1.2 Number System and Representation

1.0 COMPUTER SYSTEM

1.2 Number System and Representation

Learning Outcome

At the end of this topic, students should be able to:

d) Convert between Hexadecimal and Decimal

i) Convert Decimal to Hexadecimal
ii) Convert Hexadecimal to Decimal

e) Convert between Hexadecimal and Binary

i) Convert Binary to Hexadecimal
ii) Convert Hexadecimal to Binary

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Recap - Introduction

Decimal

Hexadecimal Numbering Binary
System
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A numbering system is a way of
representing numbers.

Recap Decimal Numbering System

Exp: 2310

 The prefix “deci-” stands for 10.
 The decimal number system is a Base 10

number system.

 There are 10 symbols that represent
quantities:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

 Each place value in a decimal number is a

power of 10.

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Decimal Number System

How 729 is represented in decimal numbers ?

102 101 100 729
7 x 100 2 x 10 9x1
700 + 20 +
9

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Recap Binary Numbering System

 The prefix “bi-” stands for 2. Exp: 10102

 The binary number system is a Base 2 number

system.

 There are 2 symbols that represent quantities,

called bits:

0, 1

 Each place value in a binary number is a power

of 2.

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Binary Number System

26 25 24 23 22 21 20
64 32 16 8 4 2 1

210 29 28 27
1024 512 256 128

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Recap Hexadecimal Numbering
System

Exp: A616

 The prefix “hexa-” stands for 6 and the prefix “deci-”

stands for 10.

 The hexadecimal number system is a Base 16

number system.

 There are 16 symbols that represent quantities:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

 Each place value in a hexadecimal number is a

power of 16.

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Hexadecimal Symbol Decimal Equivalent Binary Equivalent 9
0 0 0000
1 1 0001
2 2 0010
3 3 0011
4 4 0100
5 5 0101
6 6 0110
7 7 0111
8 8 1000
9 9 1001
A 10 1010
B 11 1011
C 12 1100
D 13 1101
E 14 1110
F 15 1111

Conversion of Numbering System

Decimal

Hexadecimal Binary

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1. Decimal to Binary - (base 10 to base 2)

• Division – Remainder Method

(Divide by 2, find the remainder) (2) Convert 15610 to binary
2 156
(1) Convert 2310 to binary
2 78 0

2 23 2 39 0
2 11 1 2 19 1
2 91

251 2 41

221 2 20

2 1 0 Ans : 101112 2 10

0 1 *Read the remainder from below 01
Ans : 100111002
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1. Decimal to Binary - (base 10 to base 2)

• Division – Remainder Method
(Divide by 2, find the remainder)
Try this: Convert 7610 to binary

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2. Binary to Decimal - (base 2 to base 10)

• Place Value Method

e.g. Convert 111012 to decimal

Place value Step 1 1 1 1 0 1

X XX XX

Step 2 16 8 4 2 1

(24 ) (23 ) (22) (21) (20)

Step 3 16 + 8 + 4 + 0 + 1

= 2910 13

2. Binary to Decimal - (base 2 to base 10)

• Place Value Method
Try this: Convert 1100112 to decimal
Step 1
Step 2
Step 3

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Try the questions given.

a) Show the conversion of Binary number given to decimal number:
1) 00111001
2) 11001
3) 1111000
4) 11000111

b) Convert the given numbers below to binary number:
1) 5610
2) 3210
3) 2610
4) 1710

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3. Decimal to Hexadecimal - (base 10 to base 16)

Hexadecimal Decimal

Symbol Equivalent

• Division – Remainder Method 00
11

(Divide by 16, find the remainder) 22
33

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e.g. Convert 7710 to hex 55
66
77

88

16 77 99
A 10

B 11
12
16 4 13 = D Read the remainder as C

04 hex number D 13
E 14
= 4D16 F 15

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3. Decimal to Hexadecimal - (base 10 to base 16)

• Division – Remainder Method
(Divide by 16, find the remainder)
Try this: Convert 9110 to hex

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4. Hexadecimal To Decimal - (base 16 to base 10)

• Place Value Method
e.g. Convert 4D16 to decimal

Step 1 4 D Decimal equivalent
Step 2 Place value
Step 3 4 13

X X

16 1
(161) (160)

Step 4 64 + 13 = 7710

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4. Hexadecimal To Decimal - (base 16 to base 10)

• Place Value Method

Try this: Convert AF1016 to decimal
Step 1
Step 2
Step 3
Step 4

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Try the questions given.

