THE NUMBER SYSTEM
Contents
1. 4 Unit 1 The Number System Example 1 Use the divisibility test to determine whether 2 416 is divisible by 2 3 4 5 6 8 9 and 10 Solution 2 416 is divisible by 2 because the unit s digit 6 is divisible by 2 2 416 is divisible by 4 because 16 the number formed by the last two digits is divisible by 4 2 416 is divisible by 8 because the number formed by the last three digits 416 is divisible by 8 2 416 is not divisible by 5 because the unit s digit is not 0 or 5 Similarly you can check that 2 416 is not divisible by 3 6 9 and 10 Therefore 2 416 is divisible by 2 4 and 8 but not by 3 5 6 9 and 10 A factor of a composite number is called a prime factor if it is a prime number For instance 2 and 5 are both prime factors of 20 Every composite number can be written as a product of prime numbers To find the prime factors of any composite number begin by expressing the number as a product of two factors where at least one of the factors is prime Then continue to factorize each resulting composite factor until all the factors are prime numbers When a number is expressed as a product of its prime factors the expression is called the prime factorization of the number 60 For example the prime factorization of 60 is Gy gt 30 6022x2x3x522 x3x5 ore 15 The prime factorization of 60 is also found by Cy 5 using a factoring tree Note that the set 2 3 5 is a set of prime factors of2. 3 IR x xis a real number Since all real numbers can be located on the number line the number line can be used to compare and order all real numbers For example using the number line you can tell that NN 4 3 lt 0 uem Example 1 Arrange the following numbers in ascending order 5 v3 d ta ANAM 0 8 X 27 6 2 M Ad v3 NP Solution Use a calculator to convert PA and D to decimals 5 6 0 83333 and 3 o 0 866025 Since 0 8 0 83 0 866025 the numbers when arranged in ascending order are 0 8 2 FAA 6 i Y AV However there are algebraic methods of comparing and ordering real numbers Here are two important properties of order Mathematics Grade 9 A third property stated below can be derived from the Trichotomy Property and the Transitive Property of Order gt For any two non negative real numbers a and b if a lt b then a lt b You can use this property to compare two numbers without using a calculator B For example let us compare z and v 2 G 25 3 3 27 6 36 2 4 36 8v 3 Since B i it follows that lt 6 2 2 1 Compare the numbers a and b using the symbol lt or gt J6 a a _ p 06 4 3 Alu b a 0 432 b 0 437 c a 0 128 b 0 123 2 State whether each set a e given below is closed under each of the following operations i addition ii subtraction iii multiplication
3. For any two real numbers a and b and for all integers n z2 a b ab Example 7 Simplify each of the following a i x b o A16 x2 Solution au b 4h6 xJ2 16 x2 2 27 multiplication 2 3 3 27 we Jg 1 a 9 90 93 3 by Theorem 1 2 29 Mathematics Grade 9 ACTIVITY 1 9 Simplify 1 1 i a Su b eI p 25 1 2 ii a RE b Bi 2 22 1 1 ii a 22 b E 7293 What relationship do you observe between a and b in i ii and iii The observations from the above Activity lead us to the following theorem Theorem 1 3 For any two real numbers a and b where b z0 and for all integers n 2 1 D 5 ae H b 1 3 6 Example 8 Simplify a 2 b PR 23 Solution eae 1 3 a A67 6 by Theorem 1 3 1 8 2 since 2 8 Vis 128 h 2 64 2 because 2 64 30 Unit 1 The Number System ACTIVITY 1 10 1 Suggest with reasons a meaning for Ji 9 1 a 22 b 2 interms of 2 3 1 2 Suggest a relation between 5 and 5 Bd 9 Applying the property a a you can write gt as 70 In general you can say 137 er i a where p and q are positive integers and a 0 Thus you have the following definition Definition 1 11 1 p a P For a 0 and p and q any two positive integers a e Ya 1 1 1 Show that a 64 4 b 2565 2 c 1253 25 2
4. 4 Exercise 1 12 Rationalize the denominator of each of the following 1 J 18 2 E Eo E GH 7 10 Bn 43 2 m cUm EE 1 2 easel Mathematics Grade 9 1 2 9 Euclid s Division Algorithm A The division algorithm ACTIVITY 1 19 1 Is the set of non negative integers whole numbers closed under division z 2 Consider any two non negative integers a and b a What does the statement a is a multiple of b mean b Isitalways possible to find a non negative integer c such that a bc If a and b are any two non negative integers then a b b 0 is some non negative integer c if it exists such that a bc However since the set of non negative integers is not closed under division it is clear that exact division is not possible for every pair of non negative integers For example it is not possible to compute 17 5 in the set of non negative integers as 17 5 is not a non negative integer 15 2 3 X5 and 20 4 X5 Since there is no non negative integer between 3 and 4 and since 17 lies between 15 and 20 you conclude that there is no non negative integer c such that 17 2 c X5 You observe however that by adding 2 to each side of the equation 15 2 3 X5 you can express it as 17 3 X5 2 Furthermore such an equation is useful For instance it will provide a correct answer to a problem such as If 5 girls have Birr 17 to share how many Birr will each girl get Examples of this sort lead to the follow
5. Unit 1 The Number System ACTIVITY 1 7 Evaluate the following 1 0 3030030003 0 1414414441 2 0 5757757775 0 242442444 3 34 V2 x 3 42 4 42x From Example 2 and Activity 1 7 you can generalize the following facts T Exercise 1 3 1 Identify each of the following numbers as rational or irrational a 2 b 234 c 0213141516 d 4081 e ORIBIL f 52 g 3m2 Aan 2 Give two examples of irrational numbers one in the form of a radical and the other in the form of a non terminating decimal 3 Foreach of the following decide whether the statement is true or false If your answer is false give a counter example to justify a The sum of any two irrational numbers is an irrational number b The sum of any two rational numbers is a rational number C The sum of any two terminating decimals is a terminating decimal d The product of a rational number and an irrational number is irrational 21 Mathematics Grade 9 IEE Real Numbers In Section 1 2 1 you observed that every rational number is either a terminating decimal or a repeating decimal Conversely any terminating or repeating decimal is a rational number Moreover in Section 1 2 2 you learned that decimals which are neither terminating nor repeating exist For example 0 1313313331 Such decimals are defined to be irrational numbers So a decimal number can be a rational or an irrational number It can be shown tha
6. 1 Show that and can each be expressed as a decimal Solution gt means 3 8 Li means 7 12 8 12 0 375 0 5833 8 3 000 12 7 0000 24 60 60 100 56 96 40 40 40 36 0 40 35 4 3 2 0375 7 205833 8 2 Mathematics Grade 9 The fraction rational number can be expressed as the decimal 0 375 A decimal like 0 375 is called a terminating decimal because the division ends or terminates when the remainder is zero The fraction ean be expressed as the decimal 0 58333 Here the digit E repeats and the division does not terminate A decimal like 0 58333 is called a repeating decimal To show a repeating digit or a block of repeating digits in a repeating decimal j number we put a bar above the repeating digit or block of digits For example 0 58333 can be written as 0 583 and 0 0818181 canbe written as 0 081 This method of writing a repeating decimal is known as bar notation The portion of a decimal that repeats is called the re petend For exar In 0 583333 20 583 the repetend is 3 In 1 777 17 the repetend is 7 In 0 00454545 0 0045 the repetend is 45 To generalize When you divide a by b one of the following two cases will occur Case The division process ends or terminates when a remainder of zero is obtained In this case the decimal is called a terminating decimal Case 2 The division process does not terminate as the remainder never becomes zero Such a decimal is cal
