Mathematics Grade 9 Unit One Numbers System PDF

Numerical Value Arabic Numeral Babylonian IRAN EE ARAL AR A444 2 « | «|v» Egypt...

Numerical Value Arabic Numeral Babylonian IRAN EE ARAL AR A444 2 « | «|v» Egyptian = Hieroglyphic Eri ITI II ITI AN DVN TV Greek Herodianic Roman 1 I I Vv X |XX |XX1I| C Ethiopian Geez THE NUMBER SYSTEM After completing this unit, you should be able to: % know basic concepts and important facts about real numbers. = 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 The real 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. BRE REVISION ON THE SET OF RATIONAL NUMBERS ACTIVITY 1.1 The diagram below shows the relationships between the sets of 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 d 0.625 e 0615 2 i Define the set of: a Natural numbers 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!  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. The set {1, 2, 3,...} describes the set of natural numbers. The set {1,2,3,...}U{...—3, —2, —1} describes the set of integers. TUT 57 is a composite number. 0 {1} N {Prime numbers} = J. oo {Prime numbers } U{Composite number} = {1, 2, 3,...}. 0 {Odd numbers} N{Composite numbers} # &. - g 48 is a multiple of 12. h Sis afactor of 72. i 621 is divisible by 3. i {Factors of 24} N {Factors of 87} ={1, 2, 3}. k {Multiples of 6} N {Multiples of 4} = {12, 24}. I 2?x3%x35 is the prime factorization of 180. 2 Given two natural numbers a and b, what is meant by: a aisafactorofb b ais divisiblebybd C ais amultiple of b From your lower grade mathematics, recall that; In this case, p is called a factor or divisor of m. We also say m is divisible by p. 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 x 207. Definition 1.1 Prime numbers and composite numbers & A natural number that has exactly two distinct factors, namely 1 and itself, is called a prime number. 4 A natural number that has more than two factors is called a composite number. Note; 1 is neither prime nor composite. 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, 5,6,8,9o0r 10. Divisibility test A number is divisible by: v' 2, ifits unit’s digit is divisible by 2. 3, if the sum of its digits is divisible by 3. NN 4, if the number formed by its last two digits is divisible by 4. 5, if its unit’s digit is either 0 or 5. 6, if it is divisible by 2 and 3. NEN 8, if the number formed by its last three digits is divisible by 8. AN 9, if the sum of its digits is divisible by 9. v' 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. 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. ¢ 2,416is 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 @& 30 60=2x2x3x5=2*x3x5. ols 5 The prime factorization of 60 is also found by oo using a factoring tree. Note that the set {2, 3, 5} is a set of prime factors of 60. Is this set unique? This property leads us to state the Fundamental Theorem of Arithmetic. i Theorem 1.1 Fundamental theorem of arithmetic Every composite number can be expressed (factorized) as a product of 3 primes. This factorization is unique, apart from the order in which the i prime factors occur. i 3 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 J 1,530+2=1765 765+3=255 255+3=85 85+5=17; and 17 is a prime number. Therefore, 1,530 =2 x 32 x 5 x 17. [1.1.2] 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 find the greatest common factor of the two numbers. 2 Given the numbers 36, 42 and 48, a find the common factors of the three numbers. b 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. » The greatest common factor of two numbers a and 4 is denoted by GCF (a, b). Example 1 Find the greatest common factor of: a 36 and 60. b 32and?27.  Unit 1 The Number System Solution: a First, make lists of the factors of 36 and 60, using sets. Let Fi and Feo be the sets of factors of 36 and 60, respectively. Then, Fs Feo Fis=1{1,2,3,4,6,9, 12, 18, 36} Feo=1{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 Figure 1.2 12 and the greatest is 12. ie, Fis NFeo={1,2,3,4,6,12} Therefore, GCF (36, 60) = 12. b Similarly, Fx» =(1,2,4,8, 16,32} and F»=1{1,3,9,27} Therefore, F3, N Fy; = {1} Thus, GCF (32,27)=1 Figure 1.3 Definition 1.2 The greatest common factor (GCF) of two or more natural numbers is 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 Compare the result of 1¢ 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. 