Basic Arithmetic TERMINOLOGY


 Conrad Nicholson
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1 Bsic Arithmetic TERMINOLOGY Absolute vlue: The distce of umber from zero o the umber lie. Hece it is the mgitude or vlue of umber without the sig Directed umbers: The set of itegers or whole umbers f,,, 0,,,, f Epoet: Power or ide of umber. For emple hs bse umber of d epoet of Ide: The power of bse umber showig how my times this umber is multiplied by itself e.g. # #. The ide is Idices: More th oe ide (plurl) Recurrig deciml: A repetig deciml tht does ot termite e.g. 0. is recurrig deciml tht c be writte s frctio. More th oe digit c recur e.g Scietific ottio: Sometimes clled stdrd ottio. A stdrd form to write very lrge or very smll umbers s product of umber betwee d 0 d power of 0 e.g is.6 # 0 8 i scietific ottio
2 Chpter Bsic Arithmetic INTRODUCTION THIS CHAPTER GIVES A review of bsic rithmetic skills, icludig kowig the correct order of opertios, roudig off, d workig with frctios, decimls d percetges. Work o sigifict figures, scietific ottio d idices is lso icluded, s re the cocepts of bsolute vlues. Bsic clcultor skills re lso covered i this chpter. Rel Numbers Types of umbers Rel umbers Urel or imgiry umbers Rtiol umbers Irrtiol umbers Itegers Itegers re whole umbers tht my be positive, egtive or zero. e.g. 0,,, Rtiol umbers c be writte i the form of frctio b where d b re itegers, b! 0. e.g.,., 0., Irrtiol umbers cot be writte i the form of frctio (tht b is, they re ot rtiol) e.g., r EXAMPLE Which of these umbers re rtiol d which re irrtiol? r,.,, 9,, 6. Solutio d r re irrtiol s they cot be writte s frctios (r is irrtiol).., 9 d 6. so they re ll rtiol. 0
3 Mths I Focus Mthemtics Etesio Prelimiry Course Order of opertios. Brckets: do clcultios iside groupig symbols first. (For emple, frctio lie, squre root sig or bsolute vlue sig c ct s groupig symbol.). Multiply or divide from left to right.. Add or subtrct from left to right. EXAMPLE Evlute 0 ] + g. Solutio 0 ( + ) 0 # 9 0 BRACKETS KEYS Use ( d ) to ope d close brckets. Alwys use them i pirs. For emple, to evlute 0 ] + g press 0 # ( + ) To evlute correct to deciml plce press : ( (.6.9 ) ' ( ) ) 0. correct to deciml plce PROBLEM Wht is wrog with this clcultio? 9 Evlute + Press 9 ' + 9 ' + Wht is the correct swer?
4 Chpter Bsic Arithmetic MEMORY KEYS Use STO to store umber i memory. There re severl memories tht you c use t the sme time y letter from A to F, or X, Y d M o the keypd. To store the umber 0 i, sy, A press 0 STO A To recll this umber, press ALPHA A To cler ll memories press SHIFT CLR Differet clcultors use differet keys so check the istructios for your clcultor. X KEY Use this key to fid the reciprocl of. For emple, to evlute.6#. press ( ().6 #. ) 0.06 (correct to deciml plces) Roudig off Roudig off is ofte doe i everydy life. A quick look t ewspper will give plety of emples. For emple i the sports sectio, ewspper my report tht fs tteded footbll mtch. A ccurte umber is ot lwys ecessry. There my hve bee ectly 9 96 people t the footbll gme, but gives ide of the size of the crowd. EXAMPLES. Roud off 69 to the erest thousd. Solutio This umber is betwee 000 d 000, but it is closer to 000. ` to the erest thousd CONTINUED
5 6 Mths I Focus Mthemtics Etesio Prelimiry Course. Write 80 to the erest hudred. Solutio This umber is ectly hlfwy betwee 800 d 900. Whe umber is hlfwy, we roud it off to the lrger umber. ` to the erest hudred I this course you will eed to roud off decimls, especilly whe usig trigoometry or logrithms. To roud umber off to certi umber of deciml plces, look t the et digit to the right. If this digit is or more, dd to the digit before it d drop ll the other digits fter it. If the digit to the right is less th, leve the digit before it d drop ll the digits to the right. EXAMPLES. Roud off 0.68 correct to deciml plce. Add to the 6 s the 8 is greter th. Solutio # ` correct to deciml plce. Roud off 0.68 correct to deciml plces. Drop off the d ll digits to the right s is smller th. Solutio # ` correct to deciml plces. Evlute 6. '. correct to deciml plces. Check this o your clcultor. Add to the 69 s is too lrge to just drop off. Solutio. 6 ' #. 0 correct to deciml plces
