The Basics. GFP - Sohar University. SET 1 - Chapter 1

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1 The Basics األساسي ات األعداد الحقيقي ة 1-1 Real Numbers Real numbers are used in everyday life to describe quantities such as speed, area, prices, age, temperature, and population. Real numbers are usually represented by symbols as in the following numbers: 2

2 The types of numbers that make up the real number system are: األعداد الطبيعية (أو أعداد الع د ) ) ( Numbers Natural or Counting = {1, 2, 3, 4, } األعداد الك لي ة ) ( Numbers Whole = {0, 1, 2, 3, } األعداد الصحيحة ) ( Integers = {, 3, 2, 1, 0, 1, 2, 3, } 3 األعداد النسبي ة ) ( Numbers Rational A number is classified as rational if it can be expressed as a fraction. The following types of numbers can be written as fractions and hence are rational numbers: 4

3 األعداد الالنسبي ة ) ( Numbers Irrational A number that cannot be written as a fraction is considered irrational. An irrational number is a decimal that doesn t infinitely repeat itself yet never terminates. The following numbers are examples on irrational numbers: 5 6

4 خط األعداد 1-2 The Number Line The set of all rational numbers combined with the set of all irrational numbers gives us the set of real numbers. The real numbers are modeled using a number line, as shown below. Each point on the line represents a real number, and every real number is represented by a point on the line. Negative numbers represent distances to the left of zero, and positive numbers are distances to the right. The arrows on the end indicate that it keeps going forever in the left and right directions. 7 Example 1: For the following problems, choose the correct answer. (i) Which of the following numbers is a positive integer? (a) (b) (c) (d) 0.26 (ii) Which of the following numbers is a negative integer? (a) (b) 3 (c) 3.86 (d) (iii) Which of the following numbers is a rational number? (a) (b) (c) (d) (iv) Which of the following numbers is an irrational number? (a) (b) (c) (d) π (v) Which of the following numbers is a natural number? (a) (b) (c) 5 (d) Solution: (i) c, (ii) d, (iii) a, (iv) d, (v) c 8

5 Example 2: For the following problems, choose the correct answer. (i) Which of the following numbers is a positive integer? (a) (b) (c) (d) 0.26 (ii) Which of the following numbers is a negative integer? (a) (b) (c) (d) (iii) Which of the following numbers is a rational number? (a) (b) (c) (d) π (iv) Which of the following numbers is an irrational number? (a) (b) (c) (d) (v) Which of the following numbers is a natural number? (a) (b) (c) (d) Solution: (i) a, (ii) c, (iii) c, (iv) b, (v) a Odd and Even Numbers األعداد الفردية Odd Numbers Odd numbers are integers not divisible by 2: Odd Numbers = {, 5, 3, 1, 1, 3, 5, } األعداد الزوجية Even Numbers Even numbers are integers divisible by 2: Even Numbers = {, 6, 4, 2, 0, 2, 4, 6, } 10

6 1-4 Prime and Composite Numbers األعداد األولية Prime Numbers A prime number is a number that has exactly two factors, it can be evenly divided by only itself and 1. Prime Numbers = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29 } The only even prime number is 2. األعداد المركبة Composite Numbers A composite number is a number divisible by more than just 1 and itself. Composite Numbers = {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, } Zero and 1 are not prime numbers or composite numbers. 11 المربعات الكاملة 1-5 Perfect Squares A perfect square is an integer that is the square of an integer. The first 15 perfect squares are: 12

7 المكعبات الكاملة 1-6 Perfect Cubes Perfect cubes are the result when integers are multiplied by themselves twice. The first 5 perfect cubes are: 13 خصائص العمليات الحسابية 1-7 Properties of Basic Operations خاصية اإلنغالق للجمع Closure Property of Addition Closure is when all results belong to the original set. If you add two even numbers, the answer is still an even number. (2 + 4 = 6), therefore, the set of even numbers is closed under addition (has closure). If you add two odd numbers, the answer is not an odd number. (3 + 5 = 8), therefore, the set of odd numbers is not closed under addition (no closure). 14

8 خاصية اإلنغالق للضرب Closure Property of Multiplication Closure is when all results belong to the original set. If you multiply two even numbers, the answer is still an even number. (2 4 = 8), therefore, the set of even numbers is closed under multiplication (has closure). If you multiply two odd numbers, the answer is an odd number. (3 5 = 15), therefore, the set of odd numbers is closed under multiplication (has closure). 15 Commutative Property of Addition خاصية اإلبدال للجمع 2+ 3 = a + b = b + a Commutative Property of Multiplication 4 7 = 7 4 a b = b a خاصية اإلبدال للضرب Associative Property of Addition خاصية اإلقتران للجمع (4 + 5) + 8 = 4 + (5 + 8) (a + b) + c = a + (b + c) Associative Property of Multiplication (3 6) 9 = 3 (6 9) (a b) c = a (b c) خاصية اإلقتران للضرب 16

9 Identity Property of Addition خاصية التطابق للجمع 5+ 0 = 5 a + 0 = a Identity Property of Multiplication 4 1 = 4 a 1 = a خاصية التطابق للضرب Inverse Property of Addition خاصية اإلنعكاس للجمع 3 + ( 3) = 0 a + ( a) = 0 خاصية اإلنعكاس للضرب Inverse Property of Multiplication 2 = 1 a a = 1 17 خاصية التوزيع Distributive Property 2(3 + 4) = 2(3) + 2(4) a(b + c) = a(b) + a(c) (2 + 3)(4 + 5) = 2(4) + 2(5) + 3(4) + 3(5) (a + b)(c + d) = a(c) + a(d) + b(c) + b(d) 18

