1.6 signed binary numbers
DESCRIPTION
1.6 Signed Binary Numbers. 1.6 Signed Binary Numbers. 1 - Sign and Magnitude representation 2 - 1’s Complement Representation 3 - 2’s Complement Representation. Notes. 1 - The previous representation are the same for positive numbers and different for negative numbers. - PowerPoint PPT PresentationTRANSCRIPT
1.6 Signed Binary Numbers
1.6 Signed Binary Numbers
1 - Sign and Magnitude representation
2 - 1’s Complement Representation
3 - 2’s Complement Representation
1 - The previous representation are the same for positive numbers and different for negative numbers
2 - For a signed binary number the most significant bit is used for representing the sign of the number
We use 0 for positive numbers and 1 for negative numbers
Notes
Example : represent +76
10 2
10 2
10 2
76 1001100 &
76 1001100 1'
0
76 1
0
0 001100 2 '
Sign Magnitude
s Complement
s Complement
Representing negative numbers in the previous three systems
1’s Complement of a negative number can be obtained by flipping all bits of the positive binary number
2’s Complement of a negative number can be obtained by adding 1 to the 1’s Complement or by flipping bits of the positive binary number after the first one from the right
Example : represent -76
10 2
10 2
10 2
76 1001100 &
76 0110011 1'
1
76 0
1
1 110100 2 '
Sign Magnitude
s Complement
s Complement
Arithmetic Addition with Comparison
The addition of two numbers in the signed mgnitude system follow the rules of ordinary arithmetic.
If the signed are the same, we add the two magnitudes and give the sum the common sign.
If the signed are different, we subtract the smaller magnitudefrom the larger and give the difference the sign of the largermagnitude. EX. (+25) + (-38) = -(38 - 25) = -13
Arithmetic Addition
Arithmetic Addition without Comparison
The addition of two signed binary number with negative numbers represented in signed 2’s complement form is obtained from the addition of the two numbers, including their signed bits. A carry out of the signed bit position is discarded (note that the 4th case).
Arithmetic Addition without Comparison
19 1110110113 1111001106 11111010
07 1111100113 1111001106 00000110
07 00000111
13 0000110106 11111010
19 00010011
13 0000110106 00000110
6
Arithmetic Subtraction
(+/-) A – (+B)= (+/-) A + (-B) (+/-) A – (-B)= (+/-) A + (+B)
Example (-6) – (-13)= +7In binary: (1111010 – 11110011)= (1111010 +
00001101) =100000111 after removing the carry out the result will be : 00000111
1.7 Binary Codes
Binary Coded Decimal (BCD)
Binary Coded Decimal (BCD)
in this system each digit is represented in 4 bits
For example : to represent in BCD
9 54
1001
10945
0100 0101
10 BCD10010100010 9 5 14
BCD Addition
Example : Evaluate the following operations in BCD System
1 – 3 + 4
2 – 4 + 8
3 - 148 + 576
3 4
7
01000111
0011BCD BCD Decimal
BCD Addition
Example : Evaluate the following operations in BCD System
1 – 3 + 4
2 – 4 + 8
3 - 148 + 576
4
8
12
10001100
0100BCD BCD
Decimal
Error
01100001001012
We must add 6 (0110) to the result
BCD
BCD Addition
Example : Evaluate the following operations in BCD System
1 – 3 + 4
2 – 4 + 8
3 - 184 + 576
1846 57
1
0001BCD 1000 0100
0101 0111 0110
Decimal
0111 0000 1010
0110 0110
0111 0110 00001 1
1
760 1
760
1 – In BCD Addition , we add (0110)=(6) if the result value was greater than (1001)=(9) or if the result was more than 4 digits
Notes
In previous Example we added 0110 when the result was
1 - greater than 9 (1001)
2 - more than 4 digits (10000)
Note : result more than 4 digit is greater than 9(1001)
Decimal Arithmetic
Addition for signed numbers
Example: (+375) + (- 240) = + 135 in BCD
• Apply 10‘s complement to the negative number only.• Addition is done by summing all digits,including the sign
digit,and discarding the end carry 0 375 +9 760 ------------ 0 135
Decimal Arithmetic
• Subtraction for signed and unsigned numbers
• Apply 10‘s complement to the subtrahend and apply addition (same as binary case)
(ex-3) is like (BCD) in the way of representing number
i.e. each digit is represented in 4 bits
Except that : each digit is firstly incremented by three
Excess-3 (ex-3)Excess-three (ex-3)is another system to represent a number
For example : to represent in ex-3
12 87
1100
10945
0111 1000
10 311000111100 94 05
ex
9 54
Gray Code
ASCII code is used to represent characters , Symbols , …
ASCII code consists of 7-bits (to represent 128 character)
ASCII character code
ASCII : American Standard Code for Information Interchange
Upper case Letters are represented by ASCII (65 : 90)
Lower case Letters are represented by ASCII (97 : 122)
# ASCII Ch
65 1000001 A
66 1000010 B
90 1011010 Z
97 1100001 a
98 1100010 b
122 1111001 z
Error Detecting Code
with even parity with odd parityASCII A 1000001 01000001 11000001ASCII T 1010100 11010100 01010100
For more information about Number Systems and Conversations between them
Check these
1 – Our Logic Book
2 - Computer Organization's Lectures
3 – Any other References