6 Chpter Bsic Arithmetic FIX KEY Use MODE or SET UP to fi the umber of deciml plces (see the istructios for your clcultor). This will cuse ll swers to hve fied umber of deciml plces util the clcultor is tured off or switched bck to orml. While usig fied umber of deciml plces o the disply, the clcultor still keeps trck iterlly of the full umber of deciml plces. EXAMPLE Clculte. '. # correct to deciml plces. Solutio. '. # # correct to deciml plces If the FIX key is set to deciml plces, the the disply will show deciml plces t ech step..'.# # If you the set the clcultor bck to orml, the disply will show the full swer of Do t roud off t ech step of series of clcultios. The clcultor does ot roud off t ech step. If it did, the swer might ot be s ccurte. This is importt poit, sice some studets roud off ech step i clcultios d the woder why they do ot get the sme swer s other studets d the tetbook.. Eercises. Stte which umbers re rtiol d which re irrtiol. () 69 (b) 0.6 (c) r (d) (e) 0. (f) 8 (g) (h) (i).% (j)
7 8 Mths I Focus Mthemtics Etesio Prelimiry Course. Evlute () 0 8' (b) # # (c) # ] ' g ' 6 (d) + # (e) 9. #. ' (f) + (g) # 8 (h) 6 ' 8 6 (i) (j). Evlute correct to deciml plces. ().6 +. ' 0. (b) ] g ' 0. (c).#.9'. (d) 8. ' 0. +.# 0. (e) ]. 6. g# ]. +. g (f). +. (g) (h) (i).. (j) Roud off 89 to the erest hudred.. Write 9 to the erest te. 6. Roud off 00 to the erest thousd.. A crowd of 0 9 specttors tteded teis mtch. Write this figure to the erest thousd. 8. A school hs 6 studets. Wht is this to the erest hudred? 9. A bk mde los to the vlue of $ 6 8 lst yer. Roud this off to the erest millio. 0. A compy mde profit of $ lst yer. Write this to the erest hudred thousd.. The distce betwee two cities is 8. km. Wht is this to the erest kilometre?. Write 0.8 correct to deciml plces.. Roud off.698 to the erest uit.. Roud off.89 to deciml plces.. Evlute. #. correct to deciml plces. 6. Evlute 8.' correct to deciml plce.. If pies re o specil t for $.8, fid the cost of ech pie. 8. Evlute.8 correct to deciml plces Evlute correct to 8 deciml plce. 0. Fid the legth of ech piece of mteril, to deciml plce, if m of mteril is cut ito equl pieces.
8 Chpter Bsic Arithmetic 9. How much will. m of tiles cost, t $.9 per m?. Divide.9 grms of slt ito equl portios, to deciml plce.. The cost of 9 peches is $.. How much would peches cost?. Evlute correct to deciml plces. ().. #.6 (b) 8. # # (c).6 +. (d) (e) ] g. Evlute 9. correct to deciml plce. DID YOU KNOW? I buildig, egieerig d other idustries where ccurte mesuremets re used, the umber of deciml plces used idictes how ccurte the mesuremets re. For emple, if. m legth of timber is cut ito 8 equl prts, ccordig to the clcultor ech prt should be 0.08 m. However, mchie could ot cut this ccurtely. A legth of. m shows tht the mesuremet of the timber is oly ccurte to the erest mm (. m is mm). The cut pieces c lso oly be ccurte to the erest mm (0.0 m or 0 mm). The error i mesuremet is relted to roudig off, s the error is hlf the smllest mesuremet. I the bove emple, the mesuremet error is hlf millimetre. The legth of timber could be ywhere betwee 0. mm d. mm. Directed Numbers My studets use the clcultor with work o directed umbers (umbers tht c be positive or egtive). Directed umbers occur i lgebr d other topics, where you will eed to remember how to use them. A good uderstdig of directed umbers will mke your lgebr skills much better. ^h KEY Use this key to eter egtive umbers. For emple, press ()
