Learning Unit 1a:

Numbers & Operations

Learning Objective
Students should be able to:
1.

Identify the features of numbers.

2.

Convert fraction into percentages and proportion.

3.

Add, subtract, multiply and divided with positive and negative

integers.

4.

Perform all the above tasks without calculator.

Natural Numbers

Numbers that are used for counting things or physical objects are called
Natural Numbers, N.
N = 1, 2, 3, 4, 5, 6, 7,
N is the set of number 1, 2, 3, 4 and so on.

The symbol is to show membership in a set. It is read is an element of

or belongs to. Example: 6 N (6 is an element of N) and 0 N (0 is not
element of N).

A set having no elements is called the empty or null set or .

Natural numbers together with 0 is called the Whole Numbers, W.

W = 0, 1, 2, 3, 4, 5, 6, 7,

The natural numbers are a proper subset () of the whole numbers, denoted N
W.

Real Numbers

The negative of natural numbers together with 0 and the natural numbers,
form the set of Integers, Z,
Z = .. 3, 2, 1, 0, 1, 2, 3,
+ Z = 1, 2, 3,
- Z = 1, 2, 3, .

Odd Number is of the form of 2n + 1, 1, 3, 5,7,

Even number is of the form of 2n, 2, 4, 6, 8,

p
Fractions are numbers that can be written in the form of
q

The set of numbers that can be written in this form are called Rational
Numbers, Q.
Q = p/q p, q Z; q 0

The vertical bar, is read such that. So, Q is the set of numbers of the form
p over q, such that p and q are integers and q is not equal to zero.

Real Numbers
Example:

1
2
6
, , 6
2
3
1

1
0.1111
9

Repeating and nonterminating

3
0.375
8

Terminating decimal

There are numbers that cannot be written as fractions and their decimal
representation neither terminate or repeat, refers as Irrational Numbers (H).
Example:
5= 2.3606797 and = 3.14159)

Summary of Real Numbers

R (real): All rational and irrational numbers
Q (rational): p/q p, q Z and q 0
Z (integer) : , -2, -1, 0 , 1, 2,
W (whole) : 0, 1, 2, 3,
N (natural):
1, 2, 3,

H (irrational):
Numbers that
cannot be written
as the ratio of two
integers; the real
number that is
not rational.
2, 7,
0.070070007
and so on.

Negative Numbers

If you subtract a bigger number from a smaller number, the result is a negative
number (-ve).
13= ?

If you multiply a positive number and a negative number, the result is a negative
number. (-ve).
( 5) * 6 = ?

If you multiply two negative numbers, the result is a positive number.

( 2) * ( 5) = ?

Symbol > or < is used to show when one number is greater or less than another
6>5
2<5

Any negative number is less than positive number, 8 < 1

Negative numbers are larger when they are closer to zero,

3<2
1>6

Absolute Value

Written as | ... |, is found ignoring the minus sign, if there is one

|6|=6
| 2| = 2

Smaller negative number has larger absolute value

|5 3| = |2| = 2
| 5 3|=| 8| = 8

Properties of Real Number

Type

Particulars

Closure

a+b;a.b

Commutative

ADDITION: a + b = b + a
MULTIPLICATION: a x b = b x a
Example:
2+5=5+2 ; 2x5=5x2
However;
2552 ; 2552

Associative

(a + b) + c = a + (b + c)
( a. b ). c = a. ( b. c )
Example:
(3+4) + 5 = 3 + (4+5) @ 3 x (4x5) = (3x4) x 5
However;
6 (4-2) (6-4) 2 @ (8 4) 2 8 (4 2)

Properties of Real Number

Type

Particulars

Distributive

a(b + c) = ab + ac
Example:
2(3+4) = 2(3) + 2(4) @ 5(86) = 5(8) 5(6)

Identity

a+0=a;1xa=a

Inverse

a + ( a) = 0 ; a x 1/a = 1

Multiplicative
Property of 0

ax0=0

Identify the following properties of real number

EXAMPLE
1

No.

Items

a)

4.7

b)

4 (7 . 6) = (4 . 7) 6

c)

(1/6 . 6)3 = 1 . 3

d)

1 . 20 = 20

e)

3(4 + 5) = 3(4) + 3(5)

f)

7 + (5 + 9) = (7 + 5) + 9

Real number
properties

Identify the following properties of real number

No.

