TOPIC 1_PART 1_STUDENT

TOPIC 1 :
NUMBER SYSEM AND EQUATIONS

1.1 Real Numbers

The Power of PowerPoint - thepopp.com

COURSE LEARNING OUTCOME

a) Acquire fundamental CLO 1
concepts and theories in
Algebra and Business
Mathematics

b) Propose solutions for real life CLO 2
problems in Algebra and
Business Mathematics
through group discussion

The Power of PowerPoint - thepopp.com

COURSE FRAMEWORK AND STUDENT LEARNING TIME (SLT)

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

diagrammatically showing   and  W  

c) Represent open, closed and semi - open intervals and
their representations on the number line.

d) Find union,  , and intersection,  , of two or more
intervals with the aid of number line.

The Power of PowerPoint - thepopp.com

a) Natural numbers are positive numbers that are used for counting :

 N = {1, 2, 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.

b) Whole numbers are natural numbers including the number zero :

Natural + zero : W = {0, 1, 2, 3, … }

The Power of PowerPoint - thepopp.com

c) Integers are whole numbers including their negatives :

 Whole + negatives : Z = {… , -2, -1, 0, 1, 2, … }

Z Z

d) Prime numbers are natural numbers greater than 1 that can only be
divided by itself and 1 only.

 Prime numbers = { 2, 3, 5, 7, … }

The Power of PowerPoint - thepopp.com

e) Rational numbers, Q are numbers that can be written in the form p
where p and q are integers and q  0 q

 Q = {-1/3 , 3/2 , 8/1 , 22/7 … }



f) fIorrrmatiopnawl hneurme pbearsn,dQqaraerneuimntbeegrserthsaatncdanqnot0be written in the

q

 Q = {   3.14159... , e  2.71828182845... ,
5  2.23606797... }

The Power of PowerPoint - thepopp.com

The difference between rational numbers and irrational numbers.

Rational Number, Q Irrational Number, Q

- decimal representations are : - decimal representations are :

 terminating decimal  non-terminating decimal
 non-repeating
3  0.375 6  0.24
8 25

 repeating 2  0.70710678..
2
0.13  0.1313...
1  1.666...  1.6 3 1.73205...
6
  3.14159...

2  0.285714285714...  0.285714 e  2.71828182845...

7 The Power of PowerPoint - thepopp.com

Relationship of number sets

The different types of real numbers can be represented by the following
Venn diagram :

e 3 R
0
Q 2 N 1,2,3,...
W
9 Z
2
1
Q

7.12



2

N  W  Z  Q ,The diagram show that :

Q  Q  RThe Power of PowerPoint - thepopp.com

EXAMPLE 1 :

Given S = {-9, 7, 1 , 2, 0, 4, 5.125125}, identify the set of
3
(a) natural numbers (b) whole numbers (c) integers
(d) rational numbers (e) irrational numbers (f) real numbers

Solution : (d)Q = {-9, 0, 4, 1
3 , 5.125125}
(a)N = {4}

(b)W = {0, 4} (e) Q = { 7, 2}
(c) Z = {-9, 0, 4}
(f) R = {-9, 7, 1, 2, 0, 4, 5.125125},
3

The Power of PowerPoint - thepopp.com

Intervals of Real Numbers

The Number Line

For example for –3.5 , 2 and  can be shown on a real number line
3

-4 -3 -2 -1 0 1 2 3 4
-3.5 2 

3

The order of real numbers is important in presenting of the intervals on
a number line.

The Order of Real Number

Symbol Description Example

a=b a equal to b 3=3
a<b a less than b 4 < 4
a>b The Power aof gProewaetrPeorintth-atnhebpopp.com 5>0

All real numbers between a and b can be written in the form of intervals

as shown in following table.

