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 SET 1 - Chapter 1 GFP - Sohar University

• 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, …}

3 SET 1 - Chapter 1 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 SET 1 - Chapter 1 GFP - Sohar University

• 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:

5 SET 1 - Chapter 1 GFP - Sohar University 6 SET 1 - Chapter 1 GFP - Sohar University

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.

7

• 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 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) 8

Solution: (i) c, (ii) d, (iii) a, (iv) d, (v) c

SET 1 - Chapter 1 GFP - Sohar University

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) 9

Solution: (i) a, (ii) c, (iii) c, (iv) b, (v) a

SET 1 - Chapter 1 GFP - Sohar University

1 - 3 Odd and Even Numbers

10

• Odd Numbers

ةيدرفلا دادعلؤا

• Even Numbers ةيجوزلا دادعلؤا

Odd Numbers = {…, ‒ 5, ‒ 3, ‒ 1, 1, 3, 5, …}

• Odd numbers are integers not divisible by 2:
• Even numbers are integers divisible by 2:

Even Numbers = {…, ‒ 6, ‒ 4, ‒ 2, 0, 2, 4, 6, …}

SET 1 - Chapter 1 GFP - Sohar University

1 - 4 Prime and Composite Numbers

11

• 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.

• Composite Numbers

ةبكرملا دادعلؤا Prime Numbers = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29 …}

• 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, …}

• The only even prime number is 2.
• Zero and 1 are not prime numbers or composite numbers.

SET 1 - Chapter 1 GFP - Sohar University 12

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:

SET 1 - Chapter 1 GFP - Sohar University

13

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 GFP - Sohar University 14

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).

SET 1 - Chapter 1 GFP - Sohar University

15

• 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 GFP - Sohar University 16

• 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) (3 × 6) × 9 = 3 × (6 × 9) (a × b) × c = a × (b × c)

• Associative Property of Multiplication

برضلل نارتقلئا ةيصاخ

SET 1 - Chapter 1 GFP - Sohar University

17

• 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 2 × = 1 a × a = 1

• Inverse Property of Multiplication

ةيصاخ ساكعنلئا برضلل

SET 1 - Chapter 1 GFP - Sohar University 18

• 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)

SET 1 - Chapter 1 GFP - Sohar University

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 19

Solution: (i) b, (ii) d, (iii) c, (iv) d

SET 1 - Chapter 1 GFP - Sohar University 20

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.

SET 1 - Chapter 1 GFP - Sohar University

21

• The table below shows the types of intervals and the ways used to describe them.

SET 1 - Chapter 1 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}

(j)

{x | x < 3}

22

Solution:

SET 1 - Chapter 1 GFP - Sohar University

23 SET 1 - Chapter 1 GFP - Sohar University 24

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

SET 1 - Chapter 1 GFP - Sohar University

25

1 - 10 Common Factors

ةكرتشملا لماوعـلا

• A common factor of two or more numbers is a number that is a factor
• f 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 GFP - Sohar University 26

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

SET 1 - Chapter 1 GFP - Sohar University

Example 5: Find the GCF of 24 and 32.

27

Solution:

The common factors are {1, 2, 4, 8} and 8 is the largest one So, the GCF of 24 and 32 is 8

SET 1 - Chapter 1 GFP - Sohar University 28

1 - 12 Multiples

تافعاضمـلا

• 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 = 3 3 2 = 6 3 3 = 9 3 4 = 12 3 5 = 15 3 6 = 18, and so on

So, the multiples of 3 are {3, 6, 9, 12, 15, 18, …}

SET 1 - Chapter 1 GFP - Sohar University

29

1 - 13 Common Multiples

ةكرتشملا تافعاضمـلا

• 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}

SET 1 - Chapter 1 GFP - Sohar University 30

1 - 14 The Lowest Common Multiple (LCM)رغصلؤا كرتشملا فعاضملا

• 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

SET 1 - Chapter 1 GFP - Sohar University

Example 6: Find the LCM of 3 and 8.

31

Solution:

The first common multiple is 24 So, the LCM of 3 and 8 is 24

SET 1 - Chapter 1 GFP - Sohar University 32

1 - 15 Order of Operations

ةيباسحلا تايلمعلا ذيفنت لسلست

• 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:

SET 1 - Chapter 1 GFP - Sohar University

Example 7: Simplify the following: (a) 6  4  3 (b) 6  2  4 (c) 6  2  4

(d) 10 – 3  6 + 102  (6  1)  4

33

Solution:

(a) 6  4  3 = 6  12 (multiplication) = 18 (then addition) (b) 6  2  4 = 3  4 (division / multiplication, left to right) = 12 (c) 6  2  4 = 12  4 (division / multiplication, left to right) = 3

SET 1 - Chapter 1 GFP - Sohar University 34

(d) 10 – 3  6 + 102  (6  1)  4 = 10 – 3  6 + 102  7  4 (brackets first) = 10 – 3  6 + 100  7  4 (powers next) = 10 – 18 + 100  28 (multiplication) = – 8 + 100  28 (addition / subtraction, left to right) = 92  28 (addition / subtraction, left to right) = 120

SET 1 - Chapter 1 GFP - Sohar University

35

1 - 16 Absolute Value

ةقلطملا ةميقلا

• 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).

SET 1 - Chapter 1 GFP - Sohar University

Example 8: Simplify the following: (a) | 5 | + 6 – 2 (b) | – 5 | + 6 – 2 (c) | 4  (– 6) + 5 | + 6

36

Solution:

(a) | 5 | + 6 – 2 = 5 + 6 – 2 = 11 – 2 = 9 (b) | – 5 | + 6 – 2 = 5 + 6 – 2 = 11 – 2 = 9 (c) | 4  (– 6) + 5 | + 6 = | – 24 + 5 | + 6 = | – 19 | + 6 = 19 + 6 = 25

SET 1 - Chapter 1 GFP - Sohar University