Number System

NUMBER SYSTEM

Subtopics

1.1 Real Numbers

1.2 Complex Numbers

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1.1 Real Numbers

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1.1 Real Numbers

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REAL NUMBERS

Prime
Numbers

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No. type Symbol Example Explanation

Natural {1,2,3,…} Counting numbers

Whole {0,1,2,…} Natural numbers with zero

Integers {0, ±1,±2, ±3,…}

Prime {2,3,5,7,11,…} Natural number that can only be
divided by itself and 1

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No. type Symbol Example Explanation
Rational
▪ Fraction
Irrational
▪ Decimals that is repeating or
terminating

Non-repeating and non-terminating
decimals

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DIFFERENTIATING RATIONAL AND IRRATIONAL NUMBERS

Example Point form Decimal representation Conclusion

1/2 0.5 terminating decimal rational
3/4 0.75 number
1/3 0.3333… or terminating decimal rational
5/11 0.4545… or number
π 3.1415926… non-terminating repeating rational
√3 1.7320508… decimal with period 3 number

non-terminating repeating rational
decimal with period 45 number

non-repeating, irrational
non-terminating decimal number

non-repeating, irrational
non-terminating decimal number

EXAMPLE 1 Identify the set of
Given

SOLUTION

Relationship of Number Set

Real Number Line

FINITE INTERVALS

Notation Inequalities Representation on number Closed
line Open
[a,b] a < x < b
(a,b) a < x < b ab
(a,b] a < x < b
[a,b) a < x < b ab
ab
ab

Half – open Half – open
(left) (right)

INFINITE INTERVALS

Notation Inequalities Representation on number
line
(-∞,b) x<b
(-∞,b] x<b b
(a,∞) x>a
[a,∞) x≥a b
a

a

Remarks: Note that these are all half – open intervals where
x is always written on the left hand side (as a subject)

EXAMPLE 2
Distinguish the type of intervals below and represent them on number line.

[1,7] Closed interval

(-2,5) Opened interval 17
-2 5

(5,∞) Opened interval

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{x : x < 0, x∈R} Right closed interval or left opened interval

or

(- ∞,0] 0

UNION AND INTERSECTION OF INTERVALS SCAN /
CLICK HERE
EXAMPLE 3 FOR VIDEO

Let A∩B https://bit.ly/3v
A = [0,6) L4VeK
B = (-1,5)

-1 0 56

A∪B

-1 0 56

EXAMPLE 4

SOLUTION

-2 1 3 66

b)

2 34

SOLUTION

-6 1 56

15

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EXAMPLE 5

Hots

SOLUTION

TRY IT YOURSELF !!

-1 0

SOLUTION

TRY IT YOURSELF !!

-1 0

IMPORTANT THINGS TO RECAP:

• Name all the numbers groups
• Represent intervals using number

line
• Solving union and intersection and

placing the answer in brackets or
inequalities forms

https://bit.ly/3ISaJqG

1.2 Complex Numbers

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Let say if we need to find:

1 2

2 = 9
⇒ = −3, = 3

Note that the roots are Real Numbers

What if we have this problem

So complex number starts here!!!

Don’t have real
solution

The answer is a COMPLEX NUMBER
(contains i)

Real part Imaginary part

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Therefore, we name a COMPLEX NUMBER as

Re(z)=a Im(z)=b
(real part) (imaginary part)

This is a Cartesian Form of a complex number

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Note that

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EXAMPLE 6
SOLUTION

3311

EXAMPLE 7

Find a and b for the following equations
a + b + (a – b)i = 6 + 4i

SOLUTION
Compare Re(z) and Im(z)

a+b=6 1 (a – b)i = 4i
(a – b) = 4
2

1+ 2

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EXAMPLE 8 TRY IT YOURSELF !!
Given w = 4 – 2i and z = -7 + 5i, find zw.

SOLUTION

Ans: zw = -18+34i

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Conjugate of a Complex Number

Note that we always change the sign in front of the imaginary part

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Rationalization

Conjugate also can be used to rationalize a fraction:
Change a fraction with irrational/ complex numbers as denominator to a real
number

Example

NOTE THAT

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EXAMPLE 9
Determine the complex number z that satisfy the equation

SOLUTION

4 = 5 … … … … … 1 − 2 = −6 … … … … . (2)

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EXAMPLE 10

TRY IT YOURSELF !!

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Polar Form

ARGAND DIAGRAM
Im (z)
P(a, b)

Re (z)

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STEPS A.M.A.P

ARGAND DIAGRAM From the given complex number, state the quadrant. SCAN/
MODULUS CLICK HERE
Find modulus of z, FOR VIDEO
ARGUMENT, Refer to the respective quadrant

arg(z) = https://bit.ly/3
sLhvsq
Note that: is always the
angle formed from positive x – Express the complex number in polar form,
axis.

POLAR FORM

ARGUMENT, Refer to the respective quadrant
Im(z)
arg(z) =
Re(z)
Note that: is always the
angle formed from positive
x – axis.

is always in radian‼

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EXAMPLE 11
Find the modulus and argument of the following complex numbers. Hence,
express them in polar form.

A

(5,-4)

M
A
P

A

M
A
P

EXAMPLE 12 Argand diagram (1st
Quadrant)
−1+5
Let z =1+ i ,
Express 2+3 them in polar form
SOLUTION

https://bit.ly/3CnpNKu

https://bit.ly/37cNZDT

https://bit.ly/3hJERbJ

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