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UNSIGNED BINARY NUMBERS In some applications, data either
positive or negative.
We concentrate on absolute value ormagnitude only.
For example :
Smallest 8-bit