محاضرة عن تمثيل الأعداد السالبة في الآلة - 1's complement - 2's complement

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
‫ج‬
5
‫الموجبة‬ ‫األعداد‬ ‫مثيل‬

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; 127]
[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

28. Coding
‫اصطالح‬ ‫ترميز‬

29. ‫المعلومات‬ ‫ترميز‬
‫وتمثيلها‬
‫ج‬
5
‫السالبة‬ ‫األعداد‬ ‫تمثيل‬

30. Negative numbers
‫السالبة‬ ‫األعداد‬

31. Negative numbers
Negative numbers ●
‫السالبة‬ ‫األعداد‬
–
‫بإشارة‬ ‫قيمة‬
–
‫الواحد‬ ‫إلى‬ ‫متمم‬
–
‫إلى‬ ‫متتم‬
2
Signed value
1’s complement
2’s Complement

32. Negative numbers/signed value
+4
-4
Sign bit: 0 => +
1 => -
‫واإلشارة‬ ‫بالقيمة‬ ‫تمثيل‬
0 1 0 0
1 1 0 0

33. Exercise
Represent the following numbers
on 5 bits in signed value
‫على‬ ‫اآلتية‬ ‫األعداد‬ ‫مثل‬
5
‫بتات‬
‫واإلشارة‬ ‫بالقيمة‬
3
,
2
,
1
,
0
- ,
1
- ,
2
- ,
3

34. Exercise
5-bit signed value
3: 00011
2: 00010
1: 00001
0: 00000
-1 : 10001
-2 : 10010
-3 : 10011

35. Exercise
●
What does the following number
represent?
●
on 5 bits in signed value
●
‫على‬ ‫العدد‬ ‫هذا‬ ‫يمثل‬ ‫ماذا‬
5
‫بتات‬
●
‫اإلشارة‬ ‫ذات‬ ‫بالقيمة‬
●
10000

36. Solution
The 5-bit number in signed value
Sign bit -0
1 0 0 0 0

37. Note
This representation is also said
Absolute value
‫المطلقة‬ ‫القيمة‬ ‫تمثيل‬ ‫أيضا‬ ‫تسمى‬ ‫الطريقة‬ ‫هذه‬

38. Arithmetic in SV
SV on 4 bits -1 + 2 = -3
1 0 0 1 -1
0 0 1 0 + 2
1 0 1 1 -3

39. Disadvantages ‫عيوب‬

Two representations for zero

+0, -0

‫للصفر‬ ‫تمثيالن‬

Calculation problem

‫الحساب‬ ‫في‬ ‫مشاكل‬

40. Negative numbers
Negative numbers ●
‫السالبة‬ ‫األعداد‬
–
‫بإشارة‬ ‫قيمة‬
–
‫الواحد‬ ‫إلى‬ ‫متمم‬
–
‫إلى‬ ‫متتم‬
2
Signed value
1’s complement
2’s Complement

41. ‫إلى‬ ‫المتمم‬
Complement to 1
+
4
-
4
Sign bit: 0 => +
1 => -
Reverse bits
‫بت‬ ‫كل‬ ‫نقلب‬
0 1 0 0
1 0 1 1

42. Exercise
Represent the following numbers
on 5 bits in 1's complement
‫على‬ ‫واحد‬ ‫إلى‬ ‫بالمتمم‬ ‫األعداد‬ ‫مثل‬
5
‫بتات‬
3
,
2
,
1
,
0
- ,
1
- ,
2
- ,
3

43. Exercise
3:00011
2:00010
1:00001
0:00000
-1: 1 1110
-2: 1 1101
-3: 1 1100

44. Exercise
●
What does the following number
represent?
●
on 5 bits in C1
●
‫الممثالن‬ ‫العددان‬ ‫هذان‬ ‫يمثل‬ ‫ماذا‬
●
‫إلى‬ ‫بالمتتم‬
1
‫على‬
5
‫بتات‬
●
11111
●
00000
●

