SET 1 - Chapter 1 - Our Math Site

1 The Basics 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 SET 1 - Chapter 1 GFP - Sohar University The types of numbers that make up the real number system are: Natural or Counting Numbers () ) ( ={1, 2, 3, 4, ..} Whole Numbers() ={0, 1, 2, 3, ..} Integers() = {.., 3, 2, 1, 0, 1, 2, 3, ..}3 SET 1 - Chapter 1 GFP - Sohar University rational Numbers() A number is classified as rationalif it can be expressed as a fraction.

2 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() A number that cannotbe written as a fractionis 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 University6 SET 1 - Chapter 1 GFP - Sohar University1 - 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 Each pointon the line represents a real number, and every real numberis represented by a pointon the line.

3 Negativenumbers represent distances to the left of zero, and positivenumbers are distances to the right. The arrowson the end indicate that it keeps going foreverin the leftand rightdirections. SET 1 - Chapter 1 GFP - Sohar UniversityExample 1:For the following problems, choose the correct answer.(i)Which of the following numbers is a positive integer?(a)(b)(c)(d) (ii)Which of the following numbers is a negative integer?(a)(b)3(c) (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) (v)Which of the following numbers is a natural number?

4 (a)(b)(c)5(d)8 Solution: (i)c, (ii)d, (iii) a, (iv)d, (v)c SET 1 - Chapter 1 GFP - Sohar UniversityExample 2:For the following problems, choose the correct answer.(i)Which of the following numbers is a positive integer?(a)(b)(c)(d) (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) (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 University1 - 3 Odd and Even Numbers 10 Odd Numbers Even Numbers Odd Numbers={.}

5 , 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 University1 - 4 Prime and Composite Numbers 11 Prime Numbers A prime number is a number that has exactly two factors, it can be evenly divided by only itself and 1. Composite Numbers 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 Numbers={4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, ..} The only even prime number is 2. Zeroand 1are not prime numbers or composite 1 - Chapter 1 GFP - Sohar University121 - 5 Perfect Squares A perfect square is an integer that is the square of an integer.

6 The first 15 perfect squares are:SET 1 - Chapter 1 GFP - Sohar University131 - 6 Perfect Cubes Perfect cubes are the result when integers are multiplied by themselves twice. The first 5 perfect cubes are:SET 1 - Chapter 1 GFP - Sohar University141 - 7 Properties of Basic Operations 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).

7 SET 1 - Chapter 1 GFP - Sohar University15 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).SET 1 - Chapter 1 GFP - Sohar University16 Commutative Property of Addition 2+ 3 =3 + 2 a + b = b + a Commutative Property of Multiplication 4 7=7 4a b = b a Associative Property of Addition (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 SET 1 - Chapter 1 GFP - Sohar University17 Identity Property of Addition 5+ 0 =5 a + 0= a Identity Property of Multiplication 4 1 =4a 1= a Inverse Property of Addition 3 + ( 3)

8 =0 a+ ( a)=02 =1a a=1 Inverse Property of Multiplication SET 1 - Chapter 1 GFP - Sohar University18 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) SET 1 - Chapter 1 GFP - Sohar UniversityExample 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?

9 (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 addition19 Solution: (i)b, (ii)d, (iii) c, (iv)d SET 1 - Chapter 1 GFP - Sohar University201 - 8 Intervals A subsetof 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, intervalscorrespond to raysand line segmentson the number line, along with the number line 1 - Chapter 1 GFP - Sohar University21 The table below shows the types of intervals and the ways used to describe 1 - Chapter 1 GFP - Sohar UniversityExample 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.

10 SET 1 - Chapter 1 GFP - Sohar University23 SET 1 - Chapter 1 GFP - Sohar University241 - 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 12are {1, 2, 3, 4, 6, 12}since 12is divisible byall of them:112=12, then 1 and 12 are factors of 12, and34=12, then 3 and 4 are factors of 12, and26=12, then 2 and 6 are factors of 12 SET 1 - Chapter 1 GFP - Sohar University251 - 10 Common Factors A common factor of two or more numbers is a number that is a factor of all them.