Numbers and number relationships LO Below is an overview of the work overed in this setion. Numbers and alulating UNIT Types of number page 2 Signifiant figures, deimal plaes and rounding page 4 Approximation and estimation page 7 Calulator work page 9 Exponents page Conversions Exhange rates page 25 Measures page 28 Other onversions page 5 Simple interest Perentages page 44 Calulating simple interest page 48 A note on alternative formulae page 52 Pattern Conjetures and ounterexamples page 62 General terms page 65 Linear patterns page 69 Non-linear patterns page 76 Compound interest and other exponential relationships Compound interest page 87 Appliations of ompound interest page 96 Exponential growth and deay page 00 2 February 2002 8 February 2002 From time to time areas of South Afria s oast witness red tides. A red tide is an exponential growth of ertain algae (usually red) in the sea. The algae use up muh of the available oxygen in the water and this often leads to the death of large numbers of fish. The photographs show the growth of a red tide near Cape Town between 2 February and 8 February 2002.
2 Numbers and number relationships LO Unit Numbers and alulating LO AS., AS.2, AS.6 In this setion, we look at different types of number, signifiant figures, deimal plaes, rounding, approximation and estimation, alulator work and exponents. Types of number The diagram gives a summary of the different types of number and a few examples of eah type of number. The diagram shows: all natural numbers are integers all integers and natural numbers are rational numbers rational numbers and irrational numbers make up real numbers. 27 2 20 Integers (Z) 0 Natural numbers (N) 5 2 4 4 4 5 45 4 Rational numbers (Q) 0 0 7 7 Real numbers (R) 4 25 22 7 4 5 Irrational numbers (Q') π 0 The different types of number inlude the following numbers: natural numbers (N) are ounting numbers (Some people inlude 0 with the natural numbers.) integers (Z) are whole numbers rational numbers (Q) are numbers that an be written as a fration a _, b where a and b are integers and b 0 irrational numbers are numbers that annot be written as a fration rational and irrational numbers make up real numbers. How to onvert a reurring deimal into a fration Convert the reurring deimal 0,6 into a fration. 0,6 = 0,6666666... Write out the deimal. 0 0,6 = 6,666666... 2 Multiply the same number of 0s as there are reurring deimals. So, 0 0,6 0,6 = 6,666666... 0,6666666... Subtrat 2. = 6 9 0,6 = 6 Divide by 9. 0,6 = 6 _ 9 = 2 _
Unit Numbers and alulating Examples Whih numbers are rational and whih are irrational? a _ b 5 d,47 2 e π f 2 g 4 2 Show that eah number is rational. a 4 b 0,587,0 9 Solutions a _ 2 : rational b = _ : rational 5 = 5 _ : rational,47 000 47 d,47 = = : rational 000 000 e f π: irrational 2 : irrational g 4 = 2 = 2 _ : rational 2 a 4 = 4 _ b 0,587 = 587 000,0 9 =,090909... 00,0 9 = 09,0909...2 There are two reurring digits, so 00,0 9,0 9 = 08 99,0 9 = 08 08,0 9 = 99 multiply by 00. Subtrat: 2. = 2 Pratie questions. Use the terms real; rational; integer; natural and irrational to identify the numbers. You may use more than one term for eah number. a 0 b 5 0,2 d 0,67 e 5_ f 0,0925 g 9,4 5 h 5 8 2 Show that the following numbers are rational. a 5 b 2 5 _ 8,492 d 0,8 e 8,4 6 f,09 4
4 Numbers and number relationships LO Between whih two integers does eah irrational number lie? a 6 b 28 2 d 8 e 4 58 f 40 < x < 4 means that x is bigger than but smaller than 4 4 Find a number for x in the interval < x < 4 so that: a x is rational b x is an integer x is real, but not rational d x is a natural number. Signifiant figures, deimal plaes and rounding Do you remember what is meant by signifiant figures, plae markers and rounding? Rules for rounding Look at the value of the digit to the right of the digit speified. If the value is 5 or higher, add one to the digit speified. If the value is lower than 5, leave the speified digit unhanged. Rules for signifiant figures Read the number from left to right. The first non-zero digit is the first signifiant figure. Any digit (inluding zero (0)) after the first signifiant figure is also a signifiant figure. Plae markers (plae value) Eah digit in a number has a meaning aording to the number system. Numbers to the left of the deimal omma measure units, tens, hundreds, thousands, and so on. Numbers to the right of the deimal omma measure tenths, hundredths, thousandths, and so on. Examples Round off,26 to two deimal plaes. 2 Write 8 26,58 to three signifiant figures. Round off 2 46 to the nearest hundred. 4 Write 0,08284 to two signifiant figures. 5 Write 50 042 to three signifiant figures. 6 Round off 0,045 to the nearest thousandth. 7 Write 0,05075 to three signifiant figures. 8 Write 0,048 to two deimal plaes.
