERNING SURDS Lt n positiv rl numrs. Tn: = ( ) = Tkin out squr ivisors A sur su s 00 is not rr s in simplii, us 00 is ivisil y t squr numr 00, so 00 n writtn s 0 : Exmpl Simpliy ts xprssions involvin surs SOLUTION 08 = 6 = 6 = 6 Exmpl 7 = 9 = 9 = Simpliy t surs in ts xprssions, tn ollt lik trms n 00 = 00 = 00 = 0 SIMPLIFYING A SURD SOLUTION + 99 = + = = = Ck t numr insi t squr root or ivisiility y on o t squrs, 9, 6,, 6, 9, 6, 8, 00,,, Continu until t numr insi t squr root sin s no mor squr ivisors (prt rom ) = 6 + = + 6 = = 9 6 = 6 6 C C Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

15 6 Cptr Numrs n surs C Exris C FOUNDATION Writ own t vlu o: Simpliy: i j 00 k 60 l 7 m 80 n 98 o 800 p 000 Simpliy: i Simpliy: Writ xprssion s sinl squr root. For xmpl, = 9 = Simpliy ully: DEVELOPMENT i CHALLENGE 7 Simpliy ully: Fin t vlu o x i: 6 8 = x 80 0 = x 0 = x Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

16 D Furtr simpliition o surs 7 D Furtr simpliition o surs Tis stion ls wit t simpliition o mor omplit suri xprssions. T usul ruls o lr, totr wit t mtos o simpliyin surs ivn in t lst stion, r ll tt is n. Simpliyin prouts o surs T prout o two surs is oun usin t intity =. It is importnt to k wtr t nswr ns urtr simpliition. Exmpl Simpliy prout. SOLUTION = 7 = = Usin inomil xpnsions = 60 = = = 70 All t usul lri mtos o xpnin inomil prouts n ppli to suri xprssions. Exmpl Expn ts prouts n tn simpliy tm. ( + )( ) ( 6 ) SOLUTION ( + )( ) = ( ) + ( ) = + 6 = + 6 ( 6) = (usin t intity (A B) = A AB + B ) = 0 = 6 0 D D Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

17 8 Cptr Numrs n surs D Exris D FOUNDATION Simpliy: ( ) i ( ) j ( 7 ) k l Simpliy: Expn: ( + ) ( ) ( ) ( ) 7(7 7 ) 6( 6 ) Simpliy ully: Expn n simpliy: DEVELOPMENT ( 0 ) 6( + ) ( + ) 6( 8 ) (9 ) 7( 7 ) 6 Expn n simpliy: ( + )( ) ( )( 7 + ) ( + )( + ) ( 6 )( 6 ) ( 7 )( 7 + ) ( )( 6 ) 7 Us t spil xpnsion ( + )( ) = to xpn n simpliy: ( + )( ) ( 7 )( + 7 ) ( + )( ) ( )( + ) ( 6 + )( 6 ) (7 )(7 + ) 8 Expn n simpliy t ollowin, usin t spil xpnsions ( + ) = + + n ( ) = +. ( + ) ( ) ( + ) ( 7 ) ( ) ( + ) ( 7 + ) ( ) i ( + 0 ) Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

18 D Furtr simpliition o surs 9 CHALLENGE 9 Simpliy ully: Us Pytors torm to in t ypotnus o t rit-nl trinl in wi t lnts o t otr two sis r: n 7 n 7 + n 7 Simpliy y ormin t lowst ommon nomintor: + + Writ own t xpnsion o ( + ). Us t xpnsion in prt to squr Hn simpliy n Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

19 0 Cptr Numrs n surs E E Rtionlisin t nomintor Wn lin wit suri xprssions, it is usul to rmov ny surs rom t nomintor, pross ll rtionlisin t nomintor. Tr r two ss. T nomintor s sinl trm In t irst s, t nomintor is sur or multipl o sur. RATIONALISING A SINGLE-TERM DENOMINATOR In n xprssion su s 7, multiply top n ottom y. Exmpl 6 Simpliy xprssion y rtionlisin t nomintor. 7 SOLUTION 7 = 7 = = 6 T nomintor s two trms = = = T son s involvs nomintor wit two trms, on or ot o wi ontin sur. T mto uss t irn o squrs intity (A + B)(A B) = A B to squr t unwnt surs n onvrt tm to intrs. RATIONALISING A BINOMIAL DENOMINATOR In n xprssion su s, multiply top n ottom y. + Tn us t irn o squrs. E Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

20 E Rtionlisin t nomintor Exmpl 7 E Rtionlis t nomintor in xprssion. + SOLUTION = + + Exris E Rwrit rtion wit rtionl nomintor. 7 7 Rwrit rtion wit rtionl nomintor. 7 + = = + = + + = 9 + = Usin t irn o squrs: ( + )( ) = ( ) + + = Usin t irn o squrs: ( )( + ) = ( ) ( ) = 9 7 FOUNDATION 7 Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

21 Cptr Numrs n surs E Simpliy xprssion y rtionlisin t nomintor Rwrit rtion wit rtionl nomintor Rwrit rtion wit rtionl nomintor i j + 7 k Simpliy xprssion y rtionlisin t nomintor. + 7 Sow tt xprssion is rtionl y irst rtionlisin t nomintors I x = +, sow tt + x = x. 9 T xprssion 6 + n writtn in t orm +. Fin n. + Expn ( x + x). Suppos tt x = i Sow tt x + x = 7. ii Us t rsult in prt to in t vlu o x + x. l DEVELOPMENT CHALLENGE Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

22 Cptr Rviw Cptr Rviw Rviw tivity Crt your own summry o tis ptr on ppr or in iitl oumnt. Cptr Multipl-oi quiz Tis utomtilly-mrk quiz is ss in t Intrtiv Txtook. A printl PDF workst vrsion is lso vill tr. Cptr rviw xris Clssiy o ts rl numrs s rtionl or irrtionl. Exprss tos tt r rtionl in t orm, wr n r intrs Us lultor to writ numr orrt to: i two iml pls ii two siniint iurs Evlut, orrt to tr siniint iurs: (.8 8.) i Simpliy: Simpliy: + ( 7) i 6 Rviw Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P

23 Cptr Numrs n surs Rviw 6 Simpliy: Expn: ( 7 ) ( 6 + ) ( ) ( 6 + ) 8 Expn n simpliy: ( + )( ) ( )( + ) ( 7 )( + ) ( 0 )( 0 + ) ( 6 + )( 6 ) ( 7 ) ( + ) ( ) 9 Writ wit rtionl nomintor: 0 Writ wit rtionl nomintor: Rtionlis t nomintor o rtion Fin t vlu o x i = x. + 0 Simpliy + y ormin t lowst ommon nomintor. + Fin t vlus o p n q su tt = p + q. Sow tt is rtionl y irst rtionlisin nomintor Unorrt r smpl ps Cmri Univrsity Prss Pnr, t l P