Number System (Matriculation)

Matriculation QS015 2014 S.Y.Chuah June 18, 2014

Chapter 1   : Number Number System System 1.1 Real Numbers (a) Define Define Natural Natural Numbers Numbers N, Whole Number W, Integers Z, Prime Numbers, Rational Numbers Q, Irrational Numbers. (b) Represent Represent rational numbers and irrational irrational numbers in decimal forms. (c) Represen Representt the relations relationship hip of nu number mber sets in a real number number system system diagramdiagrammatically. (d) Represen Representt open, closed closed and half-open half-open interv intervals and their their represen representati tations ons on the number line. (e) Simplify Simplify union , intersection line.

∪

∩ of two or more intervals with the aid of number

1.2 Complex Numbers (a) Represent Represent a complex number number in Cartesian form. (b) Define the equality equality of two complex numbers. numbers. (c) Determine Determine the conjugate of a complex nu number, mber, ¯ Z . (d) Perform Perform algebraic operations on complex complex numbers. (e) Represen Representt a complex complex number number in polar form Z  = r(cosθ + cosθ +  i sinθ) sinθ) where r < 0 and π < θ < π. π.

 −

1.3 Indices, Surds and Logarithms (a) State State the rules rules of indice indices. s. (b) Explain the meaning of surd and its conjugate and to carry out algebraic operations on surds. (c) State State the laws of logarithms logarithms.. (d) Cha Change nge the base base of logarithms logarithms..

1

1.1 Real Numbers R 1.1.1 Sets of Real Numbers Definition 1 (Real Numbers) The set of real numbers,R, comprises rational numbers and irrational numbers.





0

Definiti Definition on 1.1 Natural Natural nu number mbers, s,N, are posit positiv ivee numbers umbers that that are used used for for count countin ing: g: N = 1,2,3, .

{

···}

Definition 1.2 Whole numbers,W, are natural numbers including the number zero: W = 0,1,2,3, .

{

···}

Definition 1.3 Integers,Z, are whole numbers including their negatives: Z= ,-2,-1,0,1,2, -2,-1,0,1,2, .

{···

···}

Definition 1.4 Prime numbers are natural numbers greater than 1 that can be divided by itself and 1 only. Primenumbers = Primenumbers  = 2,3,5,7,11, 2,3,5,7,11, .

{

···}

Definition Definition 1.5 Rational numbers, numbers,Q, are numbers that can be written in the form pq  where p and q are integers and q  = 0. Q =  pq p, q  Z, q  = 0

{

 ∈  ∈

    }

   

In decimal form, rational numbers may be a  terminating decimal , such as 34 = 0.75 or a 3 = 0.27272727 , in which a group of one or more digits repeating repeating decimal , such as 11 repears indefinitely. Examples of rational numbers are 3, 34 , 25 , 0.6, 1.212121 , 6, 20.

···

 − −

···

¯ , are numbers that cannot be written in the form Definition 1.6 Irrational numbers,Q  p  where p and q are integers and q  = 0. For example, π , e and 3. q

   

Figure 1: Real Number System

√ 

Figure 2: Venn Diagram represents different types of real numbers

Exercise 1:

(a) N

⊂W (b) Z ⊂ N

Determine whether each statement is true or false.

(c)

√ 3 ∈ Q

(d) 8.2525

· · · ∈ Q¯

(e) 0.21212212 . . . / Q

∈

(f) 0.23

∈Q

1.1.2 Intervals of Real Numbers Intervals of real numbers can be illustrated using 1. Set notation denoted by . The solution to the inequality x in  set notation  as follows:

 {}

≥ 2 can be expressed

{x : x ≥ 2} It is read as :  The set of all x such that x is greater than or equal to 2 . 2. Real number line denoted by

3. Interval notation denoted by [ ], ( ), [ ) or ( ]. (a,b) - open interval [a, b] - closed interval (a,b], [a,b) - half-open interval (a, ) - infinite interval

∞

Exercise 2:

Summary of Set notation and Interval notation

Problem 1

Note  : The symbol  is not numerical. When we write [a, ), we are simply referring to the interval starting from a  and continuing indefinitely to the right.

 ∞

∞

Problem 2

Graph all real numbers x such that (i) ( 20, 5)

− − (ii) (−2, ∞)

(iii) (

−∞, −7)

(iv) [0, 6]

Problem 3

Graph each of the following on a number line. (i) All integers x such that

 −3 < x