1. The real number system includes natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Rational numbers can be expressed as fractions, while irrational numbers cannot. 2. Key concepts covered include number lines, intervals, unions and intersections of intervals, rules of indices, properties of surds, and laws of logarithms. 3. Examples are provided to illustrate solving equations involving indices, surds, and logarithms through appropriate transformations and applications of properties.

1. CHAPTER 1 : NUMBER SYSTEM AND EQUATIONS
“Man jadda wa jadda” ( whosever strive shall
succeed ) – arabic quote
||||||||||2015/2016||||||||||||
TOPIC 1: NUMBER
SYSTEM
LEARNING OUTCOMES
1.1 Real Numbers
( )
At the end of this topic, students should be able to:
a. define natural numbers ( ), whole numbers ( W ), integers ( ), prime
numbers, rational numbers ( ) and irrational numbers ( ).
b. represent the relationship of number sets in a real number system
1
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2. diagrammatically showing
W⊂ ⊂ ⊂
and
c. represent open, closed and semi-open intervals and their
representations on the number line.
d. use number line to simplify union, ∪ and intersection, ∩ of two
intervals.
1.2 Indices, Surds and
Logarithms
At the end of this topic, students should be able to:
a. state the rules of indices
b. explain the meaning of a surd and its conjugate, and to carry out
algebraic operation on surd.
c. apply the laws of logarithms such as
MNMiii
NM
N
M
ii
NMMNi
a
N
a
aaa
aaa
loglog..
logloglog.
logloglog.
=
−=





+=
and
d. To change the base of logarithm using
a
M
M
b
b
a
log
log
log =
e. solve equations involving surds, indices and logarithms.
1.1 REAL NUMBERS
2
1.1 Real Numbers 1.2 Indices, Surds and Logarithms
NUMBER SYSTEM AND EQUATIONS

3. 1. The real number system evolved over time by expanding the notion of what
we mean by the word “number.” At first, “number” meant something you could
count, like how many cows a farmer owns. These are called the natural
numbers ( ), or sometimes the counting numbers.
2. Natural Numbers ( ) { }...,5,4,3,2,1
3. At some point, the idea of “zero” came to be considered as a number. If the
farmer does not have any cow, then the number of cow that the farmer owns
is zero. We call the set of natural numbers ( ) plus the numbers zero the
whole numbers (W).
4. Whole Numbers (W) = Natural Numbers together with “zero”
{ }...,5,4,3,2,1,0
5. Integers ( ) = Whole Numbers plus negatives numbers
3
The use of three dots at the end of the list
is a common mathematical notation to
indicate that the list keeps going forever.

4. { }...,4,3,2,1,0,1,2,3,4... −−−−
6. If we add fractions to the set of Integers ( ), we get the set of Rational
Numbers ( ).
7. Prime numbers are natural numbers greater than 1 that can be divided by
itself and 1 only.
Prime Numbers { }...,11,7,5,3,2=
8. All numbers of the form q
p
, where Zqp ∈, but 0≠q are called rational
numbers.
9. When a rational number is written in the decimal form the digit(s) after the
decimal point repeats itself or terminates.
4
0≠qwhere
q
p
Rational numbers include what we
usually call fractions
Notice that the word “rational”
contains the word “ratio,” which
should remind you of fractions.
The bottom of the fraction is called the
denominator.
The top of the fraction is called the
numerator.
RESTRICTION:
The denominator cannot be zero!
(But the numerator can)
All integers can also be thought of as
rational numbers, with a denominator of 1:
()

5. Example 1:
OR
Example 2 :
5
Repeating decimal
3.1111111111…
3.12121212…
3.0000000…
3.123451234512345…
3.1234123412341234… 3.1234567812345678…
Repeating decimal
0.3 1.3 12.3
12345678.31234.3
12345.3
Terminating Decimal
5.0
2
1
=4.87559
25.0
4
1
=
725.0
40
29
=

6. 10. Now it might seem as though the set of rational
numbers would cover every possible case, but that is not so. There are
numbers that cannot be expressed as a fraction, and these numbers are
called irrational ( Q ) because they are not rational.
Example 3 :
a) ...414213562.12 =
b) ...718281828.2=e
c) ...141592654.3=π
6
This means that all the previous sets of numbers (natural
numbers, whole numbers, and integers) are subsets of the
rational numbers.