7. 5 3 5 42 875 So 3 5 3 53 lt 4 Try 3 7 3 250 653 So 3 7 3 53 4 Try 3 8 3 8 54 872 So 3 7 lt 3 53 lt 3 8 Try 3 75 3 75 52 734375 So 3 75 lt 3 53 lt 3 8 Therefore 3 53 is 3 8 to the nearest tenth B Meaning of fractional exponents ACTIVITY 1 8 1 1 State another name for 2 ii 2 What meaning can you give to 2 or 2 3 Show that there is at most one positive number whose fifth root is 2 1 By considering a table of powers of 3 and using a calculator you can define3 as 35 5 iy ar This choice would retain the property of exponents by which al 3 1 Similarly you can define 5 where n is a positive integer greater than 1 as 4 5 In 1 general you can define b for any bER and n a positive integer to be 4 b whenever lb is areal number Definition 1 10 The n power If b R and n is a positive integer greater than 1 then l b 3 b Example 5 Write the following in exponential form a V7 hb oe 2 10 Solution 1 1 1 l a JEP b gz 0 10 28 Unit 1 The Number System Example 6 Simplify 1 1 1 a 25 b SP c 64 Solution 1 a 25 25 5 Since 5 25 1 b 89 3 8 Since 2 8 1 c 649 64 2 Since 2 64 Group Work 1 5 Eoad 1 The observations from the above Group Work lead you to think that 5 x33 5 x3 y This particular case suggests the following general property Theorem Theorem 1 2 jest 1
8. 6 you may have realized that the set of irrational numbers is not closed under all the four operations namely addition subtraction multiplication and division Do the following activity and discuss your results ACTIVITY 1 13 1 Finda b if a a 23 442 and b 23 J2 b a 3 V3 and b 22443 Mathematics Grade 9 2 Fida b if a a 43 and b v3 b a 45 and b 42 3 Find ab if a a 43 1 and b V3 1 b a 2243 and b 2342 4 Find a b if a a 542 and b 23 2 b a 6V6 and b 2245 Let us see some examples of the four operations on real numbers Example 1 Add a 2V3 4342 and V2 43 Solution 243 342 V2 3 23 342 442 33 J3 2 3 5 3 V3 442 Example 2 Subtract 3 2 V5 from 3 5 2V2 Solution 3 5 22 3V2 W5 2345 2V2 3V2 5 N5 3 D 4 2 32 3 245 542 Example 3 Multiply a 243 by342 b 2X5 by 3V5 Solution a 2438 xa342 5646 b 24s x34s5 2 3x V5 30 Example 4 Divide a 8 6 by 243 b 12 6 by v2 xv3 Solution a 8 65243 E 4 2 12 12 b 12V6 4 2 x3 5 uuo Unit 1 The Number System ACTIVITY 1 14 1 Find the additive inverse of each of the following real numbers 1 a 5 b ES c 424 d 2 45 e 2 1010010001 2 Find the multiplicative inverse of each of the following real numbers a 3 b 5 BO 16 d 26 V2 e 17 p o 13 we g 3 Explain each of the following steps V6 adis x2 20 P V6 2015 20 2 NG edis PNE P 2 aS 00 3 3 V9 xJ 2 249 x
9. System Definition 1 3 For any two natural numbers a and b the least common multiple of a and b denoted by LCM a b is the smallest multiple of both a and b Example 3 Find LCM 8 9 Solution Let Mg and Mo be the sets of multiples of 8 and 9 respectively Ms 8 16 24 32 40 48 56 64 72 80 88 M 9 18 27 36 45 54 63 72 81 90 Therefore LCM 8 9 72 Prime factorization can also be used to find the LCM of a set of two or more than two numbers A common multiple contains all the prime factors of each number in the set The LCM is the product of each of these prime factors to the greatest number of times it appears in the prime factorization of the numbers Example 4 Use the prime factorization method to find LCM 9 21 24 Solution 923x323 The prime factors that appear in these i factorizations are 2 3 and 7 21 3 x7 Considering the greatest number of times 24 2 x2 X2 3 22 x3 each prime factor appears we can get 2 3 and 7 respectively Therefore LCM 9 21 24 23 x3 x7 504 ACTIVITY 1 4 1 Find a The GCF and LCM of 36 and 48 b GCF 36 48 x LCM 36 48 C 36 x 48 2 Discuss and generalize your results Mathematics Grade 9 IRE Rational Numbers About 5 000 years ago Egyptians used N O hieroglyphics to represent numbers The Egyptian concept of fractions was 1 10 100 1 000 mostly limited to fractions with numerator mus 1 The
10. d ware va f 027 g 6 h d pou j ya 3 Identify the error and write the correct solution each of the following cases a A student simplified V28 to J25 3 and then to 54 3 b Astudent simplified V72 to J44 18 and then to 44 3 C A student simplified 47x and got x 4 7 4 Simplify each of the following a 84 250 b ih6 xJ5 c 45x25 A 3 81 124 96 d ET xJ 14 e f 7 SE 3v6 E e TU A A aJo 34 Unit 1 The Number System 5 The number of units N produced by a company from the use of K units of capital and L units of labour is given by N 12VLK a What is the number of units produced if there are 625 units of labour and 1024 units of capital b Discuss the effect on the production if the units of labour and capital are doubled Addition and subtraction of radicals Which of the following do you think is correct 1 4248 410 2 43 4 8 5V2 M YIN The above problems involve addition and subtraction of radicals You define below the concept of like radicals which is commonly used for this purpose Definition 1 12 Radicals that have the same index and the same radicand are said to be like radicals For example i 345 25 and 1 5 are like radicals ii J5 and 3 5 are notlike radicals iii J1 and J are not like radicals By treating like radicals as like terms you can add or subtract like radicals and express them as a single radical On the other hand the sum of unlike radicals c
11. hieroglyphic was placed under the I I A nn 9e symbol lt gt to indicate the number as a 1 1 i i i denominator Study the examples of Egyptian fractions 3 10 20 100 Recall that the set of integers is given by Z21 3 2 1 0 1 2 3 Using the set of integers we define the set of rational numbers as follows Definition 1 4 Rational number Any number that can be expressed in the form o where a and b are integers and b O0 is called a rational number The set of rational numbers denoted by Q 1s the set described by Q saan b are integers and b oh Through the following diagram you can show how sets within rational numbers are related to each other Note that natural numbers whole numbers and integers are included in the set of rational numbers This is because integers such as 4 and can be Whole 4 j written as B and EM Rational Numbers Q Integers Z Numbers W The set of rational numbers also includes terminating and repeating decimal numbers Natural because terminating and repeating decimals acest can be written as fractions Figure 1 4 10 Unit 1 The Number System For example 1 3 can be written as and 0 29 as Mixed numbers are also included in the set of rational numbers because any mixed number 30 2x3x5 2 x can be written as an improper fraction 45 3x3x5 3 For example 2 can be written as T When a rational number is expressed as a frac
12. ii b c a 2 ba ca Unit 1 The Number System 1 Find the numerical value of each of the following 3 a Pj x x 8 x 64 b 176 24275 1584 891 c 15 104 5 rli 50 02 300 d 40 0001 3 0 00032 e 2340 125 4 0 0016 2 Simplify each of the following 1 DUC iy Ji a 2116F b 23x25 c 5 d 494 1 1 1 FF 5 32 e 2 SDN 16 2 NISI g 3 243 3 What should be added to each of the following numbers to make it a rational number There are many possible answers In each case give two answers a 5 3 b 245 C 4383383338 d 6 23456 e 10 3030003 1 2 6 Limits of Accuracy In this subsection you shall discuss certain concepts such as approximation accuracy in measurements significant figures s f decimal places d p and rounding off numbers In addition to this you shall discuss how to give appropriate upper and lower bounds for data to a specified accuracy for example measured lengths ACTIVITY 1 15 1 Round off the number 28617 to the nearest a 10 000 b 1000 c 100 2 Writethenumber i 7 864 ii 6 437 iii 4 56556555 a to one decimal place b to two decimal places 3 Write the number 43 25 to a two significant figures b three significant figures 4 The weight of an object is 5 4 kg Give the lower and upper bounds within which the weight of the object can lie 43 Mathematics Grade 9 1 Counting and measuring Counting and measuring are an integral part of our daily life Most of us do s
13. iv division a Nthesetof natural numbers b Zthe set of integers c Q the set of rational numbers d The set of irrational numbers e R the set of real numbers IEZ Exponents and Radicals A Roots and radicals In this subsection you will define the roots and radicals of numbers and discuss their properties Computations of expressions involving radicals and fractional exponents are also considered 24 Unit 1 The Number System Roots The Pythagorean School of ancient Greece focused on the study of philosophy mathematics and natural science The students called Pythagoreans made many advances in these fields One of their studies was to symbolize numbers By drawing pictures of various numbers patterns can be discovered For example some whole numbers can be represented by drawing dots arranged in squares e e o o o o0 0 oO e e o o o o0 0 eO e e o o oe 0o eO e e o o o 050 0 eO e ee eo e e o o o 056 0 eO e e e oe eo e e o o o 0o eO e e ee eo e e o0 o o 050 0o oO e e e e e eo e e o o o o0 0o eO 1 4 9 16 64 1x1 2x2 3x3 4x4 8x8 Numbers that can be pictured in squares of dots are called perfect squares or square numbers The number of dots in each row or column in the square is a square root of the perfect square The perfect square 9 has a square root of 3 because there are 3 rows and 3 columns You say 8 is a square root of 64 because 64 8 x 8 or 8 Definition 1 7 Square root For any two r