Type of Interval Interval Set Representation on the number
Notation Notation line

Closed interval [a, b] x : a  x  b a bx
(a, b) a bx
Open interval (a, b] x : a  x  b a bx
[a, b) a bx
Half-closed Or x : a  x  b
Half-opened x : a  x  b bx
interval x
x
Infinite (, b) x : x  b
b
interval (a, ) x : x  a a x
Infinite (, b]
x : x  b

interval [a, ) x : x  a

Note : An empty circle aTnhde PaodweernosfePcoiwrecrlPeoint - a
thepopp.com

EXAMPLE 2 :

Express the type of this intervals and represent in on real number line.

(a) [-1, 4] (b) (2 , 5]

(c) (2, ) (d) {x : x  0, x  R}
Solution :

a) [-1, 4] is a closed interval

b) (2, 5] is a half-opened interval -1 4
5
c) (2, ) is a infinite interval 2
2 0
d) x : x  0, x  R  ,0

is a infinite interval

The Power of PowerPoint - thepopp.com

EXAMPLE 3 :

Given A   4,0, B   2,5 and C  x : 5  x  3, x  Z

Represent A,B and C on a number line.
Solution :

C 5
B
A

-5 -4 -3 -2 -1 0 1 2 3

The Power of PowerPoint - thepopp.com

Intersection and Union operations
on intervals

 The intersection ,  of two intervals is the set of real

numbers that belong to both intervals.

 The union ,  of two intervals is the set of real

numbers that belong to one, or the other, or both of
two intervals.

The Power of PowerPoint - thepopp.com

Intersection and union operation can be perform on intervals.
For example, given A = [1 , 6) and B = (2, 4)

if A  B = [1 , 4) if A  B = (-2 , 6)

-2 1 B -2 1 A
A B

46 46

The Power of PowerPoint - thepopp.com

union intersection

 

MESYUARAT RUKUN
PIBG NIKAH

The Power of PowerPoint - thepopp.com

EXAMPLE 4 : (b) (, 5)  (1, 9)

Simplify the following using the number line

(a) [0, 5)  (4, 7)

(c) (, 0]  [0, ) (d) (0,6]  [6,9)

(e) [ (4, 2)  (0, 4] ]  [2, 2) Answer :

The Power of PowerPoint - thepopp.com a) 0,7
b) 1,5
c) , 
d) 6
e) 2, 2

EXAMPLE 5 :

Given A = (-5,0) , B = [-1,7) and C  x : 3  x  3, x  Z

By using real number line, find the following :

a) A B b) A B ' c) AC

The Power of PowerPoint - thepopp.com Answer :

a) 5, 7
b) 5, 1
c) 3, 2, 1

Exercise 1 : “PSPM PDT_2013/14”

Given A   3,2, B  1,5 and C  x : 4  x  2, x  R

(a) Represent A,B and C on a number line.

(b) Hence, find ii AC [6 marks]

i A B

Answer :

b) i. 3,5 ii. 3, 2

The Power of PowerPoint - thepopp.com

TOPIC 1 :
NUMBER SYSEM AND EQUATIONS

1.2 Indices, Surds
and Logarithms

The Power of PowerPoint - thepopp.com

COURSE FRAMEWORK AND STUDENT LEARNING TIME (SLT)

a) Express the rule of indices.

b) Explain the meaning of a surd and its conjugate.

c) Perform algebraic operations on surds.

d) State the laws of logarithms such as:

e) Change the base of logarithm using

f) Solve equations involving surds, indices and logarithms.
*Only single equations such as and equations that result to quadratic
forms (solve by using factorisation, completing the square and formula).

The Power of PowerPoint - thepopp.com

a0  1

an  1
an

1. a m  a n  a m n Rules of 1
2. a m  a n  a mn Indices
an n a

3. (a m) n  a mn  a n  a n
 b  b n
 m

ab n a n b n a n  n a m

The Power of PowerPoint - thepopp.com

NO. RULES OF INDICES EXAMPLES

1 am  an  amn 23  25 

2 am  an  amn 34  32 

 3 am n amn  23 4

4 a  bm  am  bn 2  32 

5  a m   am   3 2 
 b  bm 2

6 1 1

an  n a 162 

 7 2
m nam
83 
an 

8 an  1 23 
an

9 a0  1 70 
The Power of PowerPoint - thepopp.com

Example 1 :

Evaluate each of the following without using calculator .