7. 11. The Real Numbers = Rationals + Irrationals
When we put the irrational numbers together with the rational numbers, we
finally have the complete set of real numbers. Any number that represents an
amount of something, such as a weight, a volume, or the distance between two
points, will always be a real number. The following diagram illustrates the
relationships of the sets that make up the real numbers.
7
Irrational Numbers
1. Cannot be expressed as a ratio of integers.
2. As decimals they never repeat or
terminate

8. 12. An Ordered Set
The real numbers have the property that they are ordered, which means that
given any two different numbers we can always say that one is greater or less
than the other.
13. The Number Line
The ordered nature of the real numbers lets us arrange them along a line
(imagine that the line is made up of an infinite number of points all packed so
closely together that they form a solid line). The points are ordered so that
points to the right are greater than points to the left:
Open ( empty circle)
Close ( dense circle)
14. Interval notation and set notation
8
The arrows on the end indicate that they keep going
forever in both directions.

9. Example 4:
Type
Interval
Notation
Set Notation Number line
Closed [ ]9,6 { }ℜ∈≤≤ xxx ,96:
Open ( )9,6 { }ℜ∈<< xxx ,96:
Open ( )∞,5 { }ℜ∈> xxx ,5:
Semi-open ]9,6( { }ℜ∈≤< xxx ,96:
Semi-open )9,6[ { }ℜ∈<≤ xxx ,96:
Semi-open ]2,(−∞ { }ℜ∈≤ xxx ,2:
15. To simplify union, ∪ and intersection, ∩ of two or more intervals with the
aid of number line.
Example 5:
Question Solution
9
6 9
x
6 9
x
5
x
6 9
x
6 9
x
2
x

10. a) (−∞, −1) ∩ (−3, ∞)
b) [−6, 2) ∪ [0, ∞)
c) [ ]7,4)5,3[ ∩
d) )9,6[]6,0( ∩
e) [ ]7,0)7,4[ ∪−
Example 6:
Given A = (-6, 0], B = [-1, 11) and C = {x: -5 ≤ x ≤ 5, x ∈Z}. By
10

11. using real number line, find the following:
a) BA ∪ b) BA ∩ c) '
BA ∩ d) CA ∩
a) b)
c)
d)
1.2 INDICES, SURDS AND LOGARITHMS
11

12. 1.2.1 Indices
16. 6
2222222 =××××× ….base 2 and index is 6
17. The rules of indices:
a. 10
=p
b. n
n
p
p
1
=− c. mnmn
ppp +
=×
d. ( ) nmmn
pp = e. ( ) mmm
qppq =
f. 0≠=





q
q
p
q
p
m
mm
g. mnmn
ppp −
=÷
h. ( )2
33 23
2
or ppp =
Example 7:
Evaluate a.
3
2
64
27






b.( ) 3
4
125
−
Example 8:
Simplify the following expressions:
a.
2
124
42
1
2
24






−−
yxx
yx
b. 3
3
−
x
xx
12

13. a.
2
124
42
1
2
24






−−
yxx
yx
b. 3
3
−
x
xx
Example 9:
Simplify a. nn
n 4
1
3
2
1684 ×÷ b. nnnn 321
220105 ×÷×+
1.2.2 Surds
13

14. 18. Any number of the form n
b (cannot be written as fraction of integers) is
called a Surd i.e. 5,6,2 3
.
19. The properties of surd
a. abba =•
b. b
a
b
a
=
c. ( )baccbca +=+
eg1:
632 =•
eg2: 3933 ==•
eg1: 3
2
3
2
= eg1:
( )
37
5233532
=
+=+
Algebraic Operations involving surd ),,,( ÷×−+
20. Adding like term (+):
Example 10: Simplify a. 3235 + b. 23342235 +++
a. 3235 +
b. 23342235 +++
21. Subtracting like term ( - ):
Example 11: Simplify a. 5455 − b. 23342435 −−+
14

