Signed Integer Representation
We need to represent both positive and negative
numbers. Three systems are used for representing such numbers:
Signed magnitude
1’s complement
2’s complement
Signed Magnitude
A signed-magnitude number has a sign as its
left-most bit (also referred to as the high-order bit or the most significant
bit) while the remaining bits represent the magnitude (or absolute value) of
the numeric value.
Example 3.1:
In an
8-bit word, –1 would be represented as 10000001, and +1 as 00000001.
Here is the table
of integer representation:
What is overflow?
v Overflow can only occur when adding two numbers that have the same
sign.
v The carry-out signal from the sign-bit
position is not a sufficient indicator of overflow when adding signed numbers.
Binary Addition
Arithmetic calculations on integers that are
represented using this notation
(1) If the signs are the same, add the
magnitudes and use that same sign for the result;
(2) If the signs differ, you must determine
which operand has the larger magnitude. The sign of the result is the same as
the sign of the operand with the larger magnitude, and the magnitude must be
obtained by subtracting (not adding) the smaller one from the larger one.
Table 3.1
Example 3.2:
Question
Add
010011112 to
001000112 using
signed-magnitude arithmetic
Answer:
If there
is a carry here, we say that we have an overflow condition and the carry is
discarded, resulting in an incorrect sum. There is no overflow in this example.
In signed magnitude, the sign bit is used only
for the sign, so we can't "carry into" it. If there is a carry
emitting from the seventh bit, our result will be truncated as the seventh bit
overflows, giving an incorrect sum.
Example 3.3:
Question
Add
010011112 to
011000112 using
signed-magnitude arithmetic.
We obtain
the erroneous result of 79 + 99 = 50.
Binary Subtraction
As with addition,
signed-magnitude subtraction is carried out in a manner similar to pencil and
paper decimal arithmetic, where it is sometimes necessary to borrow from digits
in the minuend.
Example 3.4:
Question
Subtract
010011112 from
011000112 using
signed-magnitude arithmetic
We find
011000112 –
010011112 =
000101002 in
signed-magnitude representation.
If two numbers
have different signs?
Example 3.5:
Question
Add
100100112 (–19)
to 000011012
(+13) using signed-magnitude arithmetic.
Complement Representation of Numbers
The 1’s complement of a binary number can be obtained by changing
all 1s to 0s and all 0s to 1s. The 2’s complement of a binary number can be obtained by adding 1 to
its 1’s complement.
Example 3.6:
1.
2. Two’s complement addition
Decimal
value: 9
0000 0101 = +5
Binary value: 01001 + 1111 1101 = -3
1’s complement: 10110 _______________
2’s complement: 10111 0000 0010 = +2
Decimal value: -9
3. Two’s
complement subtraction
0000 0111 = +7
+1111 0100 = -12
______________
1111 1011 = -5
Here a link: http://youtu.be/HAoxqOFtJJY
Hexadecimal Addition
Example 3.7:
(Use the color code in each step to see what’s
happening)
The problem:
You are to add these numbers:
|
||||||
A
|
C
|
5
|
A
|
9
|
||
E
|
D
|
6
|
9
|
4
|
||
Carry
Over:
|
||||||
1. Add one column at a time
2. Convert to decimal & add (9 + 4 = 13)
3. Follow less than 16 rule
Decimal
13 is hexadecimal D
|
A
|
C
|
5
|
A
|
9
|
|
E
|
D
|
6
|
9
|
4
|
||
D
|
Carry
Over:
|
1
|
|||||
1. Add
next column
2. Convert
to decimal & add (10 + 9 = 19)
3. Follow
16 or larger than 16 rule
(19
– 16 = 3
carry a 1)
|
A
|
C
|
5
|
A
|
9
|
|
E
|
D
|
6
|
9
|
4
|
||
3
|
D
|
Carry
Over:
|
1
|
|||||
1. Add
next column
2. Convert
to decimal & add (1 + 5 + 6 = 12)
3. Follow
less than 16 rule, convert to hex
Decimal
12 is hexadecimal C
|
A
|
C
|
5
|
A
|
9
|
|
E
|
D
|
6
|
9
|
4
|
||
C
|
3
|
D
|
Carry
Over:
|
1
|
|||||
1. Add
next column
2. Convert
to decimal & add (12 + 13 = 25)
3. Follow
16 or larger than 16 rule
(25 – 16 = 9 carry a 1)
|
A
|
C
|
5
|
A
|
9
|
|
E
|
D
|
6
|
9
|
4
|
||
9
|
C
|
3
|
D
|
Carry
Over:
|
1
|
||||||
1. Add
next column
2. Convert
and add (1
+ 10
+ 11
= 22)
3. Follow
16 or larger than 16 rule
(22 – 16 = 6 carry a 1)
|
A
|
C
|
5
|
A
|
9
|
||
B
|
D
|
6
|
9
|
4
|
|||
6
|
9
|
C
|
3
|
D
|
Carry
Over:
|
1
|
||||||
1. Add
next column
2. Convert
and add (1
+ 0 + 0 = 1)
3. Follow
less than 16 rule
|
0
|
A
|
C
|
5
|
A
|
9
|
|
0
|
B
|
D
|
6
|
9
|
4
|
||
1
|
6
|
9
|
C
|
3
|
D
|
Hexadecimal Subtraction
Example 3.8:
Here a link: http://youtu.be/OM0exghFNAk
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