180=2>x3*x 5; 216=2° x 3% 540=2"x 3x5 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: * List the 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: 1 ‘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 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 &. Example 3 Find LCM (8, 9). Solution: Let Mg and Ms be the sets of multiples of 8 and 9 respectively. Ms ={8,16,24,32,40,48,56,64,.80,88,...} My = {9,18,27,36,45,54,63,,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: 9=3%x3=3 The prime factors that appear in these - - factorizations are 2,3 and 7. 21=3x7 Considering the greatest number of times 2MU=2x2%x2x3=2%3 each prime factor appears, we can get 2°, 3%and 7, respectively. Therefore, LCM (9, 21, 24) = 23x 32x 7 = 504. ACTIVITY 1.4 Alas 1 Find: a The GCF and LCM of 36 and 48 b GCF (36, 48) x LCM ( 36, 48) c 36x48 2 Discuss and generalize your results. » For any natural numbers a and b, GCF (a, b) X LCM (a,b) =a X b. Mathematics Grade 9 1.1.3] Rational Numbers HisTORICAL NOTE: About 5,000 years ago, Egyptians used hieroglyphics to represent numbers. = Nl ©) T The Egyptian concept of fractions was 1 10 100 1,000 mostly limited to fractions with numerator -_— a= = — 1. The hieroglyphic was placed under the | Nn nn 9) symbol — to indicate the number as a = denominator. Study the examples of i 1 4 1 1 2 3 10 20 100 Egyptian fractions. Recall that the set of integers is given by zZ={..,-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 > where a and b are integers and b # 0, is called a rational number. The set of rational numbers, denoted by Q, is the set described by Q= {aaa are integers and b # of \_ b 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 —7 can be. 4 —7 written as — and —. 1 1 The set of rational numbers also includes terminating and repeating decimal numbers because terminating and repeating decimals 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 29 100 Mixed numbers are also included in the set of rational numbers because any mixed number 30 2x3x5 2 — = = —. can be written as an improper fraction. 45 3x3x5 3 For example, 22 can be written as > ‘When a rational number is expressed as a fraction, it is often expressed in simplest form a (lowest terms). A fraction » is in simplest form when GCF (q, b) = 1. Example 1 Write in simplest form. Solution: 30 _23x5 2, (by factorization and cancellation) 45 3x3x5 3 Hence - when expressed in lowest terms (simplest form) is 2 1 Determine whether each of the following numbers is prime or composite: a 45 b 23 c 91 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 ¢ Sand? ii 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 Cc 2,470 d 144 e 12,357 4 a Is3afactorof777? b Is 989 divisible by 9? ¢ 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 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 2x 3° x 5% and2° x3 x 5°. Write three numbers that have a GCF of 7. List the first six multiples of each of the following numbers: a 7 b 5 c 14 d 25 e 150 13 Find: a LCM(12, 16) b LCM(10, 12,14) c¢ LCM(5,18) d LCM(7,10) 14 When will the LCM of two numbers be the product of the numbers? 15 Write each of the following fractions in simplest form: a _3 b 24 Cc had d 2 9 120 72 98 16 How many factors does each of the following numbers have? a 12 b 18 c 24 d 72 17 Find the value of an odd natural number x if LCM (x, 40) = 1400. 18 There are between 50 and 60 eggs in a basket. When Mohammed counts by 3s, 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? 19 The GCF of two numbers is 3 and the LCM is 180. If one of the numbers is 45, what is the other number? 20 i Let a, b, ¢, d be non-zero integers. Show that each of the following is a rational number: a c a c a_c a. c a —+-= b —— c —X— d —+— b d b d b d bd What do you conclude from these results? ii Find two rational numbers between : and 2.