Items

Real number
properties

a)

4.7

Closure

EXAMPLE

b)

4 (7 . 6) = (4 . 7) 6

Associative

c)

(1/6 . 6)3 = 1 . 3

Inverse

d)

1 . 20 = 20

Identity

e)

3(4 + 5) = 3(4) + 3(5)

Distributive

f)

7 + (5 + 9) = (7 + 5) + 9

Associative

SOLUTION

Finite Intervals
Interval Notation

a<x<b
axb
a<xb
ax<b

(a,b)
[a,b]
(a,b]
[a,b)

Real Number Line

Open
Interval

Closed
Interval

Half-Open
Interval

Half-Open
Interval

Infinite Intervals
Interval Notation Real Number Line

x<a

(-,a)
a

xa

(-,a]
a

x>a

(a, )
a

xa

[a,)
a

Intervals

EXAMPLE

2 x 6

x<2

Interval
Notation

Real Number
Line

10

[5,)

Intervals

Interval
Notation

2 x 6

[2, 6]
2

EXAMPLE
2
SOLUTION

Real Number
Line

x<2

(, 2)
2

3 < x 10

(3, 10]
3

x 5

10

[5,)
5

x2

[2, )
2

Learning Unit 1a:

Mathematical Operations

Mathematical Operations
Operations
Addition
Multiplication
Subtraction
Division

Sign
+
x or *

/ or

Mathematical Rules
When different operations occur, the rule is to

multiply and
subtracting.

divide

Operations

BEFORE

inside the brackets

performed before those outside.

Work from left to right.

adding

should

and

be

a)

EXAMPLE
3

b)

3 *11 6
3
3 * (11 6)
3

c) 9 6 5 3
d ) 9 [6 (5 3)]
e) 16 4 4
f ) 16 ( 4 4)

a)

EXAMPLE
3
SOLUTION

b)

3 *11 6
3

33 6

27

9
3

3 * (11 6) 3 * (5) 15

5
3
3
3

c ) 9 6 5 3 15 5 3 10 3 13
d ) 9 [6 (5 3)] 9 (6 8) 9 ( 2) 7
e) 16 4 4 4 4 16
f ) 16 ( 4 4) 16 16 1

Learning Unit 1a:

Fraction & Percentage

Fractions
1
2

numerator
denominator

Usually write fractions using numbers that have no common divisors (lowest

terms)

10
2*5
5

6
2*3
3

To add or subtract fractions, put them over a common denominator

3 2
3*3 2 *5
9
10
9 10 19

5 3
5 * 3 3 * 5 15 15
15
15
To multiply, simply multiply the numerator and denominator.

3
2
3 2
6
2

5
3
53
15
5
To divide, turn the divisor upside down, then multiply.

8 2
8 3
24 12

5 3
5 2 10
5

Fractions
The percentage (%) is used for divide 100.

Example : 10% of 140:

10
140
140 10 14
100

Adding 10% of 40:

10
1

40 40

40

40

40 4 44
100
10

Subtract 10% from 40:

10
1

40 40

40

40

40 4 36
100
10

1. 33 is what percent of 99?

EXAMPLE
4

2. 30 is 70% of what number?

3. Net value of discounted 17% from 232?
4. 70% of 10.30 is what?
5. Total value of mark-up of 25% on 135?

33
1.
100% 33.33%
99

EXAMPLE
4
SOLUTION

2. 30 70% x
30
x
42.86
70%
3. 232 83% 192.56
4. 70% 10.30 7.21
5. 135 125% 168.75

Learning Unit 1a:

Powers & Exponents

Powers & Exponents

If we multiply a number by itself n times, we have raised it to the n-th power

23 = 2 . 2 . 2 = 8

1,0001 = 1,000

The superscript number is called the exponent

a. 2 . 102 6 = 2. 100 6 = 194
b. (2 . 10)2 6 = 400 6 = 394
c. 2 . (10 6)2 = 2 .16 = 32
00 = 1
Multiplying powers of the same number is equivalent to adding the exponents

32 . 33 = 32+3 = 35
Dividing powers of the same number is similar as subtracting exponents

34 33 = 34-3 = 31

Roots & Fractional Exponents

A square root of x is a number that when multiplied by itself, gets back to x.

Square roots:
Cube roots:
3 a (aR)
a b if b2 = a (a 0)
if b3 =
a b
This indicates that

a. a a
or

a .

or
2

a .

a a

Square root of 25 is 5, since 52 = 25. Cube root of 8 is 2, since 23 = 8.

Since a negative number squared is positive, every positive number has two square
roots (-5 is also square root of 25)
52 = 25

(5)2 = 25

BUT a negative number has no square roots.

