REAL AND COMPLEX NUMBERS​ INDICES, SURD & LOGARITHM​

1. prepared by NASS
CHAPTER 1&2
• REAL AND COMPLEX NUMBERS
• INDICES, SURD & LOGARITHM

2. prepared by NASS

3. prepared by NASS
At the end of this chapter, students should be able to:
 Define real numbers, all the subsets of real numbers, complex
numbers, indices, surds & logarithm.
 Represent the relationship of number sets in a real number system
diagrammatically.
 Understand open, closed and half-open intervals and their
representations on the number line.
 Simplify union, and intersection of two or more intervals with the
aid of the number line.
 Perform operations on complex number.
 Simplify indices, surds & logarithm.

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REAL LINE
0 2.5 10
-9 -1
To the right, getting bigger
To the left, getting smaller

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Intervals of Real Numbers
Can be illustrated using:
Set Notation
Interval/Bracket Notation
Real number line
S
R
I

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Summary of Real Numbers Interval
Set Notation Interval Notation Real Number Line Notation
 
b
x
a
x 

:
 
b
x
a
x 

:
 
a
x
x 
:
 
a
x
x 
:
 
a
x
x 
:
 
a
x
x 
:
 

,
a
 
a
,


 
a
,


 
b
a,
 
b
a,
 

,
a
a
a
a
a
a b
a b

7. Example:
Write the following set of real numbers using a real number line and interval notation.
1. The set of real numbers less than 10. ;
2. The set of real numbers greater than or equal to 5. ;
3. The set of real numbers greater than -5 but less than or equal to 9.
;
4. The set of real numbers between 7 and 20. ;
prepared by NASS
10
 
10
,


5
 

,
5
5 9
 
9
,
5
7
20
)
20
,
7
(

8. prepared by NASS
Combining Intervals
Union : the set of real numbers that belong to either one or both of
the intervals.
A B = { x | x A or x B }
Intersection : the set of real numbers that belong to both of the
intervals.
A B = { x | x A and x B }
  




9. prepared by NASS
Example: Write each union/intersection as a single interval.
   
9
,
3
5
,
1 
   
9
,
3
5
,
1 
   


 ,
0
3
,
4
   


 ,
0
3
,
4
5 9
1 3
 
9
,
1
5 9
1 3
 
5
,
3
3
-4 0
 

 ,
4
3
-4 0
 
3
,
0

10. Is any number of the form ,which cannot be written as a fraction of two
integers is called surd.
Properties of Surds:
b
 
a
a
a
b
c
a
b
c
b
a
b
a
b
a
ab
b
a








4)
3)
2)
1)

11. chapter 1 11
Conjugate Surds
a b a b
  
RATIONALIZING DENOMINATORS
Problem arise when algebraic fraction involving surds in the denominator.
Solution:
1) Eliminate the surd from denominator by multiplying the numerator and
denominator by the conjugate of the denominator.

12. Is a set of number in form,
Where and are real numbers and . A complex number is
generally denoted by,
prepared by NASS
bi
a 
Real part, Re(z) Imaginary part, Im(z)
a b 1


i
bi
a
z 


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Equality of complex numbers .
Conjugate of complex numbers .
bi
a
z 
 bi
a
z 

is the complex number obtained by changing the sign of the
imaginary part of .
z

14. prepared by NASS
• Addition/Subtraction
• Multiplication
• Division
       i
d
b
c
a
di
c
bi
a 






      i
bc
ad
bd
ac
di
c
bi
a 





 
2
2
2
2
d
c
i
ad
bc
d
c
bd
ac
di
c
bi
a








Algebraic Operations of Complex
Numbers

15. prepared by NASS
If a is a real number and n is a positive integers,
then
Where, a = base
n = index
a
a
a
a
an




