m31126 To differentiate between rational and irrational numbers 1.1 2009/08/08 12:57:30.687 GMT-5 2009/08/08 13:07:32.103 GMT-5 gert bezuidenhout gert bezuidenhout gertb@mweb.co.za gert bezuidenhout gert bezuidenhout gertb@mweb.co.za gert bezuidenhout gert bezuidenhout gertb@mweb.co.za Mathematics and Statistics en

1. Can you remember what each of the following represents? N = { ........................................................................... } N0_{= { ........................................................................... }} Z = { ........................................................................... } R = { ........................................................................... } 2. Provide the definition for: a rational number: an irrational number: 3. How would you represent each of the following? 3.1 Rational number......................... 3.2 Irrational number ......................... 4. Complete the following table by marking relevant numbers with an X:

5. Select the required numbers from the list: −23 size 12{ { { size 8{ - 2} } over { size 8{3} } } } {}; 1 + 4 size 12{ sqrt {4} } {} ; 9+4 size 12{ sqrt {9+4} } {} ; -4 ; 1215 size 12{"12" { { size 8{1} } over { size 8{5} } } } {} ; 1+22 size 12{ { {1+ sqrt {2} } over { sqrt {2} } } } {} 5.1 Integers: 5.2 Rational numbers: 5.3 Irrational numbers: 6. Explain what you know about an equivalent fraction. 7. Provide two equivalent fractions for the following: 27 size 12{ { { size 8{2} } over { size 8{7} } } } {} = ............... = ............... 8. Provide the terms used to identify each of the following (e.g. proper fraction): 8.1 27 size 12{ { { size 8{2} } over { size 8{7} } } } {} 8.2 72 size 12{ { { size 8{7} } over { size 8{2} } } } {} 8.3 627 size 12{6 { { size 8{2} } over { size 8{7} } } } {} 8.4 0,67 8.5 0,6˙7˙ size 12{0, { dot {6}} { dot {7}}} {} 8.6 23 % Any of the above can be reduced to any of the others.

Use your pocket calculator to reduce the following fraction to a decimal number:

2. Explain how you would reduce this to a decimal number without the use of your pocket calculator. There are two methods: Method 1: .................................................. (reduce denominator to 10 / 100 / 1 000) Method 2: .................................................. (do division) (Let your educator assist you.) Do you see that the answer is the same – if the denominator cannot be reduced to multiples of 10 you have to apply the second method. 3. Now reduce each of the following to decimal numbers (round off, if necessary, to two digits): 3.1 58 size 12{ { { size 8{5} } over { size 8{8} } } } {} .................................................. 3.2 134 size 12{ { { size 8{"13"} } over { size 8{4} } } } {} .................................................. 3.3 534 size 12{5 { { size 8{3} } over { size 8{4} } } } {} .................................................. 3.4 378 size 12{3 { { size 8{7} } over { size 8{8} } } } {} .................................................. 3.5 67 size 12{ { { size 8{6} } over { size 8{7} } } } {} .................................................. 3.6 79 size 12{ { { size 8{7} } over { size 8{9} } } } {} .................................................. 4. Write the following decimal numbers as fractions or mixed numbers:(N.B.: All fractions have to be presented in their simplest form.) 4.1 6,008 .................................................. 4.2 4,65 .................................................. 4.3 0,375 .................................................. 4.4 7,075 .................................................. 4.5 13,65 .................................................. 4.6 0,125 .................................................. 5. How do we reduce fractions to recurring decimal numbers? E.g. 511 size 12{ { { size 8{5} } over { size 8{"11"} } } } {} Step 1: place a comma after the 5, i.e. 5, 0000 Step 2: carry on dividing until a pattern becomes visible - the pattern will be indicated by the recurring numbers.

Now try the following: 5.1 79 size 12{ { { size 8{7} } over { size 8{9} } } } {} 5.2 −556 size 12{ - 5 { { size 8{5} } over { size 8{6} } } } {} 5.3 31399 size 12{3 { { size 8{"13"} } over { size 8{"99"} } } } {} 6. What is noticeable about fractions that are recurring decimal numbers (with regard to the denominator)? 7. Now, before we provide the steps for reducing a recurring decimal number to a common fraction, see if you are able to write the following as fractions by making use of the information from no. 6.

