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NUMBER SYSTEM

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... complex number using the Argand diagram ... Argand Diagram ... Illustrate each of the following by using Argand diagram. z1 = 2 i. z2 = 3i. z3 = 2 i ... – PowerPoint PPT presentation

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Title: NUMBER SYSTEM


1
NUMBER SYSTEM
  • Chapter 2
  • DCT1043

2
Content
  • 2.1 Real Numbers
  • 2.2 Indices
  • 2.3 Logarithm
  • 2.4 Complex Number

3
2.1 Real Numbers
  • 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

4
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
5
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
6
Representing Real Numbers as Decimal
  • Every real number can be written as decimal
  • Repeating or terminating
  • Repeating (rational numbers)
  • 0.6666666 , 0.33333
  • Terminating (rational numbers)
  • ½0.5
  • Neither Terminate nor Repeat (irrational numbers)

7
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

8
Representing Real Numbers as Number Line
Negative direction
Positive direction
x
0
Origin
9
Exercise
  • Graph the elements of each set on a number line

10
Properties of the Real Number System
Rules of Operations
11
Properties of Negatives
Properties Involving Zero
12
Properties of Quotients
13
Exercise
  • Simplify the following algebraic expression

14
Open Closed Interval
Open Half Interval
Open Interval
Closed Interval
15
Exercise
  • Write the suitable interval for the following
    number line

a. b.
- 8
9
9
- 8
16
Exercise
  • Illustrate the following interval by using
    suitable number line

17
Exercise
  • Given
  • Simplify

18
2.2 Indices (Exponents)
  • 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

19
Exponential Notation
20
Rules of Exponents
21
Exercise
  • Without using a calculator, evaluate the
    following

22
Exercise
  • Simplify

23
Exercise
  • If
  • Express
  • in term of y

24
Radicals
Principle of Square Root
25
Surds
Operation on Surds
26
Exercise
  • Simplify the following

27
Conjugate Surds
Rationalized Surds
28
Exercise
  • Simplify the following by rationalizing the
    denominators

29
Surds Rules
30
Exercise
  • Write the following in term of surds
  • Write the following in term of index

31
Exercise
  • Simplify

32
2.3 Logarithm
  • 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

33
What is Logarithm
  • Logarithm is the power
  • WHY?

Index form
Logarithm form
34
Exercise
  • Convert the following to logarithmic form
  • Convert the following to index form

35
Exercise
  • Without using a calculator, evaluate the following

36
Common Natural Algorithms
  • Common Logarithms - Logarithms to base 10
  • Natural Logarithms - Logarithms to base e

37
Exercise
  • Solve the following equations

38
Laws of Logarithms
39
Exercise
  • Simplify

40
Exercise
  • Without using a calculator, evaluate the following

41
Exercise
  • Evaluate the following by using the change of
    base law
  • Given that loga2 0.301 and loga3 0.477, find

42
2.4 Complex Number
  • 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

43
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.

44
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
45
Exercise
  • Write the following in complex number form
  • ( a bi )

46
Operations on Complex Numbers
  • For , then
  • Adding complex numbers
  • Subtracting complex numbers
  • Multiplying complex numbers
  • Dividing complex numbers

47
Exercise
  • Given z 2 i and w 3 2i. Find
  • z w
  • z w
  • z .w
  • z / w

48
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.

49
Conjugate Complex
  • A complex conjugate of a complex number z
    a bi is z a bi
  • If z1 and z2 are complex numbers, then

50
Exercise
  • Given z 2 i and w 3 2i. Find
  • z w
  • z w
  • z .w
  • (z ) / w

51
Exercise
  • Given z 2 3i , Find
  • the value of a and b if

52
Exercise
  • Given (3 i) (a 2i) 2i.
    Find the value of a and b
  • Given
  • Find z.
  • Then find z.z

53
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
54
Exercise
  • Illustrate each of the following by using Argand
    diagram
  • z1 2 i
  • z2 3i
  • z3 2 i
  • z4 3i

55
Modulus and Argument
  • Modulus
  • Argument
  • In polar form,

56
Exercise
  • Find the modulus and argument for

57
Exercise
  • Given z1 3 4i and z2 2 3i
  • Determine z1 z2
  • argument z1 z2
  • Express z1 z2 in polar form

58
THANK YOU
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