a) Express the following numbers in hexadecimal numbers.
1) 2910
2) 2410
3) 3810
4) 6010

b) Convert the given numbers below to decimal numbers.
1) 4AF16
2) 10A 16
3) 11BC16

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5. Hexadecimal to Binary - (base 16 to base 2)

• Step 1: Convert Hexadecimal to Decimal
Place Value Method

• Step 2: Convert Decimal to Binary
Division – Remainder Method
(Divide by 2, find the remainder)

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5. Hexadecimal to Binary - (base 16 to base 2)

Step 1: Place Value Method

e.g. Convert F116 to decimal

Step 1 F 1 Decimal equivalent
Step 2 Place Value
Step 3 15 1
22
Step 4 X X

16 1
(161) (160)

240 + 1

= 24110

5. Hexadecimal to Binary - (base 16 to base 2)

Step 2: Division – Remainder Method 2 241 1
(Divide by 2, find the remainder) 2 120 0
2 60 0
e.g. Convert 24110 to binary 2 30 0
2 15 1
*Read the remainder from below 27 1
23 1
= 111100012 21 1

0 23

5. Hexadecimal to Binary - (base 16 to base 2)

Try this: Convert 1E16 to binary
1. Step 1 ?
2. Step 2 ?

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5. Hexadecimal to Binary - (base 16 to base 2)

• 1 Hex digit = 4 Bits Simplified Method

e.g. Convert 9AF16 to binary

Break up each digit Step 1 9 A F
Decimal equivalent
Step 2 9 10 15
Place value
Group all digits Step 3 8 4 2 1 8 4 2 1 8 4 2 1
23222120 23222120 23222120

Step 4 1 0 0 1 1 0 1 0 1 1 1 1

= 1001101011112 25

6. Binary to Hexadecimal - (base 2 to base 16)

• Step 1: Convert Binary to Decimal
Place value method

• Step 2: Convert Decimal to Hex
Division – Remainder Method
(Divide by 16, find the remainder)

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6. Binary to Hexadecimal - (base 2 to base 16)

Step 1: Place Value Method
e.g. Convert 101012 to decimal

Place Value Step 1 1 0 1 0 1

X X X X X

Step 2 16 8 4 2 1
(24 ) (23 ) (22) (21) (20)

Step 3 16 + 0+ 4+ 0+ 1

= 2110

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6. Binary to Hexadecimal - (base 2 to base 16)

Step 2: Division – Remainder Method
(Divide by 16, find the remainder)

e.g. Convert 2110 to hex

16 21 5 Read the remainder as
16 1 1 hex number

0 = 1516

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6. Binary to Hexadecimal - (base 2 to base 16)

Try this: Convert 111010012 to hex
1. Step 1 ?
2. Step 2 ?

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6. Binary to Hexadecimal - (base 2 to base 16)

• 4 Bits = 1 Hex digit Simplified Method

e.g. Convert 10101111012 to hex

Group to 4 bits Step 1 0 0 1 0 1 0 1 1 1101
(R to L)
8421
Place value Step 2 8 4 2 1 8 4 2 1 23222120
Decimal equivalent 23222120 23222120
13
Hex equivalent Step 3 2 11
D
Step 4 2 B

= 2BD16

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Try the questions given.

a. Convert the following hexadecimal numbers to binary numbers :
1) 11A16
2) 1CB16
3) F616
4) DB16

b. Convert the following binary numbers to hexadecimal numbers.
1) 110110012
2) 10111002
3) 1110010002

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TO BASE

FROM 10 16
BASE (Decimal) (Hexadecimal)

2 Place Value Method Gr4obuiptst(o4 Simplify Method
Bits = 1 Hex digit)
(Binary) Step 1 1 1 1 0 1
Step 2 16 8 4 2 1 (R to L)
Place value
16x1 8x1 4x1 2x0 1x1 Step 1 0 0 1 0 1 0 1 1 1 1 0 1
(24 ) (23 ) (22) (21) (20)
Step 3 16 + 8 + 4 + 0 + 1 Step 2 8 4 2 1 8 4 2 1 8 4 2 1
23222120 23222120 23222120

Step 3 2 11 13

Step 4 2 B D

= 2910 Decimal Hex equivalent
equivalent
Place value

= 2BD16

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TO BASE

FROM 2 10
BASE (Binary) (Decimal)

16 Simplify Method Place Value Method
(Hexa (1 Hex digit = 4 Bits)
decimal)

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TO BASE

FROM 2 16
BASE (Binary) (Hexadecimal)

10 1. Division – Remainder Method 1. Division – Remainder Method

(Decimal) (Divide by 2, find the remainder) (Divide by 16, find the remainder)

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