8. The following provides complete steps for reducing a recurring decimal number to a common fraction:

Suggestion: Multiply by 10 (if you have one recurring figure). Multiply by 100 (if there are 2 recurring figures), etc. 9. Now try to do no. 7.2 in the way that is discussed in no. 8.

1. What is the meaning of % (percentage)? ..................................................................... 2. If you have to reduce any fraction to a percentage, you have to reduce the denominator to 100. If this is not possible, you have to x

(This principle can be applied in any situation, e.g. when you want to reduce a test that is marked out of 15 to a mark out of 50, you need to multiply by 501 size 12{ { { size 8{"50"} } over { size 8{1} } } } {}) Reduce the following mathematics test marks from a grade 8 class to percentages (to one decimal figure, where necessary): 2.1 1720 size 12{ { { size 8{"17"} } over { size 8{"20"} } } } {} ....................................... 2.2 1940 size 12{ { { size 8{"19"} } over { size 8{"40"} } } } {} ....................................... 2.3 3850 size 12{ { { size 8{"38"} } over { size 8{"50"} } } } {} ....................................... 2.4 4560 size 12{ { { size 8{"45"} } over { size 8{"60"} } } } {} ....................................... 3. Reduce each of the following percentages to a common fraction (or a mixed number): 3.1 55 % ....................................... 3.2 15,5% ....................................... 3.3 16 12 size 12{ { { size 8{1} } over { size 8{2} } } } {}% ....................................... 3.4 623 size 12{6 { { size 8{2} } over { size 8{3} } } } {}% ....................................... 4. Each South African citizen should have access to some means of transport. Bolokanang has a community of 25 500 people. Study the accompanying table indicating the number of people that use the given means of transport and answer the questions that follow. Vehicle Number of users Bicycle 418 size 12{4 { { size 8{1} } over { size 8{8} } } } {}% Car 3 5 size 12{ { { size 8{3} } over { size 8{5} } } } {} Motorbike 0,085 4.1 Indicate how many inhabitants make use of: a) a bicycle b) a car c) a motorbike 4.2 Express the number of inhabitants that use a car as a fraction of those who travel by bicycle. 4.3 Which percentage of the inhabitants has no vehicle? 4.4 Which other means of transport do farm labourers use to get to the nearest town? 4.5 If the number of job opportunities in rural areas should increase, the fraction of citizens who use cars for transport will double. What fraction of the community will be using cars for transport under such conditions?