15. a. 5455 − b. 23342435 −−+
22. Multiplication with surd form (x):
Example 12:
Simplify
a. ( )3102 − b.( )( )5232 +−
c. ( )( )33 +− xx d. ( )( )2532332 −+
a) ( )3102 − b) ( )( )5232 +−
c) ( )( )33 +− xx d) ( )( )2532332 −+
15

16. 23. Rationalising operations ( ÷ ) .To rationalise the denominator, we
a. multiply the numerator and denominator by itself
b. multiply the numerator and denominator by its conjugate
** Conjugate: The conjugate of ba + is ba −
The conjugate of 62 + is
The conjugate of 35 − is
The conjugate of 52 +− is
The conjugate of 64 −− is
Example 13:
Rationalise a) 5
3
b) 2
4−
( )( )
( )
ba
ba
baba
−=
−=
−+
2
22
16

17. Example 14:
Rationalise a) 23
5
−
b) 13
2
+
c) 25
3
+
a)
b)
17

18. c)
18

19. 19

20. Example 15:
Simplify a. 13
1
13
1
−
+
+
b. 1
325
3324
+
+
+
a)
20

21. b)
21

22. 1.2.3 Logarithms
24. If
bax
=
, then xba =log
where 0>a
and .1≠a
25. Law of logarithms:
NMMN aaa logloglog += …..(1) NM
N
M
aaa logloglog −=





….(2)
MPM a
P
a loglog = ….(3) 01log =a
1log =aa Pa Pa
=log
26. To change the base of logarithm using a
M
M
b
b
a
log
log
log =
Example 16:
Express the following in terms of yx log,log and zlog
a. )log( 2
xy b. 





z
xy
log c. 







yz
x
log d. 





xyz
1
log
a) b)
c) d)
22

23. Example 17:
Given 59.13log2 = and 32.25log2 = , without the use of the
calculator, evaluate:
a. 9log2 b. 15log2 c. 6.0log2 d. 30log5
a. 9log2 b. 15log2
c. 6.0log2 d. 30log5
23

24. Example 18:
Evaluate the following logarithms without using table or
calculator.
a. 8log
2
1
b. ( )5125log5 c. 18log 23
a.
8log
2
1
b. ( )5125log5
24

25. c. 18log 23
25

26. Example 19:
Simplify
a. 2log
32log
b
b
b. 3log9 c. 





+





−+
45
2
log
45
35
log5log270log 10101010
a. 2log
32log
b
b
b. 3log9
c. 





+





−+
45
2
log
45
35
log5log270log 10101010
26

27. 27. Solving equations involving indices normally requires the changing of the
equations to the same base (2 terms).
Example 20:
Solve the following equations
411
93.813.273. −++
=== xxxx
cba
28. Solving equations involving indices (3 terms).
27

28.  Let Aax
= (basic)
 Substitute in the original equation…….. (quadratic equation)
 Solve the quadratic equation
 Find the original answer
 Conclusion
Example 21: Solve the equation 03)3(103 12
=+−+ xx
 Let Aax
= Ax
=3
 Substitute in the original
equation……..quadratic
equation 03103
03)3(10333
03)3(103
2
1
12
=+−
=+−••
=+−+
AA
xxx
xx
 Solve the quadratic
equation 3
3
1
0)3)(13(
==
=−−
AA
AA
 find the original answer
we know that:
3
1
=A
3=A
1
33
3
1
3
1
−=
=
=
−
x
x
x
1
33
33
1
=
=
=
x
x
x
 Conclusion The values of x are -1 or 1
Example 22: Solve the equation 1
22)2(3 +
=− xx
28

29. Example 23: Solve the equation 06222
=−+ xx
Example 24: Solve )5(26125 1−
=+ xx
29

30. 29. Solving equations involving surds normally requires the need to square
both sides of the equations. Remember to check your answers.
30. To solve surds equation, we have to look if the equations have 1, 2 or 3
surds in the equation.
xx =++ 115 one two three
125 =−+ xx one two three
xxx −=−−+ 23352 one two three
30

31. IF : ……
a. if there is only one surd, put it on one side
b. if there are 2 surds, move one to the other side
c. if there are 3 surds, make sure one of them is on one side
31. Then, we have 5 steps to solve the equation.
STEP 1: square both sides of the equation and isolate any remaining surds
STEP 2: square the equation again to remove any remaining surds
STEP 3: solve the resulting equation
STEP 4: check your answer
STEP 5 : Conclusion
Example 25:
Solve the following surd equations
a. xx =++ 115
b. 125 =−+ xx
c. xxx −=−−+ 23352
31

