Describe numbering system in binary, decimal, octal, hexadecimal and its conversion. Other presentation canbe found at http://part66.blogspot.com

1. 5.2 NUMBERING SYSTEMS

2. Many number systems are in use in digital
technology.
The most common are :
• Decimal
• Binary
• Octal
• Hexadecimal

3. 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.

4. DECIMAL EXAMPLE
• 3.1410
• 53210
• 1082410
• 64900010

5. 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

6. BINARY EXAMPLE
• 1110
• 1011110
• 1111011100
• 10000101111011

7. OCTAL SYSTEM
• The octal number system has a base of eight
• Eight possible digits: 0,1,2,3,4,5,6,7

8. OCTAL EXAMPLE
• Octal Example
• 5410
• 765421
• 1047664
• 4123170137

9. 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

10. HEXADECIMAL EXAMPLE
• BD
• 452EA
• E451B2CD3
• 35412BABE

11. NUMBERING CONVERSION
OCTAL
DECIMAL BINARY
HEXADECIMAL

12. DECIMAL TO BINARY CONVERSION
Reverse of Binary-To-Decimal Method :
20 21 22 23 24 25 26 27 28 29
1 2 4 8 16 32 64 128 256 512
• 2710 = 16+8+0+2+1
= 11011
• 18110 = 128+0+32+16+0+4+0+1
= 10110101

13. DECIMAL TO BINARY CONVERSION
Repeat Division Method :
EG : 18110
181/2 = 90 balance 1
EG : 2710
90/2 = 45 balance 0
27/2 = 13 balance 1
45/2 = 22 balance 1
13/2 = 6 balance 1
22/2 = 11 balance 0
6/2 = 3 balance 0
11/2 = 5 balance 1
3/2 = 1 balance 1
5/2 = 2 balance 1
1/2 = 0 balance 1
2/2 = 1 balance 0
1/2 = 0 balance 1
Result : 2710= 110112
Result : 18110=
101101012

14. DECIMAL TO OCTAL CONVERSION
Ex : 17710 Ex : 398510
177/8 = 22 balance 1 3985/8 = 498 balance 1
22/8 = 2 balance 6 498/8 = 62 balance 2
2/8 = 0 balance 2 62/8 = 7 balance 6
Result 17710 = 2618 7/8 = 0 balance 7
Result 398510 = 76218

15. DECIMAL TO HEXADECIMAL
Ex : 37810
378/16 = 23 balance 10 = (A)
23/16 = 1 balance 7
1/16 = 0 balance 1
Result 37810 = 17A16

16. 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

17. BINARY TO DECIMAL CONVERSION
20 21 22 23 24 25 26 27 28 29
1 2 4 8 16 32 64 128 256 512
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

18. 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

19. BINARY TO HEXADECIMAL
0 0000
1 0001
2 0010
3 0011
4 0100 EXAMPLE :
5 0101
6 0110
7 0111 101 11012 = (101) (1101)2 = 5 D16
8 1000
9 1001
A 1010 11 1001 10112 = (11) (1001) (1011)2 = 3 9 B16
B 1011
C 1100
D 1101 1011 0010 11112 = (1011) (0010) (1111)2 = B 2 F16
E 1110
F 1111

20. OCTAL TO DECIMAL CONVERSION
• Example :
• 2378 = 2(82)+ 3(81)+ 2(80) = 15910
• 95348 = 9(83)+ 5(82)+ 3(81)+ 4(80) = 495610

21. OCTAL TO BINARY CONVERSION
0 1 2 3 4 5 6 7
000 001 010 011 100 101 110 111
• Example:
• 4 7 28 = (100) (111) (010)2 = 100 111 0102
• 1 5 28 = (001) (101) (010)2 = 1 101 0102

22. HEXADECIMAL TO DECIMAL
• Example :
• 2E16 = 2(161) + 14 (160) = 4610
• 9BC316 = 9(163) + 11 (162) +12 (161) +3 (160) =
3987510

23. HEXADECIMAL TO BINARY
0 0000
1 0001
2 0010
3 0011 • 5 D16 = (101) (1101)2 =101 11012
4 0100
5 0101
6 0110 • 3 9 B16 = (11) (1001) (1011)2 =11 1001 10112
7 0111
8 1000
9 1001 • B 2 F16 = (1011) (0010) (1111)2 =1011 0010 11112
A 1010
B 1011
C 1100
D 1101
E 1110
F 1111

24. NUMBERING CONVERSION
OCTAL
Table (div 3 )
Table (div 3)
X/2
DECIMAL BINARY
(+2 x )
Table (div 4)
Table (div 4)
HEXADECIMAL

25. CONVERSION VALUE
Binary - Hexa
Power 2
0 0000
20 1 1 0001 Binary - Octal
Power 8
21 2 2 0010 0 000
22 4 80 1 3 0011 1 001
23 8 81 8 4 0100
2 010
24 16 82 64 5 0101
6 0110 3 011
25 32 83 512
7 0111 4 100
26 64 84 4096
8 1000 5 101
27 128 85 32768 9 1001
6 110
28 256 86 262144 A 1010
7 111
29 512 87 2097152 B 1011
210 1024 C 1100
D 1101
E 1110
F 1111