14. justifies each of the following statements 2 3 x 2 x3 x b s 2 z 2 oe 3 do uS mS Qs 2 MOORE In this section you will discuss operations on the set of real numbers The properties you have studied so far will help you to investigate many other properties of the set of real numbers Group Work 1 6 Factors product product written as a power 37 Mathematics Grade 9 2 Trythis Copy the following table Use a calculator to find each quotient and complete the table Division Quotient Quotient written as a power Bae Discuss the two tables i a Compare the exponents of the factors to the exponents in the product What do you observe b Write a rule for determining the exponent of the product when you multiply powers Check your rule by multiplying 3 x3 using calculator ii a Compare the exponents of the division expressions to the exponents in the quotients What pattern do you observe b Write a rule for determining the exponent in the quotient when you divide powers Check your rule by dividing T by T on a calculator 3 Indicate whether each statement is false or true If false explain a Between any two rational numbers there is always a rational number b The set of real numbers is the union of the set of rational numbers and the set of irrational numbers c The set of rational numbers is closed under addition subtraction multiplication and division excluding
15. to 0 draw an arc that intersects the number line at B The distance from the point corresponding to 0 to B is v2 units 2 To locate V5 on the number line Find two numbers whose squares have a sum of 5 One pair that works is 1 and 2 since 1 2 5 Draw a number line At the point corresponding to 2 on the number line construct a perpendicular line segment 1 unit long Draw the line segment shown from the point corresponding to 0 to the top of the 1 unit segment 0 1 2 3 Label it as c Figure 1 6 Mathematics Grade 9 Definition 1 5 Irrational number ee a An irrational number is a number that cannot be expressed as that a and b are integers and b 40 ACTIVITY 1 6 1 Locate each of the following on the number line by using geometrical construction a v3 b 4 c v6 2 Explain how 4 2 can be used to locate a 45 b 6 3 Locate each of the following on the number line a 1 V2 b 24 V2 c 3 2 Example 2 Show that3 2 is an irrational number Solution To show that 3 JA is not a sce number let us begin by assuming that 3 2 is rational i 34 2 where a and b are integers b 40 Tei gt ES Since a 3b and P are integers Why 30 is a rational number meaning that NET is rational which is false As the assumption that 3 2 is rational has led to a false conclusion the assumption must be false Therefore 3 2 is an irrational number y 20
16. 4 245 25 255 2 6 As an inequality it would be expressed as 2 45 2 5 lt 2 55 2 45 is known as the lower bound of 2 5 while 2 55 is known as the upper bound 2 50 on the other hand is written to two decimal places and therefore only numbers from 2 495 up to but not including 2 505 would be rounded to 2 50 This therefore represents a much smaller range of numbers than that being rounded to 2 5 Similarly the range of numbers being rounded to 2 500 would be even smaller Example 3 A girl s height is given as 162 cm to the nearest centimetre i Work out the lower and upper bounds within which her height can lie ii Represent this range of numbers on a number line ili If the girl s height is cm express this range as an inequality Solution i 162 cm is rounded to the nearest centimetre and therefore any measurement of cm from 161 5 cm up to and not including 162 5 cm would be rounded to 162 cm Thus lower bound 161 5 cm upper bound 162 5 cm ii Range of numbers on the number line is represented as ooo 6 e 161 161 5 162 162 5 163 iii Whenthe girl s height h cm is expressed as an inequality it is given by 161 5 xh 162 5 Unit 1 The Number System Effect of approximated numbers on calculations When approximated numbers are added subtracted and multiplied their sums differences and products give a range of possible answers Example 4 The length and width of a rectangle are 6 7 cm and 4 4 cm respect
17. 5 Seius 2 22978 aa 3 3 v2 2V5 20 42 H 2V5 245 2 Mathematics Grade 9 Let us now examine the basic properties that govern addition and multiplication of real numbers You can list these basic properties as follows v Closure property The set R of real numbers is closed under addition and multiplication This means that the sum and product of two real numbers is a real number that is for all a b R a b ER andab ER Addition and multiplication are commutative in R That is for all a b R i at b b a ii ab ba Addition and multiplication are associative in R That is for all a b c R i a b c a b c ii ab c a bc Existence of additive and multiplicative identities There are real numbers 0 and 1 such that i a 0 0 a a forallaER ii 4 121 029 fo alae R Existence of additive and multiplicative inverses i For each a R there exists a R such that a a 0 a a and a is called the additive inverse of a is 1 1 il For each non zero a R there exists R such that a x 1 xa a a a Ll TRU and is called the multiplicative inverse or reciprocal of a a Distributive property Multiplication is distributive over addition that is if a b c R then i a b c ab ac
18. 60 Is this set unique This property leads usto state the Fundamental Theorem of Arithmetic Theorem 1 1 Fundamental theorem of arithmetic Every composite number can be expressed factorized as a product of primes This factorization is unique apart from the order in which the prime factors occur You can use the divisibility tests to check whether or not a prime number divides a given number Mathematics Grade 9 Example 2 Find the prime factorization of 1 530 Solution Start dividing 1 530 by its smallest prime factor If the quotient is a composite number find a prime factor of the quotient in the same way Repeat the procedure until the quotient is a prime number as shown below Prime factors L 1 530 2 765 765 3 255 255 3 85 85 5 17 and 17 is a prime number Therefore 1 530 2 x 3x 5 x 17 IRE Common Factors and Common Multiples In this subsection you will revise the concepts of common factors and common multiples of two or more natural numbers Related to this you will also revise the greatest common factor and the least common multiple of two or more natural numbers A Common factors and the greatest common factor ACTIVITY 1 3 1 Given the numbers 30 and 45 a find the common factors of the two numbers b X find the greatest common factor of the two numbers 2 Given the numbers 36 42 and 48 a find the common factor
19. Compare the result of 1e with the GCF of the given numbers Are they the same The above Group Work leads you to another alternative method to find the GCF of numbers This method which is a quicker way to find the GCF is called the prime factorization method In this method the GCF of a given set of numbers is the product of their common prime factors each power to the smallest number of times it appears in the prime factorization of any of the numbers Example 2 Use the prime factorization method to find GCF 180 216 540 Solution Step 1 Express the numbers 180 216 and 540 in their prime factorization 180222 x 3 x 5 216225 x35 54022 x3 x 5 Step 2 As you see from the prime factorizations of 180 216 and 540 the numbers 2 and 3 are common prime factors So GCF 180 216 540 is the product of these common prime factors with the smallest respective exponents in any of the numbers GCF 180 216 540 2 x 3 36 B Common multiples and the least common multiple For this group work you need 2 coloured pencils Work with a partner Try this x Listthe natural numbers from 1 to 100 on a sheet of paper Cross out all the multiples of 10 Using a different colour cross out all the multiples of 8 Discuss Which numbers were crossed out by both colours 2 How would you describe these numbers 3 What is the least number crossed out by both colours What do you call this number Unit 1 The Number