9 0 Mths I Focus Mthemtics Etesio Prelimiry Course Addig d subtrctig To dd: move to the right log the umber lie To subtrct: move to the left log the umber lie 0 Subtrct Add You c lso do these o clcultor, or you my hve differet wy of workig these out. EXAMPLES Evlute. + Solutio Strt t d move plces to the right Solutio Strt t d move plces to the left. 0 Multiplyig d dividig To multiply or divide, follow these rules. This rule lso works if there re two sigs together without umber i betwee e.g. Sme sigs Differet sigs + +
10 Chpter Bsic Arithmetic EXAMPLES Evlute. # Solutio Differet sigs ( d + ) give egtive swer. #. ' Solutio Sme sigs ( d ) give positive swer. '. Solutio The sigs together re the sme (both egtive) so give positive swer. + Strt t d move plces to the right.. Eercises Evlute #. ]g. 8 ' #. 9.. ]8. g 0.. #. + #. # ]g. + ' 8. # 6 # 9. 8' # 0. ]g
11 Mths I Focus Mthemtics Etesio Prelimiry Course Frctios, Decimls d Percetges Coversios You c do ll these coversios o your clcultor usig the c b or S + D key. EXAMPLES. Write 0. s frctio i its simplest form. Solutio ' mes ' Covert 8 to deciml. Solutio 0. 8g. 000 So Chge.% to frctio. Solutio.. % # Write 0.6 s percetge. Multiply by 00% to chge frctio or deciml to percetge. Solutio # 00%. 6%. Write 0 g s frctio of kg i its simplest form. Solutio kg 000 g 0 g 0 g kg 000 g 0
12 Chpter Bsic Arithmetic 6. Fid the percetge of people who prefer to drik Lemo Fuzzy, if out of every 0 people prefer it. Solutio 00% # 80% 0 Sometimes decimls repet, or recur. Emple 0. f 0. There re differet methods tht c be used to chge recurrig deciml ito frctio. Here is oe wy of doig it. Lter you will discover other method whe studyig series. (See HSC Course book, Chpter 8.) EXAMPLES. Write 0. s rtiol umber. A rtiol umber is y umber tht c be writte s frctio. Solutio Let 0. f ( ) The 0. f ( ) ( ) ( ): 9 9 Check this o your clcultor by dividig by 9.. Chge 9. to frctio. Solutio Let. 999 f ( ) The f ( ) ( ) ( ): # Try multiplyig by 0. Why does t this work? CONTINUED
13 Mths I Focus Mthemtics Etesio Prelimiry Course This method voids decimls i the frctio t the ed. Aother method Let. 999 f The f ( ) d f ( ) ( ) ( ): Eercises. Write ech deciml s frctio i its lowest terms. () 0.6 (b) 0.0 (c).0 (d).8. Chge ech frctio ito deciml. () (b) 8 (c) (d). Covert ech percetge to frctio i its simplest form. () % (b).% (c) 0.% (d) 09.%. Write ech percetge s deciml. () % (b) 09% (c) 0.% (d) 6.%. Write ech frctio s percetge. () 0 (b) (c) (d) Write ech deciml s percetge. (). (b) 0. (c) 0.0 (d).9. Write ech percetge s deciml d s frctio. () % (b) % (c) 6.8% (d) 09% (e).% (f) % 8. Write these frctios s recurrig decimls. () 6 (b) 99 (c) 99 (d) 6 (e)
14 Chpter Bsic Arithmetic (f) (g) (h) 9. Epress s frctios i lowest terms. () 08. (b) 0. (c). (d). (e) 06. (f) 0. (g) 0. (h) 06. (i) 09. (j) Evlute d epress s deciml. () + 6 (b) 8 ' + (c) + (d) (e) +. Evlute d write s frctio. (). ' ] g. 8.9 (b) (c) (d). 9. (e) Agel scored out of i clss test. Wht ws her score s percetge, to the erest uit?. A survey showed tht out of 0 people wtched the ews o Mody ight. Wht percetge of people wtched the ews?. Wht percetge of kg is 0 g?. Write miutes s percetge of hour. Ivestigtio Eplore ptters i recurrig decimls by dividig umbers by, 6, 9,, d so o. C you predict wht the recurrig deciml will be if frctio hs i the deomitor? Wht bout 9 i the deomitor? Wht bout? C you predict wht frctio certi recurrig decimls will be? Wht deomitor would digit recurrig give? Wht deomitor would you hve for digits recurrig? Opertios with frctios, decimls d percetges You will eed to kow how to work with frctios without usig clcultor, s they occur i other res such s lgebr, trigoometry d surds.
