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IT1004: Data
Representation
and Organization
Negative number representation
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Sign-Magnitude Representation
• This is one method used to represent negative
numbers in Binary
• Here an extra digit is placed in front the existing
binary number to represent the sign
• If this extra digit is a '1', it means that the rest of
the digits represent a negative number
• If the extra digit is a 0', it means that the number
is a positive
Example :
+37 = 00100101 (in 8 bits)
-37 = 10100101 (in 8 bits)
What is the range of numbers that can be represented in this method?
2
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Sign-Magnitude Representation…
Example 2:
+127 = 01111111 (in 8 bits)
- 127 = 11111111 (in 8 bits)
Example 3 :
0= 00000000 (in 8 bits)
if we consider negative value of this bit stream
we get 10000000
This is also representing zero
This is a drawback of Sign-Magnitude representation
(That is two values to represent zero as +0 and -0)
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One’s complement
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• One's complement number representation is used for signed
numbers in binary format
• To obtain the 1's complement of a number:
– Get the binary format of the given number
– Complement all the bits in the binary number
• there are different representations for +0 and -0 in one's
complement.
Examples of 8-bit one's complement numbers:
BIT Pattern
Decimal Value
BIT Pattern
Decimal Value
0000 0000
+0
0000 0011
+3
1111 1111
-0
1111 1100
-3
0000 0001
+1
0001 1111
+31
1111 1110
-1
1110 0000
-31
One’s complement
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The range of 8-bit one's complement integers is -127 to +127.
Exercises: Find the 1’s complement of
1. 4510
2. 3710
3. ABC16
4. 1238
5. 458
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Addition in One’s complement
• Addition of signed numbers in one's complement is
performed using binary addition with end-around
carry.
• If there is a carry out of the most significant bit of the
sum, this bit must be added to the least significant bit
of the sum.
•
To add decimal 17 to decimal -8 in 8-bit one's complement:
0001 0001
1111 0111
1 0000 1000
1
0000 1001
(17)
(-8 )
=
(9)
810 = 0000 10002
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Two’s
complement
number
representation
Two’s complement
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• Two's complement number representation is used for
signed numbers on most modern computers.
• This notation allows a computer to add and subtract
numbers using the same operations
• We can illustrate two's complement notation as
follows:
– A fixed number of bits are used to represent numbers
– The most significant bit is called the sign bit
– This same notation is used to represent both positive and
negative numbers
Two’s complement…
Positive numbers are represented normally
• Example 1: Using a 4 bit representation
5 in 2's complement = 0101
• Example 2: Using an 8 bit representation
5 in 2's complement = 00000101
• Example 3: Using an 8 bit representation
24 base 16
= 0010 0100
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Two’s complement…
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Negative numbers Are represented using a 2's complement form
• To obtain the 2's complement of a number:
– Complement the bits
– Add one to the result
Example1 : Find the 2’s complement of the following 8
bit number
00101001
11010110 ………….. First, invert the bits
+ 00000001 ………….. Then, add 1
= 11010111
The 2’s complement of 00101001 is 11010111
Two’s complement…
Example 2 : (4 bits)
Represent -6 in 2's complement
+6 …………………… 0110
complement ……. 1001
add 1 ……………… 0001
-6 =
1010
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Example 3: (4 bits)
Represent -3 in 2's complement
+3 …………………… 0011
complement ……. 1100
add 1 ………………..0001
-3 = 1101
Example 4: (5 bits)
Represent -13 in 2's complement
+13 ………………… 01101
complement …. 10010
add 1 …………… 00001
-13 = 10011
Two’s complement…
Represent the following
numbers in two’s
complement form using
5 bits
1.
2.
3.
4.
5.
-5
-7
-4
-11
-3
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Represent the
following numbers in
two’s complement
form using 8 bits
1.
2.
3.
4.
5.
-12
-18
-21
-19
-9
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Addition and subtraction in 2's
complement notation
• Addition is performed by doing the simple binary addition of the two
numbers.
• Subtraction is accomplished by first performing the 2's complement
operation on the number being subtracted and then adding the two
numbers.
Examples: 5 bits
8 ……. 01000
+4 …… 00100
12 …… 01100
-8 ..…….. 11000
+ -4 ………. 11100
-12 ………. 10100
8 ……….. 01000
+ -4 ………. 11100
4 ……….. 00100
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Addition and subtraction in 2's
complement notation
• Since we are working with numbers contained in
a fixed number of bits, we must be able to detect
overflow following an operation.
• No overflow occurs when the value of the bit
carried into the most significant bit is the same as
the value carried out of the most significant bit.
• Overflow occurs when the value of the bit carried
into the most significant bit is not the same as the
bit carried out of the most significant bit.
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Addition and subtraction in 2's
complement notation
• Example: 4 bits
6 ……… 0110
+ 1 ……… 0001
7 ……… 0111
7 ……… 0111
+ 1 ……… 0001
8 ……… 1000
X
This means
a negative
number
Here we can get the correct answer by increasing the number of bits
15
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Addition and subtraction in 2's
complement notation
• Example: 5 bits
6 ……… 00110
+ 1 ……… 00001
7 ……… 00111
7 ……… 00111
+ 1 ……… 00001
8 ……… 01000
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Addition and subtraction in 2's
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complement notation…
Perform the following
calculations in two’s
complement method
using 5 bits
1.
2.
3.
4.
5.
10-5
-7+3
-4 +1
-11+ 8
-3+4
Perform the following
calculations in two’s
complement method using
8 bits
1.
2.
3.
4.
5.
-12+7
-18+9
22-21
-19+11
-9+5
Polynomial Evaluation
Whole Numbers (Radix = 10):
123410 = 1  103 + 2  102 + 3  101 + 4  100
With Fractional Part (Radix = 10):
36.7210 = 3  101 + 6  100 + 7  10-1 + 2  10-2
General Case (Radix = R):
(S1S0.S-1S-2)R =
S1  R1 + S0  R0 + S-1  R -1 + S-2  R-2
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