20. Express each of the following without fractional exponents and without radical signs 1 1 PEE a 814 b 9 c 64 d z 1 e 0 00032 j f 0 0016 g 41 729 3 Explain each step of the following 1 1 1 27x125 3 x3 x3 x 5 x5 x5 P 3 x5 x 3 x5 x 3 x5 P 3 x5 15 4 In the same manner as in Question 3 simplify each of the following jl 1 1 a 25x21y b 625x16 c 1024 x243 s 5 Express Theorem 1 2 using radical notation 6 Show that a 7 xsi xs b V5 xv3 2 558 1 1 c Vio V7x9 d 117 x67 11x6 7 31 Mathematics Grade 9 7 Express in the simplest form 1 1 2 11925506 d Xx 1 x95 h n 58 X27 x g d 1 93 X33 3 16 x3 4 3 5 x3 8 qia 8 Express Theorem 1 3 using radical notation 9 Simplify 1 1285 a b 45 V16 3 2 93 T 243 1a E 10 Rewrite each of the following in the form a 1 S9 jp wl 13 a b b 12 c fu 1 2 11 Rewrite the following in the form G T a 35 13 Rewrite the expressions in Question 11 using radicals b 12 Rewrite the expressions in Question 10 using radicals c d 814 9 1512 h 8 P 5 c 64 1 324 1624 a V62 1 128 E 327 x47 on d 14 Express the following without fractional exponents or radical sign NS a 15 Simplify each of the following 1 a 645 6 d 814 32 b b e c c 648 f 512 83 1 6 2 2 p23 Unit 1 The Number System Si
21. If a b q and r are positive integers such that a q xb r then GCF a b GCF b r 57 Mathematics Grade 9 Example 2 Find GCF 224 84 Solution To find GCF 224 84 you first divide 224 by 84 The divisor and remainder of this division are then used as dividend and divisor respectively in a succeeding division The process is repeated until a remainder O is obtained The complete process to find GCF 224 84 is shown below Euclidean algorithm Computation Division algorithm Application of Euclidean form Algorithm 84 224 224 2 X84 56 GCF 224 84 GCF 84 56 168 56 56 84 56 84 1 X56 28 GCF 84 56 GCF 56 28 28 56 2 x28 0 GCF 56 28 28 by inspection Conclusion GCF 224 84 28 Exercise 1 14 1 For the above example verify directly that GCF 224 84 GCF 84 56 GCF 56 28 2 Find the GCF of each of the following pairs of numbers by using the Euclidean Algorithm a 18 12 b 269 88 c 143 39 d 1295 407 e 85 68 f 7286 1684 58 Unit 1 The Number System bar notation composite number divisible division algorithm factor fundamental theorem of arithmetic greatest common factor GCF irrational number least common multiple LCM multiple perfect square prime factorization prime number 2 summary principal m root principal square root radical sign radicand rational number rationaliz
22. Numerical Value Arabic Numeral Babylonian Egyptian Hieroglyphic Greek Herodianic Roman Ethiopian Geez THE NUMBER S After completing this unit you should be able to know basic concepts and important facts about real numbers E justify methods and procedures in computation with real numbers solve mathematical problems involving real numbers Main Contents 1 1 Revision on the set of rational numbers 1 2 Thereal number system Key Terms Summary Review Exercises Mathematics Grade 9 INTRODUCTION In earlier grades you have learnt about rational numbers their properties and basic mathematical operations upon them After a review of your knowledge about rational numbers you will continue studying the number systems in the present unit Here you will learn about irrational numbers and real numbers their properties and basic operations upon them Also you will discuss some related concepts such as approximation accuracy and scientific notation 31 1 REVISION ON THE SET OF RATIONAL NUMBERS ACTIVITY 1 1 The diagram below shows the relationships between the sets of W Natural numbers Whole numbers Integers and Rational numbers Use this diagram to answer Questions 1 and 2 given below Justify your answers 1 To which set s of numbers does each of the following numbers belong a 27 b 17 c 7 3 d 0 625 e 0 615 2 i Define the set of a Natural numb
23. This is known as the number of significant figures Example 2 a Write 43 25 to 3 s f b Write 0 0043 to 1 s f Solution a We want to write only the three most significant digits However the fourth digit needs to be considered to see whether the third digit is to be rounded up or not That is 43 25 is written as 43 3 to 3 s f b Notice that in this case 4 and 3 are the only significant digits The number 4 is the most significant digit and is therefore the only one of the two to be written in the answer That is 0 0043 1s written as 0 004 to 1 s f 3 Accuracy In the previous lesson you have studied that numbers can be approximated a byrounding up b by writing to a given number of decimal place and c by expressing to a given number of significant figure In this lesson you will learn how to give appropriate upper and lower bounds for data to a specified accuracy for example numbers rounded off or numbers expressed to a given number of significant figures Mathematics Grade 9 Numbers can be written to different degrees of accuracy For example although 2 5 2 50 and 2 500 may appear to represent the same number they actually do not This is because they are written to different degrees of accuracy 2 5 is rounded to one decimal place or to the nearest tenths and therefore any number from 2 45 up to but not including 2 55 would be rounded to 2 5 On the number line this would be represented as es oo a 2
24. annot be expressed as a single radical unless they can be transformed into like radicals Example 11 Simplify each of the following a HS b 3V2 3 e h aa Solution a V2 4 8 V2 2x4 V2 42 V2 4242 42 42 2342 35 Mathematics Grade 9 b Mia AB 1 Jan 3 4 gt 8 3 42 iS v 344 x43 8 423 41 oxi Jo 9 3 V3 4243 445 2 1 4e Hd EE 645 Simplify each of the following if possible State restrictions where necessary 1 905 255 b V3x 6 c V21xJ5 d J2xxJ 8x e x f g d50y 2y h an oo j 9424 15 75 343 poca 38 b 942 552 c J45HM2 d 6 48 e i548 t viz 43 g 2x J4s0 h 53 54 23 2 i 8 24 4 J54 2 06 a 2ab b ERN j Ek Maeg 3 a Find the square of 7 2 10 Simplify each of the following i 542V6 5 2V6 i Mon NT Nat ii Jr pra Jp A p 4 4 Suppose the braking distance d for a given automobile when it is travelling v km hr is approximated by d 0 0002 uv m Approximate the braking distance when the car is travelling 64 km hr Unit 1 The Number System 1 2 5 The Four Operations on Real Numbers The following activity is designed to help you revise the four operations on the set of rational numbers which you have done in your previous grades ACTIVITY 1 12 1 Apply the properties of the four operations in the set of rational numbers to compute the following mentally if possible a 2 3 P b 3 X dl dl 9 US 9 T X 21 yi 21 e S TB qd I 9 2 State a property that
25. ation real number repeating decimal repetend scientific notation significant digits significant figures terminating decimal 1 The sets of Natural numbers Whole numbers Integers and Rational numbers denoted by N W Z and Q respectively are described by Neth 2 3l Q 2 a z bEZ b 0 W 0 L 2 Eco rdc c 0 1 7 9 ol A composite number is a natural number that has more than two factors b Aprime number is a natural number that has exactly two distinct factors 1 and itself C Prime numbers that differ by two are called twin primes d When a natural number is expressed as a product of factors that are all prime then the expression is called the prime factorization of the number 59 Mathematics Grade 9 10 11 12 13 e Fundamental theorem of arithmetic Every composite number can be expressed factorized as a product of primes and this factorization is unique apart from the order in which the prime factors occur a The greatest common factor GCF of two or more numbers is the greatest factor that is common to all numbers b The least common multiple LCM of two or more numbers is the smallest or least of the common multiples of the numbers a Any rational number can be expressed as a repeating decimal or a terminating decimal b Any terminating decimal or repeating decimal is a rational number Irrational numbers are decimal numbers that neither repeat nor terminate The set
26. division by zero d The set of irrational numbers is closed under addition subtraction multiplication and division 4 Give examples to show each of the following a The product of two irrational numbers may be rational or irrational b The sum of two irrational numbers may be rational or irrational c The difference of two irrational numbers may be rational or irrational d The quotient of two irrational numbers may be rational or irrational 5 Demonstrate with an example that the sum of an irrational number and a rational number is irrational 6 Demonstrate with an example that the product of an irrational number and a non zero rational number is irrational 38 Unit 1 The Number System Rational Irrational Real number number number Number 120220222 7 E x1 23 3 V75 H23 V75 45 120220222 3 0 132113 III Questions 3 4 5 and in particular Question 7 of the above Group Work lead you to conclude that the set of real numbers is closed under addition subtraction multiplication and division excluding division by zero You recall that the set of rational numbers satisfy the commutative associative and distributive laws for addition and multiplication If you add subtract multiply or divide except by 0 two rational numbers you get a rational number that is the set of rational numbers is closed with respect to addition subtraction multiplication and division From Group work 1
