This document discusses different number systems used in computer architecture including binary, octal, decimal, and hexadecimal. It provides examples of converting between these number systems, such as converting the decimal number 47 to binary (101111), octal (57), and hexadecimal (2F). Conversion charts and formulas are provided for changing between the number systems.

1. Digital Logic Design
NUMBER SYSTEMS
1
2
Prepared by: Mir Omranudin Abhar
Email : MirOmran@Gmail.com
Fall ,2019

2. Number systems
Number systems are the technique to represent numbers in the computer
system architecture, every value that you are saving or getting into/from
computer memory has a defined number system.
2

3. Number systems
Computer architecture supports following number systems.
1. Binary number system
2. Octal number system
3. Decimal number system
4. Hexadecimal (hex) number system
3

4. Number systems [Binary ]
A Binary number system has only two digits that are 0 and 1. Every number
(value) represents with 0 and 1 in this number system. The base of binary
number system is 2, because it has only two digits
4
0 , 1
1 ≥ ≥ 𝟎

5. Number systems [Octal]
1. Octal number system has only eight (8) digits from 0 to 7.
2. The base of octal number system is 8, because it has only 8 digits.
5
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7
7 ≥ ≥ 𝟎

6. Number systems [Decimal]
1. Decimal number system has only ten (10) digits from 0 to 9.
2. The base of decimal number system is 10, because it has only 10 digits.
6
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9
9 ≥ ≥ 𝟎

7. Number systems [Hexadecimal]
1. A Hexadecimal number system has sixteen (16) alphanumeric values
from 0 to 9 and A to F.
2. The base of hexadecimal number system is 16, because it has 16
alphanumeric values. Here A is 10, B is 11, C is 12, D is 13, E is
14 and F is 15.
7
0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , A , B , C , D , E , F
F ≥ ≥ 𝟎

8. 8
Conversion
(𝑛)10 → (𝑚)2
(𝑛)10 → (𝑚)8
(𝑛)10 → (𝑚)16
1

9. Convert Decimal to [ Binary ]
9
(47)10 = (101111)2

10. Convert Decimal to [ Octal ]
10
(47)10 = (57)8

11. Convert Decimal to [hexadecimal ]
11
(47)10 = (2𝐹)16

12. 12
Conversion
(𝑛)2 → (𝑚)10
(𝑛)2 → (𝑚)8
(𝑛)2 → (𝑚)16
2

13. Convert Binary to [Decimal ]
13
(101111)2 = (47)10
101111 2 ⇒ 1 ∗ 25
+ 0 ∗ 24
+ 1 ∗ 23
+ 1 ∗ 22
+ 1 ∗ 21
+ 1 ∗ 20
⇒ 32 + 0 + 8 + 4 + 2 + 1
⇒ (47)10

14. Convert Binary to [Octal ]
14
(101111)2 = (57)8
101111 2 ⇒ 1 ∗ 25 + 0 ∗ 24 + 1 ∗ 23 + 1 ∗ 22 + 1 ∗ 21 + 1 ∗ 20
⇒ 32 + 0 + 8 + 4 + 2 + 1
⇒ (47)10

15. Convert Binary to [hexadecimal]
15
(101111)2 = (2𝐹)16
101111 2 ⇒ 1 ∗ 25 + 0 ∗ 24 + 1 ∗ 23 + 1 ∗ 22 + 1 ∗ 21 + 1 ∗ 20
⇒ 32 + 0 + 8 + 4 + 2 + 1
⇒ (47)10

16. 16
Conversion
(𝑛)8 → (𝑚)10
(𝑛)8 → (𝑚)2
(𝑛)8 → (𝑚)16
3

17. Convert Octal to [Decimal]
17
(57)8 = (47)10
57 8 ⇒ 5 ∗ 81
+ 7 ∗ 80
⇒ 40 + 7
⇒ (47)10

18. Convert Octal to [Binary]
18
(57)8 = (101111)2
57 8 ⇒ 5 ∗ 81 + 7 ∗ 80
⇒ 40 + 7
⇒ (47)10

19. Convert Octal to [hexadecimal]
19
(57)8 = (2𝐹)16
57 8 ⇒ 5 ∗ 81 + 7 ∗ 80
⇒ 40 + 7
⇒ (47)10

20. 20
Conversion
(𝑛)16 → (𝑚)10
(𝑛)16 → (𝑚)2
(𝑛)16 → (𝑚)8
4

21. Convert hexadecimal to [decimal]
21
(2𝐹)16 = (47)10
2𝐹 16 ⇒ 2 ∗ 161 + 𝐹 ∗ 160
⇒ 32 + 15
⇒ (47)10

22. Convert hexadecimal to [decimal]
22
(2𝐹)16 = (57)8
2𝐹 16 ⇒ 2 ∗ 161 + 𝐹 ∗ 160
⇒ 32 + 15
⇒ (47)10

23. Convert hexadecimal to [binary]
23
(2𝐹)16 = (101111)2
2𝐹 16 ⇒ 2 ∗ 161 + 𝐹 ∗ 160
⇒ 32 + 15
⇒ (47)10

24. 24
Exercise
(2𝐴𝐵)16 → (? )10
(528)10 → (? )8
(110001)2 → (? )10
(𝐴0)16 → (? )8
(621)8 → (? )16
(101)10 → (? )2

25. Question
.1
VHDL
‫ست؟‬‫چی‬
.2
‫عدد‬
12.31
‫ا‬‫ر‬
‫از‬
‫مت‬‫س‬‫سی‬
‫اعداد‬
Decimal
‫به‬
‫مت‬‫س‬‫سی‬
‫اعداد‬
Binary
‫بدیل‬‫ت‬
‫مناید‬
.
25