15 6 Mths I Focus Mthemtics Etesio Prelimiry Course The emples o frctios show how to dd, subtrct, multiply or divide frctios both with d without the clcultor. The deciml emples will help with some simple multiplyig d the percetge emples will be useful i Chpter 8 of the HSC Course book whe doig compoud iterest. Most studets use their clcultors for deciml clcultios. However, it is importt for you to kow how to operte with decimls. Sometimes the clcultor c give wrog swer if the wrog key is pressed. If you c estimte the size of the swer, you c work out if it mkes sese or ot. You c lso sve time by doig simple clcultios i your hed. DID YOU KNOW? Some coutries use comm for the deciml poit for emple, 0, for 0.. This is the reso tht our lrge umbers ow hve spces isted of comms betwee digits for emple, 000 rther th,000. EXAMPLES. Evlute. Solutio Evlute Solutio ' ' # 6 '.. Evlute # 00. Move the deciml poit plces to the right. Solutio 0. 06# 00. 6
16 Chpter Bsic Arithmetic. Evlute 00. # 0.. Solutio 0. 0# Multiply the umbers d cout the umber of deciml plces i the questio.. Evlute Solutio ' Move the deciml poit plce to the left. 6. The price of $ teis rcquet icresed by %. price. Solutio ` % of $ 0. 0# $ $. Fid the ew % 0. 0 or 0 % of $. 0# $ So the price icreses by $. to $9.. $ 9.. The price of book icresed by %. If it ow costs $8.00, wht did it cost before the price rise? Solutio The ew price is % (old price 00%, plus %) % $ ` 00% $ # $6.0 So the old price ws $6.0.. Eercises. Write 8 miutes s frctio of hours i its lowest terms.. Write 0 ml s frctio of litre i its simplest form.. Evlute () + (b) (c) # (d) (e) ' 0 '
17 8 Mths I Focus Mthemtics Etesio Prelimiry Course. Fid of $ Fid of kg, i grms correct to deciml plce. 6. Trih speds of her dy sleepig, t work d etig. Wht frctio of the dy is left?. Icrese 0 g by %. 6. Decrese m by 8 %.. The cost of clcultor is ow $. If it hs icresed by.%, how much ws the old cost? 8. A tree ow mesures. m, which is 8.% more th its previous yer s height. How high ws the tree the, to deciml plce?. I get $0.00 week for csul job. If I sped o bus fres, 0 o luches d o outigs, how much moey is left over for svigs? 8. Joh grew by of his height 00 this yer. If he ws 6 cm tll lst yer, wht is his height ow, to the erest cm? 9. Evlute () (b) 9. (c) 9. # 0 (d) 0. 0# 00 (e) 0. # (f) 08. # 0. (g) 0. 0# (h). # (i) 00 (j) 6. # This moth there hs bee.9% icrese i stole crs. If 6 crs were stole lst moth, how my were stole this moth? 0. George s computer cost $00. If it hs deprecited by.%, wht is the computer worth ow? 0. Fid % of $0.. Fid 6.% of 8 ml.. Wht is.% of 9 g?. Fid.% of $89... If Kye mkes profit of $ by sellig bike for $8, fid the profit s percetge of the sellig price.
18 Chpter Bsic Arithmetic 9 PROBLEM If both the hour hd d miute hd strt t the sme positio t o clock, whe is the first time, correct to frctio of miute, tht the two hds will be together gi? Powers d Roots A power (or ide ) of umber shows how my times umber is multiplied by itself. EXAMPLES. # # 6 I the is clled the bse umber d the is clled the ide or power.. # # # # A root of umber is the iverse of the power. EXAMPLES. 6 6 sice sice sice 6 6 DID YOU KNOW? My formule use idices (powers d roots). For emple the compoud iterest formul tht you will study i Chpter 8 of the HSC Course book is A P^ + rh Geometry uses formule ivolvig idices, such s V rr. Do you kow wht this formul is for? I Chpter, the formul for the distce betwee poits o umber ple is d ( ) + ( y y ) See if you c fid other formule ivolvig idices.
