DCT1043 CHAPTER 1

1
DCT1043CHAPTER 1

• NUMBER SYSTEM

2
Content

• 1.1 Real Numbers
• 1.2 Indices
• 1.3 Logarithm
• 1.4 Complex Number

3
1.1 Real Numbers
4
Objectives

• At the end of this topic you should be able to
• Define natural numbers, whole numbers, integers,
  prime numbers, rational numbers and irrational
  numbers
• Represent rational and irrational numbers in
  decimal form
• Represent the relationship of number sets in real
  number system diagrammatically
• State clearly the properties of real numbers such
  as closure, commutative, associative,
  distributive, identity and inverse under addition
  and multiplication
• Represent the open, closed and half-open
  intervals in number line

5
The Set of Real Numbers R
Rational Numbers
Irrational Numbers
Nonterminating nonrepeating decimal number
Integers
Fractions
Terminating or repeating decimal numbers
Proper, Improper, Mixed Number
Whole Numbers
Negative Integers
Prime number
Zero
Natural/Counting Numbers/ positive integers
Composite number
6
Relationship among various sets of number
Real Numbers R
Rational Numbers Q
Irrational numbers
Irrational Numbers H
Integers Z
Whole numbers W
Natural Numbers N
7
Representing Real Numbers as Decimal

• Every real number can be written as decimal
• Repeating or terminating
• Repeating (rational numbers)
• 0.6666666 , 0.33333 , 2/7
• Terminating (rational numbers)
• ½ 0.5, 3/4
• Neither Terminate nor Repeat (irrational numbers)

8
Exercise

• Given a set of numbers
• List the numbers in the set that are
• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational Numbers
• Real numbers

9
Representing Real Numbers as Number Line
Negative direction
Positive direction
x
0
Origin
10
Exercise

• Graph the elements of each set on a number line

11
Properties of the Real Number System
Rules of Operations
12
Properties of Negatives
Properties Involving Zero
13
Properties of Quotients
14
Exercise

• Simplify the following algebraic expression

A) B) C) D)
E) F) G) H)
15
Open Closed Interval
Open Half Interval
Open Interval
Closed Interval
16
Exercise

• Write the suitable interval for the following
  number line

a. b.
- 8
9
9
- 8
17
Exercise

• Illustrate the following interval by using
  suitable number line

18
Exercise

• Given
• Simplify

19
1.2 INDICES (EXPONENTS)
20
Objectives

• At the end of this topic you should be able to
• Define indices
• State the rule of indices
• Explain the meaning of a surd and its conjugate
  and carry out algebraic operation on surd

21
Exponential Notation
22
Rule of Exponents
23
Exercise

• Without using a calculator, evaluate the
  following

24
Exercise

• Simplify the following

25
Exercise

• If
• Express
• in term of y

26
Radicals
Principle of Square Root
27
Surds
Operation on Surds
28
Exercise

• Simplify the following

29
Conjugate Surds
Rationalized Surds
30
Exercise

• Simplify the following by rationalizing the
  denominators

31
Surds Rules
nth root of a to the power of n
32
Exercise

• Write the following in term of surds
• Write the following in term of index

33
Exercise

• Simplify

34
1.3 logarithm
35
Objectives

• At the end of this topic you should be able to
• State and use the law of algorithm
• Change the base of logarithm
• Understand the meaning of ln M and log M
• Solve equations involving logarithm

36
What is Logarithm

• Logarithm is the power
• WHY?

Logarithm form log y base a equal to x
Index form a to the power of x
37
Exercise

• Convert the following to logarithmic form
• Convert the following to index form

38
Exercise

• Without using a calculator, evaluate the following

39
Laws of Logarithms
40
Exercise

• Simplify

41
Exercise

• Without using a calculator, evaluate the following

42
Exercise

• Evaluate the following by using the change of
  base law
• Given that loga2 0.301 and loga3 0.477, find

43
Common Natural Algorithms

• Common Logarithms - Logarithms to base 10
• Natural Logarithms - Logarithms to base e

44
Exercise

• Solve the following equations

45
1.4 complex Numbers
46
Objectives

• At the end of this topic you should be able to
• Define complex number
• Represent a complex number in Cartesian form
• Define the equality of two complex number
• Define the conjugate of a complex number
• Perform algebraic operations on complex number
• Find the square root of complex number
• Represent the addition and subtraction of complex
  number using the Argand diagram
• Find the modulus and argument of a complex number
• Express a complex number in polar form

47
What is Complex Number

• A number that can be expressed in the form a bi
  where a and b are real numbers and i is the
  imaginary unit.
• Imaginary unit is the number represented by i,
  where
• Imaginary number is a number that can be
  expressed in the form bi, where a and b are real
  numbers and i is the imaginary unit.
• When written in the form a bi , a complex
  number is said to be in Standard Form.

48
The Set of Complex Numbers
R
Complex Numbers C
Real Numbers R
Rational Numbers Q
Integers Z
Imaginary Numbers i
Whole numbers W
Irrational Numbers H
Natural Numbers N
49
Exercise

• Write the following in complex number form
• ( a bi )

50
Operations on Complex Numbers

• For , then
• Adding complex numbers
• Subtracting complex numbers
• Multiplying complex numbers
• Dividing complex numbers

51
Exercise

• Given z 2 i and w 3 2i. Find
• z w
• z w
• z .w
• z / w

52
Same Complex Numbers

• 2 complex numbers z1 a bi and z2 c di
  are same if a c and b d.
• Example
• Given z1 2 (3y1)i and z2 2x 7i
• with z1 z2 . Find the value of x and y.

53
Conjugate Complex

• A complex conjugate of a complex number z
  a bi is z a bi
• If z1 and z2 are complex numbers, then

54
Exercise

• Given z 2 i and w 3 2i. Find
• z w
• z w
• z .w
• (z ) / w

55
Exercise

• Given z 2 3i , Find
• the value of a and b if

56
Exercise

• Given (3 i) (a 2i) b 2i.
  Find the value of a and b
• Given
  
• Find z.
• Then find z.z

57
Argand Diagram

• Represents any complex number z a ib in terms
  of its Cartesian coordinate point P (a, b) or its
  polar coordinate

P (a, b)
P (a, b)
b
b
Imaginary number line
Imaginary number line
r z
z
a
a
Real number line
Real number line
58
Exercise

• Illustrate each of the following by using Argand
  diagram
• z1 2 i
• z2 3i
• z3 2 i
• z4 3i

59
Modulus and Argument

• Modulus
• Argument
• In polar form,

60
Exercise

• Find the modulus and argument for

61
Exercise

• Given z1 3 4i and z2 2 3i
• Determine z1 z2
• argument z1 z2
• Express z1 z2 in polar form

62
Thank You