WORKING WITH NUMBERS

Transcription

1 1 WORKING WITH NUMBERS WHAT YOU NEED TO KNOW The defiitio of the differet umber sets: is the set of atural umbers {0, 1,, 3, }. is the set of itegers {, 3,, 1, 0, 1,, 3, }; + is the set of positive itegers; is the set of egative itegers. is the set of all ratioal umbers. A ratioal umber ca be writte as a fractio usig two itegers, where the deomiator must ot be zero. Examples iclude 1, ,,. 4 is the set of all real umbers. These iclude the atural umbers, itegers ad ratioal umbers, as well as irratioal umbers. A irratioal umber caot be writte as a fractio; examples iclude π, 3, si 60. Numbers ca be writte to a fixed umber of decimal places or sigificat figures: To roud to oe decimal place, look at the digit i the secod decimal place; to roud to two decimal places, look at the digit i the third decimal place. If that digit is less tha 5, roud dow; if the digit is 5 or more, roud up. To roud to three sigificat figures, look at the digit to the right of the third sigificat figure. If that digit is less tha 5, roud dow; if the digit is 5 or more, roud up. Very small ad very large umbers ca be expressed i stadard form (or scietific otatio) a 10 k where 1 a < 10 ad k is a iteger. νa ν E To fid the percetage error, use the formula ε = 100%, where ν E is the exact value ad ν νe A is the approximate value of ν. Covertig to a larger uit meas fewer of them, so divide. Covertig to a smaller uit meas more of them, so multiply. To chage oe currecy to aother, multiply by the appropriate exchage rate. Whe commissio is charged, first work out the amout of commissio paid. The exchage rate is the applied to the origial amout commissio paid. EXAM TIPS AND COMMON ERRORS The square root of ay umber that is ot a perfect square (e.g. 4 ) or the ratio of two 16 perfect squares (e.g. ) will be irratioal. May trigoometric ratios (e.g. si 45 ) are irratioal. 5 Zeros at the begiig of a decimal (e.g ) ad at the ed of a large umber (e.g ) do ot cout as sigificat figures. The first sigificat figure of a umber is the first o-zero digit i the umber, coutig from the left. 1 Workig with umbers 1

2 1.1 DIFFERENT TYPES OF NUMBERS WORKED EXAMPLE 1.1 Write dow a umber that is: (a) a real umber but ot ratioal (b) a ratioal umber ad i (c) a ratioal umber but ot i. (a) π π is a irratioal umber as it caot be writte as a fractio. (b) 3 All itegers are ratioal umbers, e.g. 3 = 3 1 is a fractio with deomiator 1. (c) 1 Practice questios Mark each cell to idicate which umber set(s) the umber belogs to. The first row has bee completed for you cos The Ve diagram shows the sets,, ad. (a) Add labels to show which regio correspods to each set. (b) Put the followig umbers ito the correct set: 3, 3, 3.3 ad π. 3. A set cotais the elemets x such that 4 x < 6, x. Each elemet is equally likely to occur. Oe elemet is draw at radom. Fid the probability that it is i: (a) (b). 1 Workig with umbers

3 1. STANDARD FORM AND SI UNITS WORKED EXAMPLE 1. If x = cm ad y = km: (a) write x ad y i metres (b) fid xy, givig your aswer i metres squared i the form a 10 k where 1 a < 10 ad k. 1 km = 1000 m, 1 cm = 0.01 m, 1 mm = m 1 kg = 1000 g, 1 mg = g 1 m = 100 cm 100 cm = cm 1 m 3 = 100 cm 100 cm 100 cm = cm cm (a) x = = cm/m y = km 1000 m/km = m 10 m Covertig to a larger uit meas fewer of them, so divide. Covertig to a smaller uit meas more of them, so multiply. (b) xy = m m Make sure that you ca covert umbers to = 1104 m m stadard form o your calculator. However, you must ot write calculator otatio, such as 10E3, i your aswer. Practice questios Give that a = ad b = , calculate the followig, givig your aswer i the form c 10 k where 1 c < 10 ad k : (a) ab (b) a b 5. Give that x = g ad y = 0.98 hours, calculate x, givig your aswer i kg per secod i the form a 10 k where 1 a < 10 ad k. y 6. A room measures 3.1 m by 4.4 m. Fid the area of the room i: (a) m (b) cm (c) Give your aswer to (b) i the form a 10 k where 1 a < 10 ad k. If umbers are very big or very small, the GDC gives the aswers i stadard form automatically. 1 Workig with umbers 3

