EASA Part 66 Module 5.2 : Numbering System

5.2 NUMBERING SYSTEMS

http://part66.blogspot.com/

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 113/2 = 6 balance 16/2 = 3 balance 03/2 = 1 balance 11/2 = 0 balance 1

Result : 2710= 110112

EG : 18110

181/2 = 90 balance 190/2 = 45 balance 045/2 = 22 balance 122/2 = 11 balance 011/2 = 5 balance 15/2 = 2 balance 12/2 = 1 balance 01/2 = 0 balance 1

Result : 18110= 101101012

DECIMAL TO OCTAL CONVERSION

Ex : 17710

177/8 = 22 balance 122/8 = 2 balance 62/8 = 0 balance 2Result 17710 = 2618

Ex : 398510

3985/8 = 498 balance 1498/8 = 62 balance 262/8 = 7 balance 67/8 = 0 balance 7Result 398510 = 76218

DECIMAL TO HEXADECIMAL

Ex : 37810

378/16 = 23 balance 10 = (A)23/16 = 1 balance 71/16 = 0 balance 1Result 37810 = 17A16

DECIMAL TO HEXADECIMAL

Ex : 694210

6942/16 = 433 balance 14 = (E)433/16 = 27 balance 127/16 = 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 HEXADECIMAL0 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001A 1010B 1011C 1100D 1101E 1110F 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(83)+ 5(82)+ 3(81)+ 4(80) = 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 00001 00012 00103 00114 01005 01016 01107 01118 10009 1001A 1010B 1011C 1100D 1101E 1110F 1111

NUMBERING CONVERSION

DECIMAL BINARY

HEXADECIMAL

OCTAL

X/8

X/16

X/2

Y(8 x )

Y(16 x )

(+2 x )

Tabl

e (d

iv 3

) Table (div 3)

Tabl

e (d

iv 4

) Table (div 4)

CONVERSION VALUE

20 121 222 423 824 1625 3226 6427 12828 25629 512210 1024

80 181 882 6483 51284 409685 3276886 26214487 2097152

0 0001 0012 0103 0114 1005 1016 1107 111

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

Power 2

Power 8

Binary - Hexa

Binary - Octal