19 0 Mths I Focus Mthemtics Etesio Prelimiry Course POWER AND ROOT KEYS Use the d keys for squres d cubes. Use the y or ^ key to fid powers of umbers. Use the key for squre roots. Use the key for cube roots. Use the for other roots. These lws work for y m d, icludig frctios d egtive umbers. Ide lws There re some geerl lws tht simplify clcultios with idices. m m # + Proof m # ( # # f# ) #( # # f# ) mtimes times # # f # m+ times m + m m ' Proof m ' m # # f# ( m times) # # f# ( times) # # f# ( m times) m ( m ) m Proof m m m m m ( ) # # # f# ( times) m+ m+ m+ f+ m m ( times)
20 Chpter Bsic Arithmetic ( b) b Proof ( b) b# b# b# f# b ( times) ( # # f# ) #( b# b# f# b) times times b c m b b Proof c m b # # # f# ( times) b b b b # # # f# ( times) b# b# b# f# b ( times) b EXAMPLES Simplify 9. m # m ' m Solutio m # m ' m m m ( ) y Solutio ( y ) ( y ) 8y y # CONTINUED
21 Mths I Focus Mthemtics Etesio Prelimiry Course. ( y ) y 6 # Solutio ( y ) y # y y # y y y 8 + y y y y 9 y 6 8 ( ). Eercises. Evlute without usig clcultor. () # (b) + 8 (c) c m (d) (e) 6. Evlute correct to deciml plce. (). (b).06. (c). 0. (d) 9 (e). 8. #. (f) Simplify 6 9 () # # (b) y # y # y (c) # (d) w (e) (f) p y (g) y 8 # w p 6 ' ' (h) ( ) (i) ( ) (j) ( y ) (k) # ' (l) f 9 p y 6 w # w (m) w p #( p ) () 9 p 6 ' (o) 6 #( b ) (p) 9 # b (q) ( ) # ( y ) # y. Simplify 9 () # (b) # (c) m m (d) k # k ' k (e) # # (f) # m # (g) m #
22 Chpter Bsic Arithmetic p # p (h) p (i) ( ) (j) ( ) 6. Simplify () ( pq ) 8 (b) c m b (c) d b (d) ( b ) (e) ( m ) m y #( y ) (f) y 8 ( k ) (g) ( 6k ) y (h) _ y i # 8 (i) (j) e o y f p 8 # y 6 # 9 6. Evlute b whe d b.. If d y, fid the vlue 9 y of. y 8. If, b d c, b evlute s frctio. c 8 b 9. () Simplify. b 8 8 b (b) Hece evlute whe 8 b d b s frctio. 8 8 pqr 0. () Simplify. pqr 6 8 pqr (b) Hece evlute s pqr 6 frctio whe p, q 8 d r.. Evlute ( ) whe b. Evlute b b. 6 y. Evlute y y. 9. Evlute k k 9 6 b. Evlute ( b ) b. 9 whe whe 6 c m whe k. whe. d d d 6 # b 6. Evlute b s frctio # whe d b. 9 b. Evlute s frctio i b ide form whe c m d b c m. 8 ( ) b c 8. Evlute s frctio b ( ) c 6 whe, b d c. 9
23 Mths I Focus Mthemtics Etesio Prelimiry Course Negtive d zero idices Clss Ivestigtio Eplore zero d egtive idices by lookig t these questios. For emple simplify ' usig (i) ide lws d (ii) ccellig. (i) ' by ide lws (ii) # # # # # # So Now simplify these questios by (i) ide lws d (ii) ccellig. () ' (b) ' (c) ' 6 (d) ' (e) ' (f) ' (g) ' 6 (h) ' (i) ' (j) ' Use your results to complete: 0 0 Proof ' ` ' 0 0
24 Chpter Bsic Arithmetic Proof ' ` ' 0 EXAMPLES b c. Simplify e o. bc Solutio b c e o bc 0 0. Evlute. Solutio 8. Write i ide form. () (b) (c) (d) + CONTINUED
25 6 Mths I Focus Mthemtics Etesio Prelimiry Course Solutio () (b) # (c) # (d) + ( + ) ] + g. Write without the egtive ide. Solutio.6 Eercises. Evlute s frctio or whole umber. () (b) (c) (d) 0 (e) 8 0 (f) 6 (g) (h) (i) (j) 9 (k) 6 (l) (m) 0 () 6 (o) (p) 0 (q) (r) 0 (s) 8 (t). Evlute () 0 (b) c m (c) c m (d) c m 6 + y 0 (e) f p y (f) c m (g) c m (h) c m (i) (j) c m c m (k) c m
26 Chpter Bsic Arithmetic (l) 8 0 c m 9 6 (m) c m 9 () c m (o) c m (p) c m (q) c m (r) c m (s) 0 c m 8 (t) c m. Chge ito ide form. () (b) m (c) p (d) d 9 (e) k (f) (g) (h) y (i) z 6 (j) t 8 (k) (l) m 6 (m) y () ( + ) (o) ( + b ) 8 (p) (q) ( p ) + (r) ( t 9) (s) ( + ) (t) 9( + b ). Write without egtive idices. () t (b) 6 (c) y (d) 8 (e) w 0 (f) (g) m (h) (i) ] g (j) ] g (k) ] + g 6 (l) ^8y zh (m) ] k g + () ^ + yh 9 (o) b l (p) c y m 0 (q) d p (r) c m + b y (s) e o y + w z (t) e o + y
27 8 Mths I Focus Mthemtics Etesio Prelimiry Course Frctiol idices Clss Ivestigtio Eplore frctiol idices by lookig t these questios. For emple simplify (i) ` j d (ii) ^ h. () i ` j ^by ide lwsh ( ii) ^ h So ` j ^ h ` Now simplify these questios. () ^ h (b) (c) ` j (d) ^ h (e) ^ h (f) (g) ` j (h) ^ h (i) ^ h (j) Use your results to complete: Proof ` j ^ h ` ^by ide lwsh