1. Reduce each of the following compound numbers to improper fractions.This is very important in addition, subtraction, multiplication and division of fractions. 1.1 5 47 size 12{ { { size 8{4} } over { size 8{7} } } } {} ................................ 1.2 7 79 size 12{ { { size 8{7} } over { size 8{9} } } } {} ................................ 2. What is of cardinal importance before attempting to add or subtract fractions? 3. Show whether you are able to do the following: 3.1 8 - 4 37 size 12{ { { size 8{3} } over { size 8{7} } } } {} 3.2 3 19 size 12{ { { size 8{1} } over { size 8{9} } } } {} - 1 12 size 12{ { { size 8{1} } over { size 8{2} } } } {} Note this: The denominators must be similar when you add fractions together or subtract them from one another. e.g. 2 47 size 12{ { { size 8{4} } over { size 8{7} } } } {} - 1 67 size 12{ { { size 8{6} } over { size 8{7} } } } {} 2 – 1 = 1 and 47 size 12{ { { size 8{4} } over { size 8{7} } } } {} - 67 size 12{ { { size 8{6} } over { size 8{7} } } } {} ( 4 – 6 --- this is not possible. Carry one whole: 1 = 77 size 12{ { { size 8{7} } over { size 8{7} } } } {}) ( 4 + 7 = 11 --- yes, 11 – 6 = 5) Answer: 57 size 12{ { { size 8{5} } over { size 8{7} } } } {} You could also reduce compound numbers to improper fractions and make the denominators similar. e.g.. 187−137=57 size 12{ { { size 8{"18"} } over { size 8{7} } } - { { size 8{"13"} } over { size 8{7} } } = { { size 8{5} } over { size 8{7} } } } {} (18 – 13 = 5: The denominators are the same. Subtract one numerator from the other.) 4. Do the following: 4.1 4 17 size 12{ { { size 8{1} } over { size 8{7} } } } {} + 4 1642 size 12{ { { size 8{"16"} } over { size 8{"42"} } } } {} 4.2 36 － 15 611 size 12{ { { size 8{6} } over { size 8{"11"} } } } {} 4.3 18+0,625−38 size 12{ { { size 8{1} } over { size 8{8} } } +0,"625" - { { size 8{3} } over { size 8{8} } } } {} 4.4 4510+712+634 size 12{4 { { size 8{5} } over { size 8{"10"} } } +7 { { size 8{1} } over { size 8{2} } } +6 { { size 8{3} } over { size 8{4} } } } {} 4.5 7 13 size 12{ { { size 8{1} } over { size 8{3} } } } {} - 4 78 size 12{ { { size 8{7} } over { size 8{8} } } } {} 4.6 7a - a4 size 12{ { { size 8{a} } over { size 8{4} } } } {}a/₄ 4.7 9a+6ab−3b size 12{ { { size 8{9} } over { size 8{a} } } + left ( { { size 8{6} } over { size 8{ ital "ab"} } } - { { size 8{3} } over { size 8{b} } } right )} {} 4.8 - 6 + 2 67 size 12{ { { size 8{6} } over { size 8{7} } } } {} 4.9 5 - (4 49 size 12{ { { size 8{4} } over { size 8{9} } } } {} + 2 23 size 12{ { { size 8{2} } over { size 8{3} } } } {}) 4.10 3 13 size 12{ { { size 8{1} } over { size 8{3} } } } {}a- 2 12 size 12{ { { size 8{1} } over { size 8{2} } } } {}a

You did this in grade 7 – let's refresh the memory. 1. Multiplication: Important: Write all compound numbers as fractions.Then do crosswise cancellation. Try the following: 1 14 size 12{ { { size 8{1} } over { size 8{4} } } } {} × 2 12 size 12{ { { size 8{1} } over { size 8{2} } } } {} × 4 2. Division: The reciprocal plays an important role in the division of fractions. Use an example to explain this term. e.g. 13÷23 size 12{ { { size 8{1} } over { size 8{3} } } div { { size 8{2} } over { size 8{3} } } } {} Both numbers are fractions Change ÷ to the × sign and obtain the reciprocal of the denominator (fraction following the ÷ sign). Do cancellation as with multiplication. 3. Do the following: 3.1 8 ÷ 811 size 12{ { { size 8{8} } over { size 8{"11"} } } } {} 3.2 18 ÷ 78 size 12{ { { size 8{7} } over { size 8{8} } } } {} 3.3 56÷52 size 12{ { { size 8{5} } over { size 8{6} } } div { { size 8{5} } over { size 8{2} } } } {} 3.4 -2 23 size 12{ { { size 8{2} } over { size 8{3} } } } {} ÷ -1 79 size 12{ { { size 8{7} } over { size 8{9} } } } {} 3.5 6 34 size 12{ { { size 8{3} } over { size 8{4} } } } {}mn ÷ -6 m³ 3.6 −4xy3ab÷−2x3a size 12{ { { size 8{ - 4 ital "xy"} } over { size 8{3 ital "ab"} } } div { { size 8{ - 2x} } over { size 8{3a} } } } {}^-