27. e example can be generated as owe Op Example 4 Express the decimal 0 375 asa fraction Solution Let d 0 375 Wen 7 a number of non repeating digits W p 2 number of repeating di ss and k p 142 gA OA av Qi aio d dio a0 19 20 10 0 Nl ANA 10 0 375 10 x0 375 NW 990 58 335 sm 990 990 From Exam ples 1 23 and 4 you conclude the following Unit 1 The Number System Express each of the following rational numbers as a decimal a 2 b a c ii d E e SUDO f 2d 9 25 9 3 100 7 Write each of the following as a decimal and then as a fraction in its lowest term a three tenths b fourthousandths c twelve hundredths d three hundred and sixty nine thousandths Write each of the following in metres as a fraction and then as a decimal a 4mm b 6 cm and 4 mm c 56 cm and 4 mm Hint Recall that 1 metre m 100 centimetres cm 1000 millimetres mm From each of the following fractions identify those that can be expressed as terminating decimals a 5 b X c 69 d 11 13 10 64 60 80 125 12 11 Generalize your observation Express each of the following decimals as a fraction or mixed number in simplest form a 0 88 b 077 c 083 d 708 e 0 5252 f 1 003 Express each of the following decimals using bar notation a 0 454545 b 0 1345345 Express each of the following decimals without bar notation In each case use at least ten digits after the decimal poi
28. eal numbers a and b if a2 b then a is a square root of b 4 Perfect squares also include decimals and fractions like 0 09 and J Since 0 3 0 09 2 and 2 it is also true that 8 64 and 12 144 So you may say that 8 is also a square root of 64 and 12 is a square root of 144 The positive square root of a number is called the principal square root The symbol called a radical sign is used to indicate the principal square root 25 Mathematics Grade 9 The symbol 25 is read as the principal square root of 25 or just the square root of 25 and A 25 is read as the negative square root of 25 If b is a positive real number Vb isa positive real number Negative real numbers do not have square roots in the set of real numbers since a gt 0 for any number a The square root of Zero is zero Similarly since 4 64 you say that 64 is the cube of 4 and 4 is the Qube root of 64 That is written as 4 3 64 l The symbol 4 3 64 is read as the principal cube root of 64 or just the cube root of 64 3 2 27 so 4 27 3 020 so Ao 0 You may now generalize as follows 3 i Definition 1 8 The n root For any two real numbers a and b and positive integer n if a b then ais called an n root of b Example 1 a 3 is a cube root of 27 because 3 27 b 4 is a cube root of 64 because 4 64 Definition 1 9 Principal n root If b 1s any real number and n 1s a p
29. ers b Whole numbers C Integers d Rational numbers ii What relations do these sets have Figure 1 1 1 1 1 Natural Numbers Integers Prime Numbers and Composite Numbers In this subsection you will revise important facts about the sets of natural numbers prime numbers composite numbers and integers You have learnt several facts about these sets in previous grades in Grade 7 in particular Working through Activity 1 2 below will refresh your memory 2 Unit 1 The Number System ACTIVITY 1 2 1 For each of the following statements write true if the statement is correct or false otherwise If your answer is false justify by giving a counter example or reason a The set 1 2 3 describes the set of natural numbers b The set 1 2 3 U 3 2 1 describes the set of integers C 57 is a composite number d 1 Prime numbers e Prime numbers U Composite number 1 2 3 f Odd numbers f Composite numbers g 48 is a multiple of 12 h 5 is a factor of 72 621 is divisible by 3 j Factors of 24 N Factors of 87 1 2 3 k Multiples of 6 N Multiples of 4 12 24 l 2 x3 x5 is the prime factorization of 180 2 Given two natural numbers a and b what is meant by a aisafactorofb b ais divisible by b c a is a multiple of b From your lower grade mathematics recall that called a factor or divisor of m We also say m is di
30. ften write very large or very small numbers in scientific notation also called standard form 49 Mathematics Grade 9 Example 1 1 86 x 10 is written in scientific notation Number from 1 up to Times 10 to but not including 10 a power 8 735 x 10 and 7 08 x 10 are written in scientific notation 14 73 x 107 0 0863 x 10 and 3 86 are not written in stand td form scientific notation ACTIVITY 1 16 1 By what powers of 10 must you multiply 1 3 to get a 13 b 130 c 1300 Copy and complete this table 13 1 3 x10 130 1 3 x 10 1 3002 1 3 x 13 000 1 300 000 z 2 Can you write numbers between 0 and 1 in scientific notation for example 0 00013 Copy and complete the following table 13 02 1 3 x102 1 3 x 10 132 13 1 213 x10 REESE a E SIE 10 ie cttm 100 0 0013 0 00013 0 000013 0 0000013 50 Unit 1 The Number System Definition 1 13 A number is said to be in scientific notation or standard form if it is written as a product of the form a x 10 where 1 a 10 and k is an integer Example 2 Express each of the following numbers in scientific notation a 243 900 000 b 0 000000595 Solution a 243 900 000 2 439 x10 The decimal point moves 8 places to the left b 0 000000595 5 95 x10 The decimal point moves 7 places to the right Example 3 Express 2 483 X10 in ordinary deci
31. herefore these numbers are not rational numbers Such numbers are called irrational numbers In general if a is a natural number that is not a perfect square then Va is an irrational number Example 1 Determine whether each of the following numbers is rational or irrational a 0 16666 b 0 16116111611116111116 c T Solution a In0 16666 the decimal has a repeating pattern It is a rational number and can be expressed as b This decimal has a pattern that neither repeats nor terminates It is an irrational number Unit 1 The Number System C T 3 1415926 This decimal does not repeat or terminate It is an irrational number The fraction is an approximation to the value of IL It is not the exact value In Example 1 b and c lead us to the following fact gt A decimal number that is neither terminating nor repeating is an irrational number 1 Locating irrational numbers on the number line You will need a compass and straight edge to perform the following 1 Tolocate J 2 on the number line Draw a number line At the point corresponding to 1 on the number line construct a perpendicular line segment 1 unit long Draw a line segment from the point corresponding to 0 to the top of the 1 unit segment and label it as c 0 Figure 1 5 Use the Pythagorean Theorem to show that c is V2 unit long Open the compass to the length of c With the tip of the compass at the point corresponding
32. ined by dividing the upper bound of the numerator by the lower bound of the denominator So the maximum value is 54 55 35 95 i e 1 52 2 decimal places Mathematics Grade 9 Round the following numbers to the nearest 1000 a 6856 b 74245 c 89000 d 99500 Round the following numbers to the nearest 100 a 78540 b 950 c 14099 d 2984 Round the following numbers to the nearest 10 a 485 b 692 c 8847 d 4 e 83 i Give the following to 1 d p a 5 58 b 4 04 c 157 39 d 15 045 ii Round the following to the nearest tenth a 157 39 b 12 049 c 0 98 d 2 95 iii Give the following to 2 d p a 6 473 b 9 587 c 0 014 d 99 996 iv Round the following to the nearest hundredth a 16 476 b 3 0037 c 9 3048 d 12 049 Write each of the following to the number of significant figures indicated in brackets a 48599 1 s f b 48599 3 s f c 2 57 28 3 s f d 2045 2s f e 0 08562 1 s f f 0 954 2 s f g 0 00305 2 s f h 0 954 1 sf Each of the following numbers is expressed to the nearest whole number i Give the upper and lower bounds of each ii Using x as the number express the range in which the number lies as an inequality a 6 b 83 c 151 d 1000 Each of the following numbers is correct to one decimal place i Give the upper and lower bounds of each ii Using x as the number express the range in which the number lies as an inequality a 338 b 15 6 c 1 0 d 03 e 02 Each of the following numbers is correct to two significant figu