4 1.3 APPROXIMATION AND ESTIMATION WORKED EXAMPLE 1.3 (a) Write dow correct to two decimal places. I a exam questio, if a specifi c degree of (b) Write dow correct to the earest te. accuracy is ot asked for, give your aswer (c) Write dow correct to two sigificat figures. correct to three sigifi cat fi gures. (d) Calculate the percetage error if is give correct to two sigificat figures. (a) To roud to decimal places, look at the digit i the third decimal place, which is 4. It is less tha 5, so roud dow. (b) 0 Remember that 0 is a multiple of te. (c) 1.4 Fid the secod sigifi cat fi gure ad look at the digit after it. As 1 < 5, roud dow. νa νe 14. (d) ε = 100% = ν E = 1. 01% ( 3 SF) 100% Substitute the rouded value, ν A = 1.4, ad the exact value, ν E =, ito the formula for percetage error. The modulus sig simply meas that we remove ay egative sig which occurs. Practice questios (a) Write π correct to three decimal places. (b) Fid the percetage error whe π is give correct to three decimal places. 8. (a) Write dow 46 ad π correct to three sigificat figures. (b) Write dow the value of π 46, givig all the digits show o your calculator. (c) Write dow the value of π 46 usig the approximate values foud i part (a), givig all the digits show o your calculator. (d) To how may sigificat figures is your result i part (c) correct? (e) What is the percetage error i your aswer to part (c)? 4 1 Workig with umbers

5 1.4 CURRENCY CONVERSIONS WORKED EXAMPLE 1.4 The table shows the exchage rates for US dollars ad euros:? USD? EUR (a) Fid the value of p to two decimal places. 1 USD (b) What is the value of $150 i euros? 1 EUR p 1 (c) Jamie chages 300 ito dollars. She is charged 6% commissio. How may dollars does she receive? (a) 1 USD = 0.78 EUR USD = 1 EUR so p = = 1.8 DP ( ) We treat this as a equatio ad divide both sides by (b) $150 = EUR = 117 EUR Use the exchage rate 1 USD = 0.78 EUR. (c) = 18 EUR = 8 EUR = USD First work out the amout of commissio paid. The exchage rate is the applied to the remaiig amout. Use the full accuracy of the coversio rate, ot a rouded result. Practice questios The table gives the coversio rates betwee British pouds (GBP) ad Australia dollars (AUD). Fid x, ad hece fid the umber of pouds which ca be bought with 1000 AUD if there is a 4% commissio.? GBP? AUD 1 GBP AUD x A currecy office buys oe South Africa rad (SAR) for Japaese ye ad sells oe SAR for ye. (a) Blaise coverts 1000 SAR to ye for a holiday. While o holiday he speds half of this moey. O his retur he coverts the remaider back to SAR. How may SAR will he get back? (b) Robbie also coverts 1000 SAR to ye for a holiday. He the cacels his holiday ad chages all the ye back to SAR. How much has Robbie lost after the two trasactios? Express your aswer as a percetage of Robbie s origial 1000 SAR. 1 Workig with umbers 5

6 Mixed practice 1 1. If x = ad y = : (a) write y to four decimal places (b) calculate xy, givig your aswer i Decimal form meas as a ordiary umber. decimal form y (c) calculate x, givig your aswer i the form a 10k where 1 a < 10 ad k.. Look at this list of umbers: , 100π, 00, , (a) Which of these umbers is largest? (b) List all the umbers which are members of: (i) (ii). (c) What is the largest umber of sigificat figures to which all five umbers are equal? 3. A farmer wats to plat a ew forest i a field coverig a rectagular area of 3 km by 5 km. (a) Fid the area of the forest i m, givig your aswer i the form a 10 k where 1 a < 10 ad k. (b) Each tree eeds a area of 3.4 m. Fid the maximum umber of trees which could be plated. Give your aswer to the earest thousad trees. 4. A travel aget coverts dollars ad ye at a exchage rate of $1 = 103 ye. They charge 5% commissio o all trasactios. (a) If Dima coverts ye to dollars, how much will he receive? (b) If he coverts $500 to ye, how much will he receive? 5. (a) Write i the form a 10 k where 1 a < 10 ad k. (b) Write to two sigificat figures. (c) What is the percetage error whe roudig to two sigificat figures? 6. Nicole wats to covert pouds to euros. She ca choose betwee two differet offers. Offer 1: a exchage rate of 1 poud to 1.6 euros with o commissio. Offer : a exchage rate of 1 poud to 1.30 euros with 5% commissio. Which offer provides Nicole with the better deal? 6 1 Workig with umbers

7 7. Mark each cell to idicate which umber set(s) the umber belogs to. The first row has bee completed for you. Number 5 0 ta 45 ta π 8. The Earth ca be modelled by a perfect sphere with a radius of 6700 km. (a) Fid the volume of the Earth i km 3. (b) Fid the volume of the Earth i cm 3, givig your aswer i the form a 10 k where 1 a < 10 ad k. The formula for the volume of a sphere is give i the Formula booklet. (c) If the average desity of the earth is 6.7 g/cm 3, fid the mass of the Earth. Goig for the top 1 1. The value of x is quoted as 0 to the earest 10. The value of y is quoted as 1.6 to two sigificat figures. (a) Write a iequality i the form a x < b showig the rage that x ca lie i. (b) What is the smallest value 3x 7 could take? (c) Fid the largest possible percetage error if x is quoted as beig 1.5. y. The table below shows the values of differet currecies compared to oe US dollar: GBP CHF EUR JPY AUD 1 USD (a) How may US dollars ca you get with oe British poud (GBP)? (b) What is the exchage rate for euros (EUR) to Japaese ye (JPY)? (c) Camille coverts 1000 US dollars to British pouds, the Swiss fracs (CHF), the euros, the Japaese ye, the Australia dollars, ad the back to US dollars. For each trasactio, she pays 5% commissio. How much does she have left at the ed, to the earest US dollar? 1 Workig with umbers 7

