CAR 66 Module 5.2 Numbering Systems

7/31/2019 CAR 66 Module 5.2 Numbering Systems

http://slidepdf.com/reader/full/car-66-module-52-numbering-systems 1/25

5.2 NUMBERING SYSTEMS



Many number systems are in use in digital

technology.

The most common are :• Decimal

• Binary

• Octal• Hexadecimal



DECIMAL SYSTEM

• Composed of 10 numerals or symbols

• Using these symbols as digits of a number, can

express any quantity.

• Called the base-10 system because it has 10

digits.

•

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



DECIMAL EXAMPLE

• 3.1410

• 53210

•

1082410 • 64900010



BINARY SYSTEM

• There are only two symbols or possible digit

values, 0 and 1.

• This base-2 system can be used to represent

any quantity that can be represented in

decimal or other base system



BINARY EXAMPLE

• 1110

• 1011110

•

1111011100• 10000101111011



OCTAL SYSTEM

• The octal number system has a base of eight

• Eight possible digits: 0,1,2,3,4,5,6,7



OCTAL EXAMPLE

• Octal Example

• 5410

•

765421• 1047664

• 4123170137



HEXADECIMAL SYSTEM

• The hexadecimal system uses base 16.

• It uses the digits 0 through 9 plus the letters A,

B, C, D, E, and F as the 16 digit symbols.

• 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F



HEXADECIMAL EXAMPLE

• BD

• 452EA

•

E451B2CD3• 35412BABE



NUMBERING CONVERSION

DECIMAL BINARY

HEXADECIMAL

OCTAL



DECIMAL TO BINARY CONVERSION

Reverse of Binary-To-Decimal Method :

• 2710 = 16+8+0+2+1

= 11011

• 18110 = 128+0+32+16+0+4+0+1= 10110101

20 21 22 23 24 25 26 27 28 29

1 2 4 8 16 32 64 128 256 512



DECIMAL TO BINARY CONVERSION

Repeat Division Method :

EG : 2710

27/2 = 13 balance 1

13/2 = 6 balance 1

6/2 = 3 balance 0

3/2 = 1 balance 1

1/2 = 0 balance 1

Result : 2710= 110112

EG : 18110

181/2 = 90 balance 1

90/2 = 45 balance 0

45/2 = 22 balance 122/2 = 11 balance 0

11/2 = 5 balance 1

5/2 = 2 balance 1

2/2 = 1 balance 01/2 = 0 balance 1

Result : 18110=

101101012



DECIMAL TO OCTAL CONVERSION

Ex : 17710

177/8 = 22 balance 1

22/8 = 2 balance 62/8 = 0 balance 2

Result 17710 = 2618

Ex : 398510

3985/8 = 498 balance 1

498/8 = 62 balance 2

62/8 = 7 balance 6

7/8 = 0 balance 7

Result 398510 = 76218



DECIMAL TO HEXADECIMAL

Ex : 37810

378/16 = 23 balance 10 = (A)

23/16 = 1 balance 7

1/16 = 0 balance 1

Result 37810 = 17A16



DECIMAL TO HEXADECIMAL

Ex : 694210

6942/16 = 433 balance 14 = (E)

433/16 = 27 balance 1

27/16 = 1 balance 11 = (B)

1/16 = 0 balance 1

Result 37810 = 1B1E16



BINARY TO DECIMAL CONVERSION

110112

= 24+23+02+21+20

= 16+8+0+2+1= 2710

101101012

= 27+06+25+24+03+22+01+20

= 128+0+32+16+0+4+0+1

= 18110

20 21 22 23 24 25 26 27 28 29

1 2 4 8 16 32 64 128 256 512



BINARY TO OCTAL CONVERSION

0 1 2 3 4 5 6 7 000 001 010 011 100 101 110 111

• Example:

• 100 111 0102 = (100) (111) (010)2 = 4 7 28

• 1 101 0102 = (001) (101) (010)2 = 1 5 28



BINARY TO HEXADECIMAL

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

9 1001

A 1010

B 1011

C 1100

D 1101

E 1110

F 1111

EXAMPLE :

101 11012 = (101) (1101)2 = 5 D16

11 1001 10112 = (11) (1001) (1011)2 = 3 9 B16

1011 0010 11112 = (1011) (0010) (1111)2 = B 2 F16



OCTAL TO DECIMAL CONVERSION

• Example :

• 2378 = 2(82)+ 3(81)+ 2(80) = 15910

•

95348 = 9(8

3

)+ 5(8

2

)+ 3(8

1

)+ 4(8

0

) = 495610



OCTAL TO BINARY CONVERSION

• Example:

• 4 7 28 = (100) (111) (010)2 = 100 111 0102

• 1 5 28 = (001) (101) (010)2 = 1 101 0102

0 1 2 3 4 5 6 7 000 001 010 011 100 101 110 111



HEXADECIMAL TO DECIMAL

• Example :

• 2E16 = 2(161) + 14 (160) = 4610

• 9BC316 = 9(163) + 11 (162) +12 (161) +3 (160) =

3987510



HEXADECIMAL TO BINARY

• 5 D16 = (101) (1101)2 =101 11012

•

3 9 B16 = (11) (1001) (1011)2 =11 1001 10112

• B 2 F16 = (1011) (0010) (1111)2 =1011 0010 11112

0 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 01107 0111

8 1000

9 1001

A 1010

B 1011

C 1100

D 1101

E 1110

F 1111



NUMBERING CONVERSION

DECIMAL BINARY

HEXADECIMAL

OCTAL

X/2(+2 x )

T a b l e

( d i v

3 ) T

a b l e ( d i v

3 )

T a b l e

( d

i v

4 )

T a b

l e ( d i v

4 )



CONVERSION VALUE

20 1

21 2

22 4

23 8

24 1625 32

26 64

27 128

2

8

25629 512

210 1024

80 1

81 8

82

6483 512

84 4096

85 32768

8

6

26214487 2097152

0 000

1 001

2 010

3 011

4 100

5 101

6 110

7 111

0 00001 0001

2 0010

3 0011

4 0100

5 01016 0110

7 0111

8 1000

9 1001

A 1010B 1011

C 1100

D 1101

E 1110

F 1111

Power 2

Power 8

Binary - Hexa

Binary - Octal

Documents

CAR 66 Module 5.2 Numbering Systems