2 b  8  2
3
a 83

 27 

The Power of PowerPoint - thepopp.com a 4 b 9

4

Example 2 :

Simplify each of the following :

    3 x y
x2  y2
a y2 2  y3
b t 3  5 t 2 c

a 1 19 c x2 y2
yx
b t10

The Power of PowerPoint - thepopp.com

Example 3 :

Simplify

a. 4n 2n 1n b. 5n1 10n  202n  23n

83 16 4

Answer :
a) 2 b) 5

The Power of PowerPoint - thepopp.com

Solving equations involving indices

Only 2 terms 3 terms or more

Same base Only 2 terms Use substitution to
get a quadratic
Comparing base & Not same base equation
index of the terms
Taking “log”of Apply rules of
both sides of the indices
equation

The Power of PowerPoint - thepopp.com

Only 2 terms Example 4 : Answer, x  3
2
Same base 1) 3x  27x1
Answer, x  1 or x  3
Comparing base & 2)  1  3x x2  32x1
index of the terms 9  Answer, x  6 or x  2

3) 7x2  4962x  0

The Power of PowerPoint - thepopp.com

Only 2 terms Example 5 : Answer, x  2.807
Answer, x  0.3067
Not same base 1) 2x  7 Answer, x  2.585

Taking “log”of 2) 312x  4x
both sides of the
equation

3) 2x1  3

The Power of PowerPoint - thepopp.com

3 terms or more Example 6: Answer, x  0 or x  3

 1) 2x  8 2x  9

Use substitution to 2) 32x1  3x3  3x  9
get a quadratic
equation Answer, x  1 or x  2

Apply rules of 3) 9x  20  3x2
indices
Answer, x  1.465 or x  1.262

The Power of PowerPoint - thepopp.com

ENHANCEMENT QUESTION:

 Solve the equation 32x  5 31x  9

The Power of PowerPoint - thepopp.com Answer, x  0.112

MEANING OF SURDS & ITS CONJUGATE

 Surd is an irrational number and expressed in terms

aof root sign n

 Example = 7, 5, 11

NOTE..!!! Let’s pronounce correctly

__a___ is not a perfect n a is nth root of a
a is square root of a
square number.
Eg : 4, 9 3 a is cube root of a
4 a is fourth root of a

The Power of PowerPoint - thepopp.com 15

PROPERTIES OF SURDS 

a b  ab

a a 
bb

a b  c b  a  c b 3

 4The Power of PowerPoint - thepopp.com

Additional rules

a  a  aa  a2  a

 

Remark!!!

 ab  a  b

The Power of PowerPoint - thepopp.com

REMEMBER ..!!! (for expansion)

1. a  ba  b  a2  b2

2. a  bc  d   ac  ad  bc  bd

3. a  b2  a2  b2  2ab
4. a  b2 a2  b2  2ab

The Power of PowerPoint - thepopp.com

Algebraic operations involving surd

5 32 3 7 3 Adding like 5 53 5 2 5
term
5 32 24 33 2 5 34 2 3 3 2
9 35 2 Subtracting 2 33 2
like term

Multiplication

Rationalising

The Power of PowerPoint - thepopp.com

Multiplication a  b2  a2  b2  2ab
a  ba  b  a2  b2

 i. 2 10 3
ii.  2 3 2  5

iii. 2 2  32 2  3

 iv. 2 Answer :
2
3 i 2 5  3 2
ii 2 2 13
The Power of PowerPoint - thepopp.com iii5
iv5  2 6

Rationalising

THE CONJUGATE OF SURDS

 Just change sign at the operation

 To rationalise the denominator

•Eg :