Unit Numbers and alulating 5 Solutions,26 =, to two deimal plaes. The value in the seond deimal plae is 2. It is rounded up to beause the number that follows it (6) is higher than 5. 2 8 26,58 = 8 0 to three signifiant figures. The value of the third signifiant figure is 2. It is rounded up to beause the number that follows it (6) is higher than 5. 2 46 = 2 400 to the nearest hundred. The 4 is in the plae for hundreds, as it is the third plae to the left of the deimal omma. It is not rounded up beause the number that follows it () is less than 5. 4 0,08284 = 0,08 to two signifiant figures. Zeroes immediately to the right of the deimal omma in this ontext are not signifiant. The 2 is rounded up to beause the number that follows it (8) is higher than 5. 5 50 042 = 50 000 to three signifiant figures. The third signifiant figure is the seond zero (0). It is not rounded up beause the number that follows it is less than 5. Zeroes to the left of the deimal omma are always signifiant. 6 0,045 = 0,05 to the nearest thousandth. The 4 is in the plae for thousandths, as it is the third plae to the right of the deimal omma. It is rounded up to 5 beause the number that follows it is 5. 7 0,05075 = 0,0508 to three signifiant figures. A zero immediately to the right of the deimal omma is signifiant in this ase, but zeroes in a deimal are signifiant. The third signifiant figure is 7. It is rounded up to 8 beause the number that follows it is 5. Sometimes d.p. and s.f. are used as abbreviations for deimal plae and signifiant figure. 8 0,048 = 0,0 to two deimal plaes. There is a zero in the seond deimal plae. It is not rounded up beause the number that follows it (4) is less than 5. The zero (0) must be inluded in the answer.
6 Numbers and number relationships LO Pratie questions.2 Write eah number orret to two deimal plaes. a,8690 b 207,0428 0,9 d 0,20528 e 2 95,8449 f 0,0028 g 0,0972 h 0,9999 i 99,997 2 Write eah number orret to three signifiant figures. a 4 02 b 26,45 24,905 d 478 629,09 e 0,086 f 2,007 g 0,00285 h 50,042 i 0,00002 Write the number 25 67,02045 orret to: a three signifiant figures b two deimal plaes four deimal plaes d four signifiant figures e three deimal plaes f two signifiant figures. 4 Write eah number orret to the number of deimal plaes or signifiant figures indiated. a b d e f g h i j k l 0,0058 to two signifiant figures 59,45 to three signifiant figures 26,94 to one signifiant figure 5,97 to two deimal plaes 7,04 to two deimal plaes 2,95 to two deimal plaes 25 97 to two signifiant figures 0,52 to three deimal plaes 0,005792 to three signifiant figures 76 82 to three signifiant figures 2 998 to two signifiant figures 25,572 to three signifiant figures m,786 to four signifiant figures n o p q r 0,429 to two signifiant figures,45942 to three deimal plaes 8,24 to three signifiant figures 45,507 to three deimal plaes 20,04 to three signifiant figures
Unit Numbers and alulating 7 Approximation and estimation Work through the examples to see if you remember how to do approximation and estimation. Examples Estimate the answer to eah problem. 27, 729 8, 68 +, 25 a 8,46 5,87 b 7, 04, 0452 2 4, 62, 756 Without using a alulator, estimate the value of: 8, 2 + 5, 9 Solutions You an obtain an estimated answer by rounding off eah number to make the alulation easier. Usually we use one or two signifiant figures. a 8,46 5,87 0 6 60 b 27, 729 28 7, 04 7 4 8, 68 +, 25, 0452 0 0 20 + 0 4, 62, 756 4 4 2 8, 2 + 5, 9 8 + 6 6 24 2 _ (0,67 to two deimal plaes) Compare this with the alulated answer of 0,65 (to two deimal plaes). Pratie questions. Use rounding off to estimate the answers. a 5,86,27 b 6,5,46 28,84 d 2,72 2,4 e,92 04,7 f 5,79 22,47 g 286, 7 92 27, 45 h 7, 2 i 947 58, 8 5, 58 j 0, 585 k 2, 7, 42 695, l 0, 68