28 Chpter Bsic Arithmetic 9 EXAMPLES. Evlute () (b) 9 Solutio () (b) 9 9. Write i ide form. Solutio ( ). Write ( + b) Solutio + b + b ( ) without frctiol idices. Puttig the frctiol d egtive idices together gives this rule. Here re some further rules. m m ( ) m Proof m m m ` j ^ h ^ m h m m
29 0 Mths I Focus Mthemtics Etesio Prelimiry Course c m b b b l Proof c m b c m b b ' b b # b b b l EXAMPLES. Evlute () 8 (b) (c) c m Solutio () 8 ( 8) ( or 8 ) 6 (b)
30 Chpter Bsic Arithmetic (c) c m c m 8 8. Write i ide form. () (b) ( ) Solutio () (b) ( ) ( ) ( ). Write r without the egtive d frctiol idices. Solutio r r r DID YOU KNOW? Nicole Oresme ( 8) ws the first mthemtici to use frctiol idices. Joh Wllis (66 0) ws the first perso to epli the sigificce of zero, egtive d frctiol idices. He lso itroduced the symbol for ifiity. Do Iteret serch o these mthemticis d fid out more bout their work d bckgrouds. You could use keywords such s idices d ifiity s well s their mes to fid this iformtio.
31 Mths I Focus Mthemtics Etesio Prelimiry Course. Eercises. Evlute () 8 (b) (c) 6 (d) 8 (e) 9 (f) 000 (g) 6 (h) 6 (i) 6 (j) (k) (l) 8 (m) 0 8 () (o) (p) 8 (q) 6 (r) 9 (s) (t) 8 6. Evlute correct to deciml plces. () (b).8 (c). +. (d) (e) #. (f) Write without frctiol idices. () y (b) y (c) (d) ( + ) (e) ( ) (f) ( 6 q + r) (g) + ( ). Write i ide form. () (b) t y (c) (d) 9 (e) s + (f) t + (g) ( y) (h) ( + ) (i) ( ) (j) y + (k) + (l) y (m) ( ) +. Write i ide form d simplify. () (b) (c) (d) (e)
32 Chpter Bsic Arithmetic 6. Epd d simplify, d write i ide form. () ( + ) (b) ( + b)( b) (c) fp + p p (d) ( ) + ( + ) (e). Write without frctiol or egtive idices. () b ( ) (b) ( y ) (c) (d) (e) + 6 ( ) ( + y) 6 ( + 8 ) 9 Scietific ottio (stdrd form) Very lrge or very smll umbers re usully writte i scietific ottio to mke them esier to red. Wht could be doe to mke the figures i the bo below esier to red? DID YOU KNOW? The By of Fudy, Cd, hs the lrgest tidl chges i the world. About tos of wter re moved with ech tide chge. The diosurs dwelt o Erth for yers util they died out yers go. The width of oe plt cell is bout m. I 00, the totl storge cpcity of dms i Austrli ws litres d households i Austrli used litres of wter. A umber i scietific ottio is writte s umber betwee d 0 multiplied by power of 0. EXAMPLES. Write i scietific ottio. Solutio # 0 8 Write the umber betwee d 0 d cout the deciml plces moved.. Write. # 0 s deciml umber. Solutio. # 0. ' Cout plces to the left.
33 Mths I Focus Mthemtics Etesio Prelimiry Course SCIENTIFIC NOTATION KEY Use the EXP or # 0 key to put umbers i scietific ottio. For emple, to evlute.# 0 '.# 0, press. EXP '. EXP () DID YOU KNOW? Egieerig ottio is similr to scietific ottio, ecept the powers of 0 re lwys multiples of. For emple,.# 0. # 0 6 SIGNIFICANT FIGURES The cocept of sigifict figures is relted to roudig off. Whe we look t very lrge (or very smll) umbers, some of the smller digits re ot sigifict. For emple, i footbll crowd of 9 96, the 6 people re ot relly sigifict i terms of crowd of bout 0 000! Eve the 6 people re ot sigifict. Whe compy mkes profit of $ 0.8, the mout of 8 cets is ot ectly sigifict sum! Nor is the sum of $.8. To roud off to certi umber of sigifict figures, we cout from the first ozero digit. I y umber, ozero digits re lwys sigifict. Zeros re ot sigifict, ecept betwee two ozero digits or t the ed of deciml umber. Eve though zeros my ot be sigifict, they re still ecessry. For emple, 0, 00, 000 d ll hve sigifict figures but re very differet umbers! Scietific ottio uses the sigifict figures i umber. EXAMPLES 000. # 0 ( sigifict figures) # 0 ( sigifict figures) # 0 ( sigifict figures) Whe roudig off to sigifict figures, use the usul rules for roudig off.