SURD ITS
CONJUGATE

2 5

32

3  2 7The Power of PowerPoint - thepopp.com

To rationalise the denominator, we

Rationalise the denominator of the following
fractions:

a) 5 b) 3  2
10 2 3

Answer : 10 Answer : 5  2 6
2
The Power of PowerPoint - thepopp.com

c) 3  2 10  3  2 10 Check denominator d) 4  5
3  10 3  10 • its pair of 2 2 1 2

conjugate

Answer : 58 Answer : 9  3 2

The Power of PowerPoint - thepopp.com

Solving equations involving surds

The Power of PowerPoint - thepopp.com

Solving equations involving surds

STEPS : STEPS :

 Put it on one sides  Move one to the other
 Square both sides to sides

isolate surds  Square both sides to
 Rearrange /simplify isolate surds
 Factorise if get quadratic
 Rearrange /simplify
equation  Square both sides
 Check your answer  Factorise if get

(substitute value into quadratic equation
original equation)  Check your answer

STEPS :
 Make sure one of them is on one side
 Let

A = A2=

B = B2=

C = C2=

AB=
 Square both sides and expand
 SubstituteTbhaeckPAo,wB,eCr&ofAPBowerPoint - thepopp.com

Example 1: a  b2  a2  b2  2ab
a  b2  a2  b2  2ab
Solve the following equation :

1) 5x 1 1  x STEPS : (1 surd)

 5x 1 2  x 12  Put it on one sides
 Square both sides
5x 1 x2  2x 1
x2 7x  0 to isolate surds
 Rearrange /simplify
xx  7  0  Factorise if get

x0 , x7 quadratic equation
 Check your answer

(substitute value
into original
equation)

The Power of PowerPoint - thepopp.com

Check the answers 5x 1 1 x

Substitute x = 0 into Substitute x = 7 into
the equation the equation

LHS : 501 1 LHS : 571 1

 11  36 1

2 7

 0 RHS  7 RHS

ThuThse,Potwhereof aPonwesrPwoinet -rthiespoppx.co=m 7

2) 5x 1  x  2  1 a  b2  a2  b2  2ab
a  b2  a2  b2  2ab

Answer : x  2 STEPS : (2 surds)

 Move one to the
other sides

 Square both sides
to isolate surds

 Rearrange /simplify
 Square both sides
 Factorise if get

quadratic equation
 Check your answer

The Power of PowerPoint - thepopp.com

3) x  6  x  3  2x  5 a  b2  a2  b2  2ab
a  b2  a2  b2  2ab

Let : STEPS : (3 surds)
 Make sure one of
A x6 A2  x  6
them is on one
B x3 B2  x 3
side
C  2x 5 C2  2x 5  Let

AB  x  6x  3 A = A2=
B = B2=
AB C C = C2=

A  B2  C2 AB=
 Square both sides
A2  B2  2AB  C2
and expand
 Substitute back

A,B,C & AB

The Power of PowerPoint - thepopp.com

x  6 x  3 2 x2  9x 18  2x  5

4  2 x2  9x 18

 42  2
2

x2  9x 18

 16  4 x2  9x 18

x2  9x 14  0 So, x= -2 is the only the answer

x  2x  7  0

x  2 x  7

Check the answers

The Power of PowerPoint - thepopp.com

a 1
a0
N 0

The Power of PowerPoint - thepopp.com

TYPES OF LOGARITHMIC

Common Natural
Logarithms Logarithms

 Is log to the base ’10’  Is log to the base ‘e’

 Write as : ‘log10’ or ‘log’  Write as : ‘loge’ or ‘ln’

Example : log 5 Example :

log10 5 a)loge 5 = ln 5
b)logex =ln x

The Power of PowerPoint - thepopp.com

LAWS OF LOGARITHMS

1) loga MN = loga M + loga N product
rule
log (M+N) log M + log N

2) loga M = loga M – loga N quotient
N rule

log M  log M - log N

log N

log M  M Cannot be
log N simplified
NThe Power of PowerPoint - thepopp.com