10 Example 3: For the following problems, choose the correct answer. (i) Which of the following numbers is an odd number? (a) 532 (b) 261 (c) 1114 (d) 1826 (ii) Which of the following numbers is an even number? (a) 209 (b) 245 (c) 3665 (d) 9376 (iii) Which of the following numbers is a perfect square? (a) 7 (b) 8 (c) 9 (d) 10 (iv) Which property is expressed in (2 + 7) + 5 = 2 + (5 + 7) (a) Commutative property of addition (b) Inverse property of multiplication (c) Associative property of multiplication (d) Associative property of addition Solution: (i) b, (ii) d, (iii) c, (iv) d 19 الفترات 1-8 Intervals A subset of the real line is called an interval if it contains at least two numbers and contains all the real numbers lying between any two of its elements. Geometrically, intervals correspond to rays and line segments on the number line, along with the number line itself. 20

11 The table below shows the types of intervals and the ways used to describe them. 21 Example 4: Use the number line representation to represent the following intervals: (a) ( 2, 5) (b) [ 3, 4] (c) [ 7, 1) (d) (1, 6] (e) ( 3, ) (f) [4, ) (g) (, 2) (h) (,4] (i) {x 2 x < 6} (j) {x x < 3} Solution: 22

12 23 العوامل 1-9 Factors Factors of a whole number are all whole numbers that it can be divided by exactly. In other words, any two whole numbers are factors of the product produced by multiplying them. The factors of 12 are {1, 2, 3, 4, 6, 12} since 12 is divisible by all of them: 1 12 = 12, then 1 and 12 are factors of 12, and 3 4 = 12, then 3 and 4 are factors of 12, and 2 6 = 12, then 2 and 6 are factors of 12 24

13 العوامل المشتركة 1-10 Common Factors A common factor of two or more numbers is a number that is a factor of all them. For example, to find the common factors of 12 and 18 we need to write all the factors of each of them and then find which of these factors are factors of 12 and 18 at the same time: So, the common factor of 12 and 18 are {1, 2, 3, 6} 25 العامل المشترك األكبر (GCF) 1-11 The Greatest Common Factor The greatest common factor (GCF) of two or more numbers is the largest factor that is common to these numbers. For the previous example, 6 is the largest factor of all factors common to 12 and 18: So, the GCF of 12 and 18 is 6 26

14 Example 5: Find the GCF of 24 and 32. Solution: The common factors are {1, 2, 4, 8} and 8 is the largest one So, the GCF of 24 and 32 is 8 27 المضاعفات 1-12 Multiples Multiples of a number are all products produced by multiplying the number by the other whole numbers. A whole number is a multiple of itself since it is equal to 1 times that number. A whole number is a multiple of its factors. The multiples of 3 can be found by multiplying 3 by 1, 2, 3, 4, 5, 6, and as shown below: 3 1 = = = = = = 18, and so on So, the multiples of 3 are {3, 6, 9, 12, 15, 18, } 28

15 المضاعفات المشتركة 1-13 Common Multiples Common multiples of two or more numbers are all multiples that are common to these numbers. To find the common multiples of two or more numbers, the multiples of each number are listed and then the common ones are specified as the common multiples. For example, to find the first three common multiples of 3 and 5, we write the multiples of each of them: So, the first three common multiples of 3 and 5 are {15, 30, 45} 29 المضاعف المشترك األصغر (LCM) 1-14 The Lowest Common Multiple The lowest common multiple (LCM) of two or more numbers is the smallest (first) factor that is common to these numbers. For the previous example, 15 is the first multiple of all multiples which are common to 3 and 5: Therefore, the LCM of 3 and 5 is 15 30

16 Example 6: Find the LCM of 3 and 8. Solution: The first common multiple is 24 So, the LCM of 3 and 8 is تسلسل تنفيذ العمليات الحسابية 1-15 Order of Operations When a problem contains a number of operations, a certain order should be followed in carrying out these operations. This order is commonly referred to as BODMAS which is the acronym of Brackets, Order, Division, Multiplication, Addition and Subtraction and as shown below: 32

17 Example 7: Simplify the following: (a) (b) (c) (d) (6 1) 4 Solution: (a) = 6 12 (multiplication) = 18 (then addition) (b) = 3 4 (division / multiplication, left to right) = 12 (c) = 12 4 (division / multiplication, left to right) = 3 33 (d) (6 1) 4 = (brackets first) = (powers next) = (multiplication) = (addition / subtraction, left to right) = (addition / subtraction, left to right) =

18 القيمة المطلقة 1-16 Absolute Value The absolute value (or modulus) of a real number is the non-negative (positive) value of that number. The absolute value is usually represented by writing the number inside two vertical bars. Absolute value is used when considering the magnitude of something regardless to whether it is positive or negative. On the number line, the absolute value of a number is the distance from that number to the origin (zero). 35 Example 8: Simplify the following: (a) (b) (c) 4 ( 6) Solution: (a) = = 11 2 = 9 (b) = = 11 2 = 9 (c) 4 ( 6) = = = = 25 36