The Basics 1 -1 Real Numbers - PDF document

The Basics تاّيـساـسلؤا 1 -1 Real Numbers ةّيقيقحلا دادعلؤا • 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 GFP - Sohar University SET 1 - Chapter 1

• The types of numbers that make up the real number system are:  Natural or Counting Numbers ( ) ةيعيبطلا دادعلؤا ) ْدـَعلا دادعأ وأ ( = {1, 2, 3, 4, …}  Whole Numbers ( ) دادعلؤاةّيلـُكلا = {0, 1, 2, 3, …}  Integers ( ) ةحيحصلا دادعلؤا = {…, ‒ 3, ‒ 2, ‒ 1, 0, 1, 2, 3, …} SET 1 - Chapter 1 3 GFP - Sohar University  Rational Numbers ( ) ةّيبسنلا دادعلؤا • 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 GFP - Sohar University SET 1 - Chapter 1

 Irrational Numbers ( ) دادعلؤاةّيبسنلبلا • 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: SET 1 - Chapter 1 5 GFP - Sohar University 6 GFP - Sohar University SET 1 - Chapter 1

1 - 2 The Number Line ا طخدادعلؤ • 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. SET 1 - Chapter 1 7 GFP - Sohar University 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 GFP - Sohar University SET 1 - Chapter 1

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) 0.25487 (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 SET 1 - Chapter 1 9 GFP - Sohar University 1 - 3 Odd and Even Numbers  Odd Numbers ةيدرفلا دادعلؤا • Odd numbers are integers not divisible by 2: Odd Numbers = {…, ‒ 5, ‒ 3, ‒ 1, 1, 3, 5, …}  Even Numbers ةيجوزلا دادعلؤا • Even numbers are integers divisible by 2: Even Numbers = {…, ‒ 6, ‒ 4, ‒ 2, 0, 2, 4, 6, …} 10 GFP - Sohar University SET 1 - Chapter 1

1 - 4 Prime and Composite Numbers  Prime Numbers ةيلولؤا دادعلؤا • 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.  Composite Numbers ةبكرملا دادعلؤا • 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. SET 1 - Chapter 1 11 GFP - Sohar University 1 - 5 Perfect Squares ةلماكلا تاعبرملا • A perfect square is an integer that is the square of an integer. • The first 15 perfect squares are: 12 GFP - Sohar University SET 1 - Chapter 1

1 - 6 Perfect Cubes ةلماكلا تابعكملا • Perfect cubes are the result when integers are multiplied by themselves twice. • The first 5 perfect cubes are: SET 1 - Chapter 1 13 GFP - Sohar University 1 - 7 Properties of Basic Operations ةيباسحلا تايلمعلا صئاصخ  Closure Property of Addition عمجلل قلبغنلئا ةيصاخ • 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 GFP - Sohar University SET 1 - Chapter 1

 Closure Property of Multiplication برضلل قلبغنلئا ةيصاخ • 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). SET 1 - Chapter 1 15 GFP - Sohar University  Commutative Property of Addition عمجلل لادبلئا ةيصاخ 2+ 3 = 3 + 2 a + b = b + a  Commutative Property of Multiplication برضلل لادبلئا ةيصاخ 4 × 7 = 7 × 4 a × b = b × a  Associative Property of Addition عمجلل نارتقلئا ةيصاخ (4 + 5) + 8 = 4 + (5 + 8) ( a + b ) + c = a + ( b + c )  Associative Property of Multiplication برضلل نارتقلئا ةيصاخ (3 × 6) × 9 = 3 × (6 × 9) ( a × b ) × c = a × ( b × c ) 16 GFP - Sohar University SET 1 - Chapter 1

 Identity Property of Addition عمجلل قباطتلا ةيصاخ 5+ 0 = 5 a + 0 = a  Identity Property of Multiplication برضلل قباطتلا ةيصاخ 4 × 1 = 4 a × 1 = a  Inverse Property of Addition عمجلل ساكعنلئا ةيصاخ 3+ (–3) = 0 a + (– a ) = 0  Inverse Property of Multiplication ةيصاخ ساكعنلئا برضلل 2 × = 1 a × a = 1 SET 1 - Chapter 1 17 GFP - Sohar University  Distributive Property عيزوتلا ةيصاخ 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 GFP - Sohar University SET 1 - Chapter 1

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 SET 1 - Chapter 1 19 GFP - Sohar University 1 - 8 Intervals تارـتـفـلا • 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 GFP - Sohar University SET 1 - Chapter 1

• The table below shows the types of intervals and the ways used to describe them . SET 1 - Chapter 1 21 GFP - Sohar University 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} { x | x < 3} (j) Solution: 22 GFP - Sohar University SET 1 - Chapter 1

SET 1 - Chapter 1 23 GFP - Sohar University 1 - 9 Factors لماوعـلا • 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 GFP - Sohar University SET 1 - Chapter 1

1 - 10 Common Factors ةكرتشملا لماوعـلا • 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} SET 1 - Chapter 1 25 GFP - Sohar University 1 - 11 The Greatest Common Factor (GCF) ربكلؤا كرتشملا لماعلا • 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 GFP - Sohar University SET 1 - Chapter 1