34 Chpter Bsic Arithmetic EXAMPLES. Roud off 9 0 to sigifict figures. Solutio to sigifict figures Remember to put the 0 s i!. Roud off to sigifict figures. Solutio to sigifict figures. Roud off.96 9 to sigifict figures. Solutio to sigifict figures.8 Eercises. Write i scietific ottio. () 800 (b) (c) (d) (e) (f) (g) 900 (h) 60 (i) (j) Write i scietific ottio. () 0.0 (b) (c) 0.00 (d) (e) (f) (g) (h) 0. (i) (j) Write s deciml umber. () 6. # 0 (b) 8. # 0 (c) 9. # 0 (d) 6. # 0 6 (e) # 0 (f). # 0 (g) 9. # 0 (h) 8. # 0 8 (i) # 0 6 (j) # 0. Roud these umbers to sigifict figures. () 980 (b) 9 60 (c) 0 (d) (e).9 (f) (g).69 (h).8 (i) (j) 9.0
35 6 Mths I Focus Mthemtics Etesio Prelimiry Course. Evlute correct to sigifict figures. (). 6# 0. (b) 8. ' (c). +. 9#. 6 (d).. 6. Evlute. # 0 # 9. # 0, givig your swer i scietific ottio. 8. # 0. Clculte d write. # 0 your swer i stdrd form correct to sigifict figures. Ivestigtio A logrithm is ide. It is wy of fidig the power (or ide) to which bse umber is rised. For emple, whe solvig 9, the solutio is. The is clled the bse umber d the is the ide or power. You will ler bout logrithms i the HSC course. The is clled the bse umber d the is the ide or power. If y the log y. The epressio log 9 mes the power of tht gives 9. The solutio is sice 9.. The epressio log 6 mes the power of tht gives 6. The solutio is sice 6. C you evlute these logrithms?. log. log. log log 6. log 6. log. log 8. log 9. log 0. log
36 Chpter Bsic Arithmetic Absolute Vlue Negtive umbers re used i mths d sciece, to show opposite directios. For emple, tempertures c be positive or egtive. But sometimes it is ot pproprite to use egtive umbers. For emple, solvig c 9 gives two solutios, c!. However whe solvig c 9, usig Pythgors theorem, we oly use the positive swer, c, s this gives the legth of the side of trigle. The egtive swer does t mke sese. We do t use egtive umbers i other situtios, such s speed. I sciece we would tlk bout vehicle trvellig t 60k/h goig i egtive directio, but we would ot commoly use this whe tlkig bout the speed of our crs! Absolute vlue defiitios We write the bsolute vlue of s ) whe $ 0 whe 0 We c lso defie s the distce of from 0 o the umber lie. We will use this i Chpter. EXAMPLES. Evlute. Solutio sice $ 0 CONTINUED
37 8 Mths I Focus Mthemtics Etesio Prelimiry Course. Evlute. Solutio ]g sice 0 The bsolute vlue hs some properties show below. Properties of bsolute vlue b # b e.g. # # 6 e.g. ]g 9 e.g. e.g. b b e.g. + b # + b e.g. + + but + + EXAMPLES. Evlute +. Solutio Show tht + b # + b whe d b. Solutio LHS mes Left Hd Side. LHS + b +
38 Chpter Bsic Arithmetic 9 RHS + b + + Sice + b # + b RHS mes Right Hd Side.. Write epressios for without the bsolute vlue sigs. Solutio whe $ 0 i.e. $ $ ] g whe 0 + i.e. Clss Discussio Are these sttemets true? If so, re there some vlues for which the epressio is udefied (vlues of or y tht the epressio cot hve)? y + y $ 0 Discuss bsolute vlue d its defiitio i reltio to these sttemets.