45. Solution
The next 5-bit number
in complement to 1
Sign bit -0
1 1 1 1 1

46. Arithmetic in C1
‫واحد‬ ‫إلى‬ ‫المتمم‬ ‫في‬ ‫الحساب‬
C1 on 4 bits -1 + 2 = 0
1
1 1
1 1 0 -1
1 0 + 2
1 0 0 0 0 -3

47. Disadvantages

Two representations for zero

+0, -0

‫للصفر‬ ‫تمثيالن‬

Calculation problem

‫الحساب‬ ‫في‬ ‫مشاكل‬

48. Exercise

Calculate in 1's complement over 5 bits

‫على‬ ‫واحد‬ ‫إلى‬ ‫المتمم‬ ‫في‬ ‫احسب‬
5
‫بتات‬

(-2) +4 =?

how to correct the result

‫النتيجة‬ ‫نصحح‬ ‫كيف‬

49. Exercise
●
●
1
1 1
1 1 0 1 -2
1 0 0 + 4
1 0 0 0 0 1 1

50. Exercise


1
1 1
1 1 0 1 -2
1 0 0 + 4
0 0 0 0 1 1
1 +
0 0 0 1 0

51. Negative numbers
Negative numbers ●
‫السالبة‬ ‫األعداد‬
–
‫بإشارة‬ ‫قيمة‬
–
‫الواحد‬ ‫إلى‬ ‫متمم‬
–
‫إلى‬ ‫متتم‬
2
Signed value
1’s complement
2’s Complement

52. Complement to 2 ‫إلى‬ ‫المتمم‬
+
5
-
5
Sign bit: 0 => +
1 => -
Reverse bits and add 1
‫إضافة‬ ‫ثم‬ ‫البتات‬ ‫قلب‬
1
0 1 0 1
1 0 1 0
1 0 1 1

53. Exercise
Represent the following numbers
on 5 bits in 2's complement
‫إلى‬ ‫المتمم‬ ‫في‬ ‫مثل‬
2
‫على‬
5
‫بتات‬
3
,
2
,
1
,
0
- ,
1
- ,
2
- ,
3

54. Exercise
3: 00011
2: 00010
1: 00001
0: 00000
-1: 11111
-2: 11110
-3: 11101

55. Exercise
●
What does the following number
represent ?
●
on 5 bits in C2
●
‫إلى‬ ‫المتمم‬ ‫في‬ ‫ـة‬
‫ل‬‫ممث‬ ‫ـداد‬
‫ع‬‫األ‬ ‫ـل‬
‫ث‬‫تم‬ ‫ماذا‬
2
●
‫على‬
5
‫بتات‬
11111
00000

56. Solution
1 1 1 1 1

57. Solution
The following 5-bit number
in 2's complement
11111-1 = 11110 (reversed)
- 00001
1 1 1 1 1

58. Arithmetic in C2
●
●
1
1 1
1 1 1 0 -2
1 0 0 + 4
1 0 0 0 1 0 2

59. exercise
Calculate in C2 on 6 bits
11110 - 1010

60. exercise
Calculate in C2 on 6 bits
11110 - 1010

61. Intervals

An 8-bit interval can represent

Unsigned value [0; 255]

Signed value [-127; 127]

1's complement [-127, 127]

2's complement [-128; 127]

62. Intervals

n-bit interval can represent

Unsigned value [0; 2 n
-1]

Signed value [-(2 n-1
-1); 2 n-1
-1]

1's complement [-(2 n-1
-1); 2 n-1
-1]

2's complement [-2 n-1
; 2 n-1
-1]

63. Exercise

Give the intervals in Signed Value and C2
for

16 bit

64. Exercise

Give the intervals in Signed Value and C2
for

16 bits = [-(2 15
-1); 2 15
-1]

SV, C1 = [-32,767; 32,767]

C2 = [-32,768; 32,767]

65. Overflow