Learning outcomes(LOs) LO 1 Numbers, Operations and RelationshipsThe learner will be able to recognise, describe and represent numbers and their relationships, and to count, estimate, calculate and check with competence and confidence in solving problems. Assessment standards(ASs) We know this when the learner: 1.2 recognises, classifies an represents the following numbers to describe and compare them: 1.2.2 decimals, fractions and percentages; 1.2.5 additive and multiplicative inverses; 1.2.6 multiples and factors; 1.2.7 irrational numbers in the context of measure­ment (e.g. π size 12{π} {}and square and cube roots of non-perfect squares and cubes); 1.3 recognises and uses equivalent forms of the rational numbers listed above; 1.6 estimates and calculates by selecting and using operations appropriate to solving problems that involve: 1.6.1 rounding off; 1.6.2 multiple operations with rational numbers (including division with fractions and decimals); 1.7 uses a range of techniques to perform calculations, including: 1.7.1 using the commutative, associative and distributive properties with rational numbers; 1.7.2 using a calculator; 1.9 recognises, describes and uses: 1.9.1 algorithms for finding equivalent fractions; 1.9.2 the commutative, associative and distributive properties with rational numbers (the expecta­tion is that learners should be able to use these properties and not necessarily to know the names of the properties).

ACTIVITY 1 1. Natural numbers Counting numbers Integers Real numbers 2. ab size 12{ { {a} over {b} } } {}; b ≠ 0 2 size 12{ sqrt {2} } {} 3.1 Q Q ¹ 4. size 12{ {2} wideslash {7} } {} 0 1 size 12{ sqrt {1} } {} 3 size 12{ sqrt {3} } {} 9 3 size 12{ nroot { size 8{3} } {9} } {} 8 3 size 12{ nroot { size 8{3} } {8} } {} 2,47 1, 45 size 12{ sqrt {1,"45"} } {} size 12{ sqrt { {4} wideslash {8} } } {} size 12{ sqrt { {"16"} wideslash { sqrt {9} } } } {} Rational √ √ √ √ √ √ Irrational √ √ √ √ 1 + 4 size 12{ sqrt {4} } {}; -4 −23 size 12{ { { - 2} over {3} } } {}; 12 15 size 12{ { {1} over {5} } } {} 9+4 size 12{ sqrt {9+4} } {}; 1+22 size 12{ { {1+ sqrt {2} } over { sqrt {2} } } } {} 6. Equal in value 7. 414 size 12{ { {4} over {"14"} } } {} = 624 size 12{ { {6} over {"24"} } } {} etc Proper fraction Inproper fraction Mixed number Decimal number Recurring decimal number Percentage ACTIVITY 2 1. 2,15 0,625 3,25 5,75 2,875 6,0007 size 12{ { {6,"000"} over {7} } } {} = 0,8571 . . . ≈ 0,86 7,0009 size 12{ { {7,"000"} over {9} } } {} = 0,777 . . . = 0, 7 size 12{ {7} cSup { size 8{ cdot } } } {} or 0,8 6 81000 size 12{ { {8} over {"1000"} } } {} = 6 1125 size 12{ { {1} over {"125"} } } {} 4 65100 size 12{ { {"65"} over {"100"} } } {} = 4 1320 size 12{ { {"13"} over {"20"} } } {} 3751000 size 12{ { {"375"} over {"1000"} } } {} = 38 size 12{ { {3} over {8} } } {} 7 751000 size 12{ { {"75"} over {"1000"} } } {} = 7 340 size 12{ { {3} over {"40"} } } {} 13 65100 size 12{ { {"65"} over {"100"} } } {} = 13 1320 size 12{ { {"13"} over {"20"} } } {} 1251000 size 12{ { {"125"} over {"1000"} } } {} = 18 size 12{ { {1} over {8} } } {} 5.1 7,0009 size 12{ { {7,"000"} over {9} } } {} = 0, 7 size 12{ {7} cSup { size 8{ cdot } } } {} 5.2 -5,8 3 size 12{ {3} cSup { size 8{ cdot } } } {}5,0006 size 12{ { {5,"000"} over {6} } } {} = 0,8333 . . . 