 ...
n times

16. 1)
2)
3)
4)
5)
prepared by NASS
Rules/Law of Indices
n
m
n
m
a
a
a 

  mn
n
m
a
a 
n
m
n
m
a
a
a 

  0
, 
 b
b
a
ab m
m
m
0
, 







b
b
a
b
a
m
m
m
0
;
1



a
a
a n
n
6)
0
a
;
1
a0


7)
8)
0
a
;
a
a n m
n
m



17. prepared by NASS
Definition: The logarithm of any number of a
given base is equal to the power to
which the base should be raised to get the
given number.
From indices, a, x and n are related such that
Then, x is said to be the logarithm of n wrt the
base a.
n
ax

log𝑎 𝑛 = 𝑥

18. 1) 6)
2) 7)
3)
4)
5)
prepared by NASS
Rules/Law of Logarithm
n
log
m
log
)
mn
(
log a
a
a 

n
log
m
log
n
m
log a
a
a 







m
log
n
m
log a
n
a 
a
log
m
log
m
log
b
b
a 
0
1
loga 
1
a
loga 
n
a n
a
log


19. MATHEMATICS 1 SEQUENCE AND SERIES
• Centre for Foundation Studies

20. Sequence and Series
•Sequence and
series
•Arithmetic
progression
•Geometric
progression
•Binomial expansion

21. Sequence and series
•Defn of sequence
•Types of sequence
•General term of
sequence
•Defn of series
•Types of series

22. Sequence
Definition
• Sequence- A set of order numbers.
• Terms – the numbers which form
the sequence, denoted by T1,
T2,T3,…….
Type
• Finite – containing a finite number of
terms.
•Infinite – have an unlimited number of
terms. General Term

23. General term of a sequence
• The nth term of the sequence of even
numbers
Tn =2n
• The nth term of the sequence of odd
numbers
Tn =2n-1
• The nth term
of
1 1 1 1
, , ,
3 5 9 17
1
,..... is given by Tn  n
2 1

24. Series
n
Definition
• The sum of the terms in a
sequence.
• Finite series
Ti  T1 T2 T3 ....Tn  Sn
i1
• Infinite series
 T1  T2  T3  ...  S

 Ti
i1

25. Arithmetic Progression
• Arithmetic progression can be either
arithmetic sequence or arithmetic
series.
• Arithmetic sequence
- A sequence whose consecutive terms
have a
constant difference.
- a, a+d, a+2d,…….,a+(n-1)d.
-The first term- ‘a’
-Fixed difference – ‘d’ also known as
common difference. Can be positive or

26. Arithmetic series
Definition
• The sum of arithmetic
sequence
2 2
n n
S 
n
2a  (n 1)d  or S 
n
(a  l)
l  a  (n 1)d
• Arithmetic mean for two numbers
- If a,b,c is an arithmetic sequence, then
b is arithmetic mean of a and c.
b 
a  c
2

27. Arithmetic series
If a and b are two numbers
and A1,A2,A3,…..,An are
arithmetic means between
a and b, then a,
A1,A2,A3,…..,An ,b are in
arithmeticsequence.

28. Geometric progression
sequenc
e
n
where r is common ratio
a is the first term
Geometric series – The sum of a
geometric
Geometric sequence – A sequence in
which the ratio of any two consecutive
terms is a constant. . T  arn1
a(1rn
)
When r  1, use Sn 
or
When r  1, use Sn 
1  r
a(rn
1)
r 1

29. Geometric mean
• If a,b,c is a geometric sequence, then
the geometric mean of a and c is b,
where
b2
 ac
b   ac
• Sum to infinity of the geometric series
where r  1
1  r
a

S 

30. Application of arithmetic and geometric series
• An engineer has an annual salary of RM24,000
in his first year. If he gets a raise of RM3,000
each year, what will his salary be in his tenth
year? What is the total salary earned for 10
years of work?
• CFS launched a reading campaign for students
on the first day of July. Students are asked to
read 8 pages of a novel on the first day and
every day thereafter increase their daily
reading by one page. If Saddam follows this
suggestion, how many pages of the novel will

31. Application of arithmetic and geometric series
• Aida deposits RM5,000 into a bank that
pays an interest rate of 5% per annum. If
she does not withdraw or deposit any
money into his account, find the total
savings after 10 years.
• Each year the price of a car depreciates
by 9% of the value at the beginning of the
year. If the original price of the car was
RM60,000. Find the price of the car after
10 years.

32. Binomial expansion
Denoted
by
Binomial
theorem
 a  b  n
 
abn

n
an
b0

n
an1
b1
..............
n
a0
bn
0 1 
where
n!
k!(nk)!
n
n
ab  a b
nk k
k
n
n
n

 k
 
k0  
     
 


33. General term
• If n is a negative integer or a rational number, then
•1 xn
1
n
x
nn1x2

nn1n2x3
......
• 1 2! 3!
• provided -1 x 1 or x  1
General term-The (r+1)th term of
the expansion of (a+b)n is denoted
by Tr+1 .
T r1

n
anr
br
 r
 