8 SEQUENCES AND SERIES WHAT YOU NEED TO KNOW The otatio for sequeces ad series: u represets the th term of the sequece u. S deotes the sum of the first terms of the sequece. A arithmetic sequece has a costat differece, d, betwee terms. If the first term is u 1, the: u = u 1 + ( ) d S = [ u + 1) d ] = ( u + u ) A geometric sequece has a costat ratio, r, betwee terms: 1 u ur 1 u1(rr S = 1 ) u1( 1 r = ), r 1 r 1 1 r Practical problems ivolvig growth or decay, such as questios about iterest or depreciatio, ca be solved usig geometric sequeces. k r The formula for compoud iterest is FV = PV + 1 k 100 where FV = future value, PV = preset value, = umber of years, k = umber of compoudig periods per year, r% = omial aual rate of iterest. EXAM TIPS AND COMMON ERRORS Questios ofte cotai parts askig you to fid a term of a sequece ad a sum of a sequece. May questios o sequeces ad series ivolve formig ad the solvig simultaeous equatios. You may eed to use the list feature o your calculator to solve problems ivolvig sequeces. You oly ever eed to use the first sum formula for geometric sequeces. u u3 Sometimes it is useful to remember that u u1 u 3 u = ad r = = = u1 u Always check whether a questio is askig for the iterest or the total amout. Your calculator may have a fiace applicatio which ca be helpful. 8 Sequeces ad series

9 .1 ARITHMETIC SEQUENCES AND SERIES WORKED EXAMPLE.1 The third term of a arithmetic sequece is 15 ad the sixth term is 7. (a) Fid the teth term. (b) Fid the sum of the first te terms. (c) The sum of the first terms is 550. Fid the value of. (a) u3 = u 1 + d 15 (1) u6 = u d 7 () () (1) 3d = 1, so d = 4 u 1 = 15 4 = 7 Hece u10 = u 1 + 9d = = (b) S u 1 + u = 5( = 50 ( ) 10 1 u1 0 ( ) (c) 550 = ( 1 ( 1) ) = ( + ( )) From GDC, the solutios are = 50 or = 5.5; but must be a positive iteger, so = 50. We eed to fi d the fi rst term ad the commo differece. Write dow the give iformatio i the form of simultaeous equatios ad the solve. There are two formulae for the sum of a arithmetic sequece. Sice we kow the fi rst ad last terms, we use S = ( u + u ). u We eed to fi d, which will be the oly ukow i the other sum formula. Form a equatio ad solve it usig a GDC. Practice questios.1 1. The fifth term of a arithmetic sequece is 8 ad the eighth term is 17. (a) Fid the twetieth term. (b) Fid the sum of the first twety terms.. The first four terms of a arithmetic sequece are 8, 7.5, 7, 6.5. (a) Fid the teth term. (b) Which term is equal to zero? (c) The sum of the first terms is equal to 65. Fid the value of. Sequeces ad series 9

10 . GEOMETRIC SEQUENCES AND SERIES WORKED EXAMPLE. Cosider the geometric sequece, 6, 18, (a) Write dow the commo ratio. (b) Which is the first term whose value exceeds 1000? (c) Fid the sum of the first 10 terms. (a) r = 3 (b) The sequece cotiues as, 54, 16, 486, 1458, So the 7th term is the first to exceed ( ) = (c) S 10 = Write dow meas that little or o calculatio should be ecessary. You do ot eed to show your workig. 1 Use the rule u = u 1 r with u 1 = ad r = 3, i.e. u = 3 1, ad the list fuctio o your GDC to exted the sequece. This could also be doe usig a graph or table. We ca use the formula for the sum of a u geometric sequece: 1( r S = 1 ). r 1 Practice questios. 3. Cosider the geometric sequece 5, 10, 0, 40,... (a) Fid the teth term. (b) What is the value of the first term to exceed 3000? (c) Fid the sum of the first twelve terms. 4. The fifth term of a geometric sequece is 18 ad the sixth term is 51. (a) Fid the commo ratio ad the first term. (b) Which term has a value of 3 768? (c) How may terms are eeded before the sum of all the terms i the sequece exceeds ? 5. The sum of the first six terms of a geometric sequece is 1365 times its first term. Fid the commo ratio. 10 Sequeces ad series