39 0 Mths I Focus Mthemtics Etesio Prelimiry Course.9 Eercises. Evlute () (b) (c) 6 (d) 0 (e) (f) (g) (h) 8 (i) (j). Evlute () + (b) (c) + (d) # (e) + (f) # 6 (g) + # (h) (i) (j) +. Evlute b if () d b (b) d b (c) d b (d) d b (e) d b.. Write epressio for () whe 0 (b) whe 0 (c) whe 0 (d) whe 0 (e) whe 0 (f) whe 0 (g) + whe (h) + whe (i) whe (j) whe.. Show tht + b # + b whe () d b (b) d b (c) d b (d) d b (e) d b. 6. Show tht whe () (b) (c) (d) (e) 9.. Use the defiitio of bsolute vlue to write ech epressio without the bsolute vlue sigs () + (b) b (c) + (d) y 6 (e) + 9 (f) (g) k + (h) (i) + b (j) p q 8. Fid vlues of for which. 9. Simplify where! Simplify d stte which vlue cot be.
40 Chpter Bsic Arithmetic Test Yourself. Covert () 0. to frctio (b) % to deciml (c) 8 to deciml (d) 8.% to frctio (e) 0.0 to percetge (f) to percetge. Evlute s frctio. () (b) (c) 9. Evlute correct to sigifict figures. () (b). 0. (c) (d) (e).. # 0 8. # Evlute () (b) (c) + # 8 (d) [( + ) #( ) ] ' 8 (e) + 9 (f) (g) ' 6. Simplify () # ' (b) ( y ) (c) ( ) b 9 b 6 (d) d b (e) e o 6 b 0 6. Evlute () 8 6 (b) # (c) 9' (d) + 0 (e) # 6. Evlute () (b) 6 (c) (d) s frctio 8 (e) (f) (g) 9 s frctio (h) 6 (i) ]g 0 (j) 8. Simplify 9 () ' (b) _ y i 6 6 (c) p # p ' p (d) b 9 ^ h (e) ( ) y 0 y 9. Write i ide form. () (b) (c) + y (d) +
41 Mths I Focus Mthemtics Etesio Prelimiry Course (e) + b (f) (g) (h) (i) ( + ) (j) m 9 0. Write without frctiol or egtive idices. () (b) (c) ( + ) (d) ( y) (e) (t ) (f) ( + b) (g) (h) b (i) ( + ) (j). Show tht + b # + b whe d b.. Evlute b whe 9 d b.. If c m d b, evlute b s frctio.. Icrese 60 ml by 6%.. Joh speds of his hour dy sleepig d t work. () How my hours does Joh sped t work? (b) Wht frctio of his dy is spet t work or sleepig? (c) If he speds hours wtchig TV, wht frctio of the dy is this? (d) Wht percetge of the dy does he sped sleepig? 6. The price of cr icresed by %. If the cr cost $ 00 previously, wht is its ew price?. Rchel scored 6 out of 80 for mths test. Wht percetge did she score? 8. Evlute 8, d write your swer i scietific ottio correct to deciml plce. 9. Write i ide form. () (b) y 6 (c) + (d) ( ) (e) y 0. Write i scietific ottio. () (b) Covert to frctio. () 0. (b) 0.. Write without the egtive ide. () (b) ( + ) (c) c m b. The umber of people ttedig footbll mtch icresed by % from lst week. If there were 080 people t the mtch this week, how my tteded lst week?. Show tht + b # + b whe d b.
42 Chpter Bsic Arithmetic Chllege Eercise. Simplify c8 + m' c m Simplify Arrge i icresig order of size: %, 0.0, 0.,. 99. Mrk speds of his dy sleepig, of the dy etig d of the dy 0 wtchig TV. Wht percetge of the dy is left?. Write 6 s rtiol umber. 6. Epress. ' 0. 0 i scietific ottio correct to sigifict figures.. Vih scored out of 0 for mths test, 9 out of for Eglish d out of 0 for physics. Fid his verge score s percetge, to the erest whole percetge. 8. Write. s rtiol umber. 9. The distce from the Erth to the moo is 8. # 0 km. How log would it tke rocket trvellig t. # 0 km hto rech the moo, to the erest hour?. Show tht ( k ) k + ( k + + ).. Fid the vlue of i ide form if bc c m, b c m d c c m.. Which of the followig re rtiol umbers:, 0.,, r,., 0,?. The percetge of slt i L of wter is 0%. If 00 ml of wter is dded to this miture, wht percetge of slt is there ow?. Simplify + for!! Evlute deciml plces. correct to. Write g s percetge of. kg Evlute. +. # 0 correct to sigifict figures.. # 0 +. # 0 9. Evlute d ( 69. # 0) epress your swer i scietific ottio correct to sigifict figures. 0. Prove + b # + b for ll rel, b. 8. #. 0. Evlute '. sigifict figures. correct to
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