33. ing theorem called the Division Algorithm Theorem 1 4 Division algorithm Let a and b be two non negative integers and b 0 then there exist unique non negative integers q and r such that a q Xb tr withO sr lt ob In the theorem a is called the dividend q is called the quotient b is called the divisor and r is called the remainder Example 1 Write a in the form b Xq r where 0 r b a If a 47andb 27 b Ifa lllandb 3 c Ifa 5andb 8 56 Unit 1 The Number System Solution a 6 b 37 c 0 7 47 ait 815 42 9 KF 5 21 5 q 6andr 5 2l q 0andr 5 47 7 6 5 0 52 8 0 45 q 37 andr 0 111 23 37 0 Exercise 1 13 For each of the following pairs of numbers let a be the first number of the pair and b the second number Find q and r for each pair such that a b xq r where 0 Sr lt b a 72 11 b 16 9 c 11 18 d 106 13 e 176 21 f 2539 B The Euclidean algorithm ACTIVITY 1 20 Given two numbers 60 and 36 1 Find GCF 60 36 2 Divide 60 by 36 and find the GCF of 36 and the remainder 3 Divide 36 by the remainder you got in Step 2 Then find the GCF of the two remainders that is the remainder you got in Step 2 and the one you got in step 3 4 Compare the three GCFs you got 5 Generalize your results The above Activity leads yow to another method for finding the GCF of two numbers which is called Euclidean algorithm We state this algorithm as a theorem Theorem 1 5 Euclidean algorithm
34. ively Find their sum Solution Ifthe length 6 7 cm and the width w 4 4 cm Then 6 65 S lt 6 75 and 4 35 sw lt 4 45 The lower bound of the sum is obtained by adding the two lower bounds Therefore the minimum sum is 6 65 4 35 that is 11 00 The upper bound of the sum is obtained by adding the two upper bounds Therefore the maximum sum is 6 75 4 45 that is 11 20 So the sum lies between 11 00 cm and 11 20 cm Example 5 Find the lower and upper bounds for the following product given that each number is given to 1 decimal place 3 4 X 7 6 Solution If x 3 4 and y 7 6 then 3 35 x lt 3 45 and 7 55 sy lt 7 65 The lower bound of the product is obtained by multiplying the two lower bounds Therefore the minimum product is 3 35 X7 55 that is 25 2925 The upper bound of the product is obtained by multiplying the two upper bounds Therefore the maximum product is 3 45 X7 65 that is 26 3925 So the product lies between 25 2925 and 26 3925 Example 6 Calculate the upper and lower bounds to gt gt given that each of the numbers is accurate to 1 decimal place Solution 54 5 lies in the range 54 45 lt x lt 54 55 36 0 lies in the range 35 95 lt x lt 36 05 The lower bound of the calculation is obtained by dividing the lower bound of the numerator by the upper bound of the denominator So the minimum value is 54 45 36 05 i e 1 51 2 decimal places The upper bound of the calculation is obta
35. led a repeating decimal Expressing terminating and repeating decimals as fractions Example 2 Express each of the following decimals as a fraction in its simplest form lowest terms a 085 b 13456 Unit 1 The Number System Solution 100 85 17 a 0 85 20 85 x 2 100 100 20 Why 4 e HA b 1 3456 1 3456 x10000 10 _13456 841 Yn I 10000 1 3456 X 10 10000 625 For example if d 2 128 then n 3 10 x2 128 2128 266 7 AKNV 2 128 EV 10 1000 125 Example 3 Express each of the following decimals aSa fraction ratio of two integers j a 07 NM b o5 0AN Solution a Let d 0 7 0777 then 10d 7 7711 N Y multiplying dy 10 because 1 digit repeats Subtract d 20 777 A to elinfitale the repeating part 0 777 iod 945 YOY Mon d M 9d Cg suDrractitig expression B from expression 1 eS iN i I N dividing both sides by 9 Hence 0 7 IN MA N 9 N A b Let d 025 0 252525 Then 100d 25 2525 multiplying d by 100 because 2 digits repeat 4 E p Mathematics Grade 9 100d 25 252525 subtracting ld from 100d eliminates the ld 0 252525 repeating part 0 2525 99d 25 25 ANI Y 99 ANY P iy 2 T4 da N So 025 222 47 lop In Example 3a one digit repeats So you multiplied d by Qu uos 3b ive di digits repeat So you multiplied d by 100 XN A P9 d The algebra used in the abov
36. mal notation Solution 2 483 x10 2 483 x100 000 248 300 Example 4 The diameter of a red blood cell is about 7 4 X10 cm Write this diameter in ordinary decimal notation 1 10 000 So the diameter of a red blood cell is about 0 00074 cm Calculators and computers also use scientific notation to display large numbers and small numbers but sometimes only the exponent of 10 is shown Calculators use a space before the exponent while computers use the letter E gt The calculator display 5 23 06 means 5 23 X10 5 230 000 The following example shows how to enter a number with too many digits to fit on the display screen into a calculator Example 5 Enter 0 00000000627 into a calculator Solution First write the number in scientific notation 0 00000000627 6 27 x10 Then enter the number 7 4 X0 0001 0 00074 Solution 7 4 x10 7 4 xc 7 4 x 6 27 lexp 9 giving 6 27 09 Decimal Scientific Calculator Computer notation notation display display 250 000 2 5 x 10 2 5 0 5 2 5E 5 0 00047 4 7 x107 4 7 04 4 7E 4 Mathematics Grade 9 Exercise 1 10 1 Express each of the following numbers in scientific notation a 0 00767 b 5 750 000 000 c 0 00083 d 400 400 e 0 054 2 Express each of the following numbers in ordinary decimal notation a 4 882 x10 b 1 19 x10 c 2 021 x10 3 Express the diameter of an electron which is about 0 0000000000004 cm i
37. mplification of radicals ACTIVITY 1 11 Evaluate each of the following and discuss your result in groups H 305 b 3 c 4 5 d xs e 2 f 3c Does the sign of your result depend on whether the index is odd or even Can you give a general rule for the result of Va where a is a real number and i n is an odd integer il nis an even integer To compute and simplify expressions involving radicals it is often necessary to distinguish between roots with odd indices and those with even indices Veo sra F V3 M Vx Ax 4 2Y 3103 22 hr 24 Example 9 Simplify each of the following ON an o gt Ww a Jy b NT i e C A25x d x Solution a yp p SITE A BF c lt V25 selise d V2 e de Lf A radical Wa is in simplest form if the radicand a contains no factor that can be expressed as an n power For example4 54 is not in simplest form because 3 isa factor of 54 Using this f ct and the radical notations of Theorem 1 2 and Theorem 1 3 you can simplify radicals 33 Mathematics Grade 9 Example 10 Simplify each of the following a 448 b xii E ap 8l Solution a 448 416 3 2416 xJ 3 2443 b 9x3 3 9 x3 23 27 23 32 6x _ f6 4 16 2 c 4 L4 L4 x42 p x42 4 9 T 81 is 4 81 3 1 Simplify each of the following a 8 b 5j 3 c 3v8x d 486 1 e 4512 f 3V27 xy g 4405 2 Simplify each of the following if possible State restrictions where necessary 2 x 55 b x56 c 4L
38. n scientific notation 1 2 8 Rationalization ACTIVITY 1 17 Find an approximate value to two decimal places for the following 1 E N 2 In calculating this the first step 1s to find an approximation of V2 in a reference book or other reference material It is 1 414214 In the calculation of A 1 is divided by 2 1 414214 which is a difficult task However evaluating a as CUM 0 707107 is easy Jz i 1 l Since is equivalent to E How you see that in order to evaluate an expression NI with a radical in the denominator first you should transform the expression into an equivalent expression with a rational number in the denominator The technique of transferring the radical expression from the denominator to the numerator is called rationalizing the denominator changing the denominator into a rational number The number that can be used as a multiplier to rationalize the denominator is called the rationalizing factor This is equivalent to 1 Unit 1 The Number System For instance if Jn is an irrational number then can be rationalized by multiplying Vn it by x So 3 is the rationalizing factor Example 1 Rationalize the denominator in each of the following 543 p a d 8 5 V3 3 2 Solution a The rationalizing factor is a aL s 558 553 5 545 s c m ays avs Js 8405 g5 S6 8 b The rationalizing factor is v3 WF 6 _ 6 1 v3 _6v3 6 B WS Ww dm B oo