Learning Unit 1a:

Index

Index
Definition
1) an = a x a x a x .. x a

1
2) a = n
a
-n

3) a0 = 1; 00 is undefined

Index
Properties of Integer Indices
1)

am x an = am+n

2)

(am)n = amn

3)

(ab)m = ambm

4)

a

b

5)

am
mn

a
an

am
m ;b 0
b

Using Index Properties

EXAMPLE
5

1)

(2a 3b 2 ) 2

2)

a
5
b
3

3)
4)

4 x 3 y 5
6 x 4 y 3
m 3m3

2
n

Using Index Properties

EXAMPLE
5
SOLUTION

1)

( 2 a 3b 2 ) 2 2 2 a 6 b 4

2)

a
5
b
3

a6
4
4b

a 6 b10
10 6
b
a

3 5
3 ( 4 )
4
x
y
2
x
2x
3)

8
4 3
3 5
6x y
3y
3y

3 3
4) m m
n 2

m 3 3

2
n

m0
2
n

1
2
n

1
4
n

Using Fractional Indices

EXAMPLE
6

1)

2
3

1
3

1
2

2) (3x )(2 x )
3)

1
2

1
2

1
2

1
2

(u 2v )(3u v )

Using Fractional Indices

EXAMPLE
2
3

6
SOLUTION

1
3 2

1) 8 (8 ) 2 2 4
1
3

1
2

2) (3x )(2 x ) 6 x
1
2

1
2

1
2

1 1

3 2

6x
1
2

5
6

1
2

1
2

3) (u 2v )(3u v ) 3u 5u v 2v

Find the value of x.

EXAMPLE
7

1 x

1)

2)

4 x 3 8

3)

2 2
x

1x

Find the value of x.

EXAMPLE

1)

1 x 3
x 2

7
SOLUTION

91 x 93

2)

4 x 3 8

2
2

x 3

23

2 2 x 6 23
2x 6 3
2x 9
x

9
2

Find the value of x.

EXAMPLE
7
SOLUTION

3) 2 x 21 x 3
2
3 0
x
2
u 2x
2x

2
3 0
u
u 2 3u 2 0
u

(u 1)(u 2) 0
u 1; u 2

2 x 1 20
x0
2 x 2 21
x 1

Learning Unit 1a:

Complex Numbers

Complex Numbers

The equation X2 = -1 has no real solutions, since the square of any real number is
positive. But if we apply the principle of square roots we get x = 1 and x = - 1.

This result what we called the set of imaginary number. The imaginary unit i represents
the number whose square is -1: i 2 = 1 and i = 1

Multiplying powers of the same number is equivalent to adding the exponents.

1 = I
3 = 3 x 1 = 3i
4 = 4 x 1 = 2i

Imaginary numbers are in the form of ai where a is a real number.

i 3 = (i 2) (i) = (1)i = i
i 4 = (i 2) (i 2) = (1)(1) = 1

Imaginary numbers can be added or subtracted.

2i + 3i = 5i
8i 2i = 6i

Multiplication and division of two imaginary numbers gives a real number.

9i
3
4i x 2i = 8i2 = 8
3i

Algebraic Operations on Complex Numbers

Addition and Subtraction

(2 + 3i) + (2 3i)
= (2 + 3) + (3 1)i
= 5 + 2i

(5 + 4i) (2 + 3i)
= (5 2) + (4 3)i
=3+i

Multiplication

(2 + 3i) (4 2i)
= 8 4i + 12i 6i2
= 8 + 8i 6(1)
= 14 + 8i

(2 + 3i) (2 4i)
= 4 8i + 8i 16i2
= 4 16(1)
= 20

Division

Multiplying the numerator and the denominator by the conjugate of the

denominator,

3 i 3 i 1 i 3 2i i 2 4 2i

2i
2
1 i 1 i 1 i
1 i
2

Learning Unit 1a:

Surds

Surds
1
n

a n a
Example:

1
3

6 3 6

Properties of Surds

2)
3)

xn x

1)
n

xy x n y
n

n
n

x
y

Surds
Simplifying Surds
1)
2)
3)

12 4 3 4 3 2 3
5

(3x 2 y )5 3 x 2 y
12 x 3 y 5 z 2

(4 x 2 y 4 z 2 )(3 xy )
(2 xy 2 z ) 2 (3 xy )
(2 xy 2 z ) 2 (3xy )
2 xy 2 z 3 xy

Surds
Combining Like Terms
5 3 4 3 (5 4) 3 9 3

Multiplication with Surd Forms

2 ( 10 3)
2 10 2 3
20 3 2
2 5 3 2