تمثيل الأعداد الموجبة

1. Coding and representation
of information
Dr. Taha Zerrouki
t.zerrouki@univ-bouira.dz
MI, semester 1
University of Bouira

2. •Program
• Number systems
•
‫التعداد‬ ‫أنظمة‬ ‫في‬ ‫الحساب‬
•
Representation of positive and negative
numbers
•
‫والسالبة‬ ‫الموجبة‬ ‫األعداد‬ ‫تمثيل‬

3. ‫المعلومات‬ ‫ترميز‬
‫وتمثيلها‬
‫جزء‬
5.1
‫الموجبة‬ ‫األعداد‬ ‫تمثيل‬

4. Number representation
‫األعداد‬ ‫تمثيل‬

5. Binary ‫الثنائي‬ ‫النظام‬

7. Bit/ byte(byte)
‫بت‬
‫بايت‬
=
8
‫بت‬
Octet = 8 bits

8. Binary ‫ثنائي‬
Decimal ‫عشري‬
Values in Decimal ‫العشرية‬ ‫النظام‬ ‫في‬ ‫القيم‬
Byte representation ‫البايت‬ ‫تمثيل‬
Number of bits ‫البتات‬ ‫أرقام‬

9. ‫األعداد‬
●
‫الرياضيات‬ ‫في‬
●
‫محدودة‬ ‫غير‬ ‫األعداد‬
●
]-∞; +∞[
●
‫حاسوب‬ ‫حاسبة‬ ‫اآللة‬
) / (

10. Calculator
What is the maximum number that can be
represented on an 8-digit calculator?
‫ذات‬ ‫حاسبة‬ ‫على‬ ‫كتابته‬ ‫يمكن‬ ‫عدد‬ ‫أكبر‬ ‫هو‬ ‫ما‬
8 ‫أرقام؟‬

11. ‫مجال‬
Interval
On an 8-digit calculator we can represent the
interval from [0, to 9999 9999]
‫المجال‬ ‫تمثيل‬ ‫يمكن‬
[0, to 99 999 999]

12. ‫مجال‬ ‫سعة‬
Size of an interval

What is the interval that can be
represented on a calculator with N decimal
digits

‫ذات‬ ‫حاسبة‬ ‫على‬ ‫تمثيله‬ ‫يمكن‬ ‫الذي‬ ‫المجال‬ ‫ما‬
‫ن‬
‫ا؟‬‫ًي‬‫عشر‬ ‫ا‬‫رقًم‬
N

13. Size of an interval
99 = 102
-1

999 = 103
-1

[ 0, 1010
-1]

[ 0, 10n
-1]
99
999
9 999 999 999
Base 10

14. Size of an interval
(77)8 = 82
-1

(777)8 = 83
-1

[ 0, 810
-1]

[ 0, 8n
-1]
77
777
7 777 777 777
Base 8

15. Size of an interval
[ 0, Bn
-1]
Base B
N

16. Size of an interval
(11)2 = 22
-1

(111)2 = 23
-1

[ 0, 28
-1]

[ 0, 2n
-1]
11
111
1111 1111
Base 2

17. Size of an interval Base 2
N bits
The basic unit is the byte
N bits => [ 0, 2n
-1]

18. Size of an interval
[0; 28
-1]
1111 1111
Base 2
[0; 255]
[0; 210
-1]
11 1111 1111
[0; 1023]

19. Exercise
What is the maximum number that can be
represented by
‫بـ‬ ‫تمثيله‬ ‫يمكن‬ ‫عدد‬ ‫أكبر‬ ‫ما‬
16 bit
24 bit
32 bit

20. Size of an interval Base 2
[0; 216
-1]
1111 1111 1111 1111
[0; 65,535]
[0; 224
-1]
1111 1111 1111 1111 1111 1111
[0; 16,777,215]

21. Size of an interval Base 2
[0; 232
-1]
1111 1111 1111 1111 1111 1111 1111 1111
[0; 4,294,967,295]

22. Solution
16 bits => [0; 65,535]
24 bits => [0; 16,777,215]
32 bits => [0; 4,294,967,295]

23. Question?

What does the result of

9999 9999 * 9

On a simple 8 digit calculator?

‫ضرب‬ ‫نتيجة‬ ‫ما‬

9999
9999
*
9

‫ذات‬ ‫بسيطة‬ ‫حاسبة‬ ‫على‬
8
‫أرقام‬

24. Overflow ‫طفح‬

25. 8-bit addition
1000 0000
+0100 0000
-----------------
1100 0000

26. 8-bit addition
Overflow
‫طفح‬
1000 0000
+1000 0000
-----------------
1 0000 0000

27. Overflow