39. nt a 013 b 0305 c 0 381 Verify each of the following computations by converting the decimals to fractions a 0 275 0 714 0 989 b 0 6 1 142857 0 476190 Mathematics Grade 9 1 2 2 Irrational Numbers Remember that terminating or repeating decimals are rational numbers since they can be expressed as fractions The square roots of perfect squares are also rational numbers For example V4 is a rational number since 4 2 Similarly 40 09 is a rational number because J0 09 0 3 1s a rational number If x 4 then what do you think is the value of x x 244 Therefore x is a rational number What if x 2 3 In Figure 1 4 of Section 1 1 3 where do numbers like J 2 and V5 fit Notice what happens when you find 4 2 and V5 with your calculator z If you first press the button 2 and then the square root Study Hint button you will find 2 on the display Most calculators round answers but some ie V2 2V 2 1414213562 truncate answers i e J5 5 V 2 236067977 they cut off at a certain point ignoring subsequent digits Note that many scientific calculators such as Casio ones work the same as the written order i e instead of pressing 2 and then the V button you press the V button and then 2 Before using any calculator it is always advisable to read the user s manual Note that the decimal numbers for J2 and 45 do not terminate nor do they have a pattern of repeating digits T
40. number can also be approximated to a given number of decimal places d p This refers to the number of figures written after a decimal point Example 4 a Write 7 864 to 1 d p b Write 5 574 to 2 d p Solution a The answer needs to be written with one number after the decimal point However to do this the second number after the decimal point also needs to be considered If it is 5 or more then the first number is rounded up That is 7 864 is written as 7 9 to 1 d p 44 Unit 1 The Number System b The answer here is to be given with two numbers after the decimal point In this case the third number after the decimal point needs to be considered As the third number after the decimal point is less than 5 the second number is not rounded up That is 5 574 1s written as 5 57 to 2 d p Note that to approximate a number to 1 d p means to approximate the number to the nearest tenth Similarly approximating a number to 2 decimal places means to approximate to the nearest hundredth C Significant figures Numbers can also be approximated to a given number of significant figures s f In the number 43 25 the 4 is the most significant figure as it has a value of 40 In contrast the 5 is the least significant as it only has a value of 5 hundredths When we desire to use significant figures to indicate the accuracy of approximation we count the number of digits in the number from left to right beginning at the first non zero digit
41. o for various reasons and at various occasions For example you can count the money you receive from someone a tailor measures the length of the shirt he she makes for us and a carpenter counts the number of screws required to make a desk Counting The process of counting involves finding out the exact number of things For example you do counting to find out the number of students in class The answer is an exact number and is either correct or if you have made a mistake incorrect On many occasions just an estimate is sufficient and the exact number is not required or important Measuring If you are finding the length of a football field the weight of a person or the time it takes to walk down to school you are measuring The answers are not exact numbers because there could be errors in measurements 2 Estimation In many instances exact numbers are not necessary or even desirable In those conditions approximations are given The approximations can take several forms Here you shall deal with the common types of approximations A Rounding If 38 518 people attend a football game this figure can be reported to various levels of accuracy To the nearest 10 000 this figure would be rounded up to 40 000 To the nearest 1000 this figure would be rounded up to 39 000 To the nearest 100 this figure would be rounded down to 38 500 In this type of situation it is unlikely that the exact number would be reported B Decimal places A
42. of real numbers denoted by IR is defined by R x x is rational or x is irrational The set of irrational numbers is not closed under addition subtraction multiplication and division The sum of an irrational and a rational number is always an irrational number For any real number b and positive integer n gt 1 1 b b Whenever lb is a real number For all real numbers a and b 40 for which the radicals are defined and for all integers n gt 2 ya i fab Vaile i f b ds A number is said to be written in scientific notation standard notation if it is written in the form a X10 where 1 Sa lt 10 and k is an integer Let a and b be two non negative integers and b 0 then there exist unique non negative integers q and r such that a q Xb r with O xr b If a b q and r are positive integers such that a q Xb r then GCF a b GCF b r Unit 1 The Number System Review Exercises on Unit 1 Determine whether each of the following numbers is divisible by 2 3 4 5 6 8 9 or 10 a 533 b 4 299 c 111 Find the prime factorization of a 150 b 202 c 63 Find the GCF for each set of numbers given below a 16 64 b 160 320 480 Express each of the following fractions or mixed numbers as a decimal a i b c 5t d 3 Express each of the following decimals as a fraction or mixed number in its simplest form a 0 65 b 005 c 016 d 2454 e 002 Arrange each of the following sets of rati
43. onal numbers in increasing order ana 100 2 30 2 11 16 67 3 18 27 100 be 3523215233253 Write each of the following expressions in its simplest form a 180 b ue c 450 d 243 4342 180 196 Give equivalent expression containing fractional exponents for each of the following 13 a V15 b va b c JE Y d 4 Express the following numbers as fractions with rational denominators a b 2 c 2 ad ge S2 3 dep 16 61 Mathematics Grade 9 10 11 12 13 14 15 62 Simplify a 7455 22 b_ 2 V5 12 5 c 2 6 354 d 2 34 V7 2V7 If V5 2 236 and V10 3 162 find the value of V10 20 40 5 v80 If XS x H6 y find the values of x and y 3 2 43 Express each of the following numbers in scientific notation a 7 410 00 b 0 0000648 c 0 002056 d 12 4 x 10 Simplify each of the following and give the answer in scientific notation 4 2 a 10 x105x27 b A C 0 00032 x 0 002 The formula d 3 56Vh km estimates the distance a person can see to the horizon where h is the height of the eyes of the person from the ground in metre Suppose you are in a building such that your eye level is 20 m above the ground Estimate how far you can see to the horizon d Figure 1 11
44. ositive integer greater than 1 then the principal n root of b denoted by 4b is defined as the positive n root of b if b gt 0 4b the negative n root of b if b lt 0 and n is odd 0 if b 0 Unit 1 The Number System Example 2 a 16 2 because 2 16 b 0 04 0 2 because 0 2 0 04 c 1000 10 because 10 1000 Numbers such as 423 3 35 and 3 10 are irrational numbers and cannot be written as terminating or repeating decimals However it is possible to approximate irrational numbers as closely as desired using decimals These rational approximations can be found through successive trials using a scientific calculator The method of successive trials uses the following property Example 3 Find a rational approximation of 4 43 to the nearest hundredth Solution Use the above property and divide and average on a calculator Since 6 36 lt 43 lt 49 7 6 lt J43 lt 7 Estimate 4 43 to tenths J43 6 5 Divide 43 by 6 5 6 615 6 5 43 000 Average the divisor and the quotient cones 6 558 Divide 43 by 6 558 6 557 6 558 43 000 Now you can check that 6 557 lt 43 lt 6 558 Therefore 43 is between 6 557 and 6 558 It is 6 56 to the nearest hundredth Example 4 Through successive trials on a calculator compute 4 53 to the nearest tenth 27 Mathematics Grade 9 Solution 33227 lt 53 lt 64 4 That is 3 lt 53 lt 4 So 3 lt 3 53 4 Try 3
45. product of the numbers Write each of the following fractions in simplest form a _3 b 24 c 48 d 72 9 120 72 98 How many factors does each of the following numbers have a 12 b 18 c 24 d 72 Find the value of an odd natural number x if LCM x 40 1400 There are between 50 and 60 eggs in a basket When Mohammed counts by 3 s there are 2 eggs left over When he counts by 5 s there are 4 left over How many eggs are there in the basket The GCF of two numbers is 3 and the LCM is 180 If one of the numbers is 45 what is the other number i Let a b c d be non zero integers Show that each of the following is a rational number a c a c Cc b d b What do you conclude from these results a d b d afa S19 SIES 1 3 ii Find two rational numbers between 3 and 5 d Unit 1 The Number System 12 THE REAL NUMBER SYSTEM IEX Representation of Rational Numbers by Decimals In this subsection you will learn how to express rational numbers in the form of fractions and decimals ACTIVITY 1 5 1 a What do we mean by a decimal number b Give some examples of decimal numbers 2 How do you represent and i as decimals 3 Can you write 0 4 and 1 34 as the ratio or quotient of two integers Remember that a fraction is another way of writing division of one quantity by another Any fraction of natural numbers can be expressed as a decimal by dividing the numerator by the denominator Example