32. Solution (a):
xx =++ 115
only one surd, put it
on one side
115 −=+ xx
square both sides of
the equation and
isolate any remaining
surds
( ) ( )
1215
115
2
22
+−=+
−=+
xxx
xx
square the equation
again to remove any
remaining surds
-no more surd
solve the resulting
equation
70
)7(0
70
1215
2
2
==
−=
−=
+−=+
xx
xx
xx
xxx
check your answer
Refer xx =++ 115
when 0=x
)(0
0
2
11
11)0(5
NAx
RHSLHS
RHS
LHS
=
≠
=
=
+=
++
Refer xx =++ 115
when 7=x
7
7
7
136
11)7(5
=
=
=
=
+=
++
x
RHSLHS
RHS
LHS
Conclusion After checking, 7=x is the only solution.
Solution (b):
125 =−+ xx
32

33. 2 surds, move one to the other side 125
125
+=+
=−+
xx
xx
square both sides of the equation and
isolate any remaining surds
x
x
xxx
xx
=
=
++=+
+=+
12
224
1225
)1()25( 22
square the equation again to remove
any remaining surds
( )22
12 x=
solve the resulting equation
x=144
check your answer
Refer 125 =−+ xx
when 144=x
144
1
11213
12169
14425144
=
=
=
=−=
−=
−+
x
RHSLHS
RHS
LHS
Conclusion After checking, 144=x
Solution (c):
xxx −=−−+ 23352
3 surds, make sure one of
them is on one side
xxx −=−−+ 23352
square both sides of the
equation and isolate any
remaining surds
( ) ( )22
23352 xxx −=−−+
Let
33

34. xCxC
xBxB
xAxA
−=−=
−=−=
+=+=
22
3333
5252
2
2
2
so
( ) ( )
*..........................2 222
22
CBABA
CBA
=+−
=−
1596
1515663352
thatknowWe
2
2
+−−=
−+−=−+=
xx
xxxxxAB
15963
159626
1596223532
2331596252
*into
2
2
2
2
+−−=
+−−=
+−−=−+++−
−=−++−−−+
xx
xx
xxxxx
xxxxx
square the equation again to
remove any remaining surds
15969 2
+−−= xx
solve the resulting equation
2
2
1
0)2)(12(
0)232(3
0696
091596
2
2
2
−==
=+−
=−+
=−+
=+−+
xx
xx
xx
xx
xx
check your answer
Refer
xxx −=−−+ 23352
when 2
1
=x
2
1
5.1
224744.1
5.16
=
=
=
=
−
x
RHSLHS
RHS
LHS
Refer xxx −=−−+ 23352
when 2−=x
)(2
24
291
NAx
RHSLHS
RHS
LHS
−=
≠
==
−=−
34

35. Conclusion
After checking, 2
1
=x .
32. Solving equations involving logarithms normally requires the changing of
the equations in the form of logarithms with same base.
• same base
• different base
Example 26:
Solve each of the following equations
a. 63 =x
b. 13loglog 1010 +=x
a. 63 =x
b. 13loglog 1010 +=x
35
Check your
final answer

36. Example 27:
Solve each of the following equations
a. 72 =x
b. )5(log1)2(log 2
2
2 ++=+ xx … same base
Solution:
36

37. Solution:
Example 28:
Solve the equation
)1(log8loglog2log 3333 −+=+ xx ………………same base
Solution:
37

38. Example 29:
Solve the equation
a) 5.24loglog4 =+ xx ………….. Different base
b) 1)(log3log2 3 −= xx ………….. Different base
Solution:
38

39. Solution:
39

40. # 4ln4log =e # # 1lnlog == eee # # # xe x
=ln
Example 30:
Solve
a. 1)2ln( =+ x
b. )17ln()2ln(22ln 2
+=−+ xx
c. 123 ln24
=+−
xex x
40
Take Note
That

41. Solution:
41

42. ||||||| END OF CHAPTER 1 : 2015/2016 ||||||||
42