8 Numbers and number relationships LO 2 Estimate the answer to eah alulation. a π 6,4 2 b 4, 8 2, 5, 69 024, 74, 9, 8 + 54, 5 47, d 2, 6 e 42, 62 + 99, 85 796 + 72 2, 67 f 48, 4 g 857 +, 05, 5, 68 9, 47 h 2, 24 i 25, 75 + 86, 82 5, 7 9, 47 27, + 0, j 0, 2 96 Now use your alulator to find the answer to eah alulation, orret to two deimal plaes. Exerise books ost R7,9 eah. a Estimate the ost of eight exerise books. b Calulate, to the nearest 0, the ost of eight exerise books. 4 Bhule runs a ross-ountry rae at an average speed of 6,24 m/s. a Estimate how far Bhule would have run after six minutes. b Estimate the time it takes Bhule to run 467 m. 5 The learners in a Grade 0 lass sit in a line, one behind the other. Eah learner needs 4 m of spae. a Estimate the length of the line if there are no gaps between the learners. b If there is a spae of 5 m between learners, estimate the length of the line. 6 A retangle has sides of 2,7 m and 5,84 m. a Estimate the perimeter of the retangle. b Estimate the area of the retangle. Calulate the area of the retangle, orret to three signifiant figures. 7 Tanya is plaing square ards on a tray. The ards have sides of length 4,2 m. Estimate and then alulate how many ards she will use to over the tray, if the tray measures 56,7 m 88,2 m. 8 A learner bought 8 pakets of sweets for her lass. Eah paket osts R,87. She shared the sweets out among learners. She alulates that it ost her R,57 per learner. a Without using a alulator, use estimation to hek whether this answer is approximately orret. b Use a alulator to find the orret ost per learner, to the nearest ent.
Unit Numbers and alulating 9 Use BOMDAS to help you remember the orret order of operations. B brakets O of M multiply D divide A add S subtrat Calulator work The keystrokes used in this setion will work for most alulators. The exat order of keystrokes may be different on your alulator and you may find sequenes for your alulator that use fewer keystrokes. Use brakets when neessary to make sure a alulation will be orret. Examples Use your alulator to find the answers. a + 4 2 b 596, + 87, 85, 62, 2, + 8, d 5 π 2 + 7 2 Calulate the value of the fration below to the nearest integer. 8 000 24 + 00 ( ) Solutions If you are not given instrutions about the auray of your answer, give your answer to two deimal plaes or three signifiant figures. a Use to obtain an answer of. Calulators follow the BOMDAS rule, so there is no need to insert brakets. b Use to obtain an answer of 4,27 to two deimal plaes. Beause your alulator follows the BOMDAS rule, it would divide,87 by 8,5 before doing the addition and subtration. This is why brakets are neessary in this problem. Use them to group the top line and the bottom line of the fration. Use to obtain an answer of 0,775 to three signifiant figures. d Use to obtain an answer of 2,96 to two deimal plaes. Use the or key on your alulator to obtain π and x. On some alulators, you an press the key or key to obtain 5. 2 Use to obtain an answer of 958,05..., whih is 958 to the nearest integer.
0 Numbers and number relationships LO Memory funtion The memory funtion on your alulator makes it possible for you to store answers to alulations so that you an use them again later. This saves time and also helps you work more aurately. You should know the following keys: adds a number to the number stored in the memory subtrats a number from the number stored in the memory stores a number in the alulator memory displays the number stored in memory lears the memory. Some alulators also have an most reent alulation only. button that displays the answer for the Examples A afé orders the fizzy drinks shown in the table. Number ordered Size Prie 8 2 l R,5,5 l R9,27 47 500 ml R6,25 6 50 ml R4,50 a b Determine the total ost of the afé s order. Calulate the total volume of fizzy drinks on order. 2 A ompany makes vitamin supplements. It needs to divide 250 ml of vitamin B solution into 7 doses. a Calulate how muh solution there will be in eah dose, orret to two deimal plaes. b The ompany pakages its vitamin supplements in boxes of 250 doses. How muh vitamin B solution should be ordered to make up ten boxes? Jafat is investigating a number pattern, whih is given by the formula below. n + 4 n He starts with n = and alulates the answer. He then uses his answer from the formula and lets this answer be n and puts it bak into the formula. He repeats this proess, substituting five times altogether. Write down Jafat s five answers, rounding off to three signifiant figures where neessary.