46. res i Give the upper and lower bounds of each ii Using x as the number express the range in which the number lies as an inequality a 42 b 0 84 c 420 d 5000 e 0 045 Unit 1 The Number System 10 11 12 13 Calculate the upper and lower bounds for the following calculations if each of the numbers is given to 1 decimal place a 9 5 X7 6 b 11 0 x 15 6 eom 32 0 254 4 9 6 4 8 2 2 6 The mass of a sack of vegetables is given as 5 4 kg a Illustrate the lower and upper bounds of the mass on a number line b Using M kg for the mass express the range of values in which it must lie as an inequality The masses to the nearest 0 5 kg of two parcels are 1 5 kg and 2 5 kg Calculate the lower and upper bounds of their combined mass Calculate upper and lower bounds for the perimeter of a school football field shown if its dimensions are correct to 1 decimal place 109 7 m 48 8m Figure 1 9 Calculate upper and lower bounds for the length marked x cm in the rectangle shown The area and length are both given to 1 decimal place X Area 223 2 cm Figure 1 10 Scientific Notation Standard form In science and technology it is usual to see very large and very small numbers For instance The area of the African continent is about 30 000 000 km The diameter of a human cell is about 0 0000002 m Very large numbers and very small numbers may sometimes be difficult to work with or write Hence you o
47. s gem because 4 w a 2 s 23 AP AWS _ V4 c The rationalizing factor is V 3 i i 2 If a radicand itself is a fraction ores E then it can be written in the equivalent form B so that the procedure described above can be applied to rationalize the denominator Therefore RENE 6 6 8T 5 de In general Mathematics Grade 9 Exercise 1 11 Simplify each of the following State restrictions where necessary In each case state the rationalizing factor you use and express the final result with a rational denominator in its lowest term 2 42 542 12 5 n a pi g c a e V V2 v6 4 10 V27 18 3 i 9 3 20 af f 2 dE hc i j 233 j a 3 4 V5 P m r d VN v4 f More on rationalizations of denominators SIN ACTIVITY 1 18 Find the product of each of the following 1 2 W3 2 V3 2 5 43V2 5 3v2 3 S JS E You might have observed that the results of all of the above products are rational numbers NN J Soe This leads you to the following conclusion Unit 1 The Number System Example 2 Rationalize the denominator of each of the following 5 3 J6 4342 a Si Solution 1 a The rationalizing factor is 1 2 s _ 5042 54540 1 42 d V2 0 2 p ya _5 5V2 __ NNN 1 2 6 30 b The rationalizing factor is Vo 342 We osa seg mox pesa 46 2 vef O 3y2 3 V6 AXR 1 AQ V a WAR 342 V6 A gt
48. s of the three numbers b X find the greatest common factor of the three numbers Given two or more natural numbers a number which is a factor of all of them is called a common factor Numbers may have more than one common factor The greatest of the common factors is called the greatest common factor GCF or the highest common factor HCF of the numbers Example 1 Find the greatest common factor of a 36and60 b 32and27 The greatest common factor of two numbers a and b is denoted by GCF a b Unit 1 The Number System Solution a First make lists of the factors of 36 and 60 using sets Let F36 and Feo be the sets of factors of 36 and 60 respectively Then Fas 1 2 3 4 6 9 12 18 36 Feo 1 2 3 4 5 6 10 12 15 20 30 60 You can use the diagram to summarize the information Notice that the common factors are shaded in green They are 1 2 3 4 6 and 12 and the greatest is 12 i e F36 N Feo 1 2 3 4 6 12 Therefore GCF 36 60 12 b Similarly F32 1 2 4 8 16 32 and Por 241 3 9 27 Therefore F3 N F5 1 Thus GCF 32 27 21 r more natural numbers that hav Definition 1 2 The greatest common factor GCF of two or more natural numbers 1s the greatest natural number that is a factor of all of the given numbers Group Work 1 2 Mathematics Grade 9 2 a Compare the result of 1c with the GCF of the given numbers Are they the same b
49. t every decimal number be it rational or irrational can be associated with a unique point on the number line and conversely that every point on the number line can be associated with a unique decimal number either rational or irrational This is usually expressed by saying that there exists a one to one correspondence between the sets C and D where these sets are defined as follows C P P is a point on the number line D d d is a decimal number The above discussion leads us to the following definition Definition 1 6 Real numbers A number is called a real number if and only if it is either a rational number or an irrational number The set of real numbers denoted by R can be described as the union of the sets of rational and irrational numbers R x xisa rational number or an irrational number The set of real numbers and its subsets are shown in the adjacent diagram reat numbers Rational Numbers Irrational Numbers From the preceding discussion you can JZ see that there exists a one to one T correspondence between the set R and Jeg 1 0100100010 the set C P P is a point on the number line Figure 1 8 22 Unit 1 The Number System It is good to understand and appreciate the existence of a one to one correspondence between any two of the following sets 1 D x xis a decimal number 2 Pz x xisapoint on the number line
50. tion it is often expressed in simplest form lowest terms A fraction E is in simplest form when GCF a 5 1 Example 1 Write in simplest form 30 2x D6 2 45 3xb6 3 Solution by factorization and cancellation Hence A when expressed in lowest terms simplest form is gt 1 Determine whether each of the following numbers is prime or composite a 45 b 23 a 9il d 153 2 Prime numbers that differ by two are called twin primes i Which of the following pairs are twin primes a 3and5 b 13and 17 c Sand7 il List all pairs of twin primes that are less than 30 3 Determine whether each of the following numbers is divisible by 2 3 4 5 6 8 9 or 10 a 48 b 153 c 2 470 d 144 e 12 357 4 a Is3afactorof 777 b Is 989 divisible by 9 C Is 2 348 divisible by 4 5 Find three different ways to write 84 as a product of two natural numbers 6 Find the prime factorization of a 25 b 36 C 117 d 3 825 Mathematics Grade 9 10 11 12 13 14 15 16 17 18 19 20 Is the value of 2a 3b prime or composite when a 11 and b 7 Write all the common factors of 30 and 42 Find a GCF 24 36 b GCF 35 49 84 Find the GCF of 2 x3 x 5 and 2 x3 x5 Write three numbers that have a GCF of 7 List the first six multiples of each of the following numbers a Ji b 5 c 14 d 25 e 150 Find a LCM 12 16 b LCM 10 12 14 c LCM 15 18 d LCM 7 10 When will the LCM of two numters be the
51. visible by p In this case p is Similarly q is also a factor or divisor of m and m is divisible by q Mathematics Grade 9 For example 621 is a multiple of 3 because 621 3 X207 Definition 1 1 Prime numbers and composite numbers A natural number that has exactly two distinct factors namely 1 and itself 1s called a prime number A natural number that has more than two factors is called a composite number Note 1 is neither prime nor composite Work 1 1 1 List all factors of 24 How many factors did you find 2 The area of a rectangle is 432 sq units The measurements of the length and width of the rectangle are expressed by natural numbers Find all the possible dimensions length and width of the rectangle 3 Find the prime factorization of 360 The following rules can help you to determine whether a number is divisible by 2 3 4 5 6 8 9 or 10 Divisibility test A number is divisible by v 2 if its unit s digit is divisible by 2 3 if the sum of its digits is divisible by 3 4 if the number formed by its last two digits is divisible by 4 5 if its unit s digit is either O or 5 6 if it is divisible by 2 and 3 8 if the number formed by its last three digits is divisible by 8 SRS ONU oe 9 if the sum of its digits is divisible by 9 Y 10 if its unit s digit is 0 Observe that divisibility test for 7 is not stated here as it is beyond the scope of your present level
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