5.3 3, 1 size 12{ {1} cSup { size 8{ cdot } } } {}3 size 12{ {3} cSup { size 8{ cdot } } } {}13,000099 size 12{ { {"13","0000"} over {"99"} } } {} = 0,1313 . . . 7.1 39 size 12{ { {3} over {9} } } {} = 13 size 12{ { {1} over {3} } } {} 7.2 4599 size 12{ { {"45"} over {"99"} } } {} = 511 size 12{ { {5} over {"11"} } } {} 7.3 23990 size 12{ { {"23"} over {"990"} } } {} 7.4 3900 size 12{ { {3} over {"900"} } } {} = 1300 size 12{ { {1} over {"300"} } } {} 9. 0, 4 size 12{ {4} cSup { size 8{ cdot } } } {}5 size 12{ {5} cSup { size 8{ cdot } } } {} = x x = 0,4545 . . .  100 x = 45,4545 . . . –  99 x = 45 x = 4599 size 12{ { {"45"} over {"99"} } } {} = 511 size 12{ { {5} over {"11"} } } {} ACTIVITY 3 2.1 17x520x5 size 12{ { {"17"x5} over {"20"x5} } } {} = 85% 2.2 1940 size 12{ { {"19"} over {"40"} } } {} x 1001 size 12{ { {"100"%} over {1} } } {} = 47,5% 2.3 38x250x2 size 12{ { {"38"x2} over {"50"x2} } } {} = 76% 2.4 4560 size 12{ { {"45"} over {"60"} } } {} x 1001 size 12{ { {"100"%} over {1} } } {} = 75% 3.1 55100 size 12{ { {"55"} over {"100"} } } {} = 1120 size 12{ { {"11"} over {"20"} } } {} 3.2 15,5100 size 12{ { {"15",5} over {"100"} } } {} = 0,155 = 1551000 size 12{ { {"155"} over {"1000"} } } {} = 31200 size 12{ { {"31"} over {"200"} } } {} 3.3 33200 size 12{ { {"33"} over {"200"} } } {} 3.4 2030 {0 size 12{ { {2 { {0}}} over {"30 {"{0}}} } } {} = 230 size 12{ { {2} over {"30"} } } {} 4.a) 33800 size 12{ { {"33"} over {"800"} } } {} x 255001 size 12{ { {"25500"} over {1} } } {} size 12{ approx } {} 1 052 b) 35 size 12{ { {3} over {5} } } {} x 255001 size 12{ { {"25500"} over {1} } } {} = 15 300 c) 851000 size 12{ { {"85"} over {"1000"} } } {} x 255001 size 12{ { {"25500"} over {1} } } {} = 2 167,5 size 12{ approx } {} 2 168 (14,5) 153001052 size 12{ { {"15300"} over {"1052"} } } {} = 7650526 size 12{ { {"7650"} over {"526"} } } {} = 3825263 size 12{ { {"3825"} over {"263"} } } {} 25 500 – 18 520 = 6 980 4.4 4.5 35 size 12{ { {3} over {5} } } {} x 21 size 12{ { {2} over {1} } } {} = 65 size 12{ { {6} over {5} } } {} = 115 size 12{1 { {1} over {5} } } {} ACTIVITY 4 1.1 397 size 12{ { {"39"} over {7} } } {} 1.2 709 size 12{ { {"70"} over {9} } } {} 2. Numbers must be the same 3.1 347 size 12{3 { {4} over {7} } } {} 3.2 22−918 size 12{2 { {2 - 9} over {"18"} } } {} = 120−918 size 12{1 { {"20" - 9} over {"18"} } } {} = 11118 size 12{1 { {"11"} over {"18"} } } {} 4.1 297 size 12{ { {"29"} over {7} } } {} + 18442 size 12{ { {"184"} over {"42"} } } {} = 174+18442 size 12{ { {"174"+"184"} over {"42"} } } {} = 35842 size 12{ { {"358"} over {"42"} } } {} = 82242 size 12{8 { {"22"} over {"42"} } } {} = 81121 size 12{8 { {"11"} over {"21"} } } {} 4.2 21 - 611 size 12{ { {6} over {"11"} } } {} = 20511 size 12{"20" { {5} over {"11"} } } {} 0,125 + 0,625 – 0,375 = 0,375 1710+10+1520 size 12{"17" { {"10"+"10"+"15"} over {"20"} } } {} = 173520 size 12{"17" { {"35"} over {"20"} } } {} = 181520 size 12{"18" { {"15"} over {"20"} } } {} = 1834 size 12{"18" { {3} over {4} } } {} 33−2124 size 12{3 { {3 - "21"} over {"24"} } } {} = 21124 size 12{2 { {"11"} over {"24"} } } {} {} 28 a 2 − a 4 size 12{ { {"28"`a rSup { size 8{2} } - a} over {4} } } {} 4.7 + (6−3aab) size 12{\( { {6 - 3a} over { ital "ab"} } \)} {} = 9b+6−3aab size 12{ { {9b+6 - 3a} over { ital "ab"} } } {} 4.8 −61 size 12{ { { - 6} over {1} } } {} + 207 size 12{ { {"20"} over {7} } } {} = −42+207 size 12{ { { - "42"+"20"} over {7} } } {} = −227 size 12{ { { - "22"} over {7} } } {} = −317 size 12{ - 3 { {1} over {7} } } {} 5 – 64+69 size 12{ left (6 { {4+6} over {9} } right )} {} = 5 – 6109 size 12{6 { {"10"} over {9} } } {} = 5 – 719 size 12{7 { {1} over {9} } } {} = – 649 size 12{ { {"64"} over {9} } } {} = 45−649 size 12{ { {"45" - "64"} over {9} } } {} = −199 size 12{ { { - "19"} over {9} } } {} = −219 size 12{ - 2 { {1} over {9} } } {} 10a3 size 12{ { {"10"a} over {3} } } {} – 5a2 size 12{ { {5a} over {2} } } {} = 20a−15a6 size 12{ { {"20"a - "15"a} over {6} } } {} = 5a6 size 12{ { {5a} over {6} } } {} ACTIVITY 5 1. 514 size 12{ { {5} over { {} rSub { size 8{1} } { {4}}} } } {} x 52 size 12{ { {5} over {2} } } {} x 411 size 12{ { { { {4}} rSup { size 8{1} } } over {1} } } {} = 252 size 12{ { {"25"} over {2} } } {} = 1212 size 12{"12" { {1} over {2} } } {} 3.1 81 size 12{ { {8} over {1} } } {} ÷ 811 size 12{ { {8} over {"11"} } } {} = 811 size 12{ { { { {8}} rSup { size 8{1} } } over {1} } } {} x 1181 size 12{ { {"11"} over { { {8}} rSub { size 8{1} } } } } {} = 11 3.2 181 size 12{ { {"18"} over {1} } } {} x 87 size 12{ { {8} over {7} } } {} = 1447 size 12{ { {"144"} over {7} } } {} = 2047 size 12{"20" { {4} over {7} } } {} 3.3 5163 size 12{ { { { {5}} rSup { size 8{1} } } over { { {6}} rSub { size 8{3} } } } } {} x 2151 size 12{ { { { {2}} rSup { size 8{1} } } over { { {5}} rSub { size 8{1} } } } } {} = 13 size 12{ { {1} over {3} } } {} 3.4 −8131 size 12{ { { - { {8}} rSup { size 8{1} } } over { { {3}}"" lSub { size 8{1} } } } } {} x −93162 size 12{ { { - { {9}} rSup { size 8{3} } } over { { {1}} { {6}} rSub { size 8{2} } } } } {} = 32 size 12{ { {3} over {2} } } {} = 112 size 12{1 { {1} over {2} } } {} 3.5 279mn4 size 12{ { { { {2}} { {7}} rSup { size 8{9} } ital "mn"} over {4} } } {} x 1−62m3 size 12{ { {1} over { - { {6}}"" lSub { size 8{2} } m rSup { size 8{3} } } } } {} = −9n8m2 size 12{ { { - 9n} over {8m rSup { size 8{2} } } } } {} 3.6 −42xy31ab size 12{ { { - { {4}} rSup { size 8{2} } ital "xy"} over { { {3}}"" lSub { size 8{1} } { {a}}b} } } {} x 3a−2x size 12{ { { { {3}} { {a}}} over { - { {2}} { {x}}} } } {} = 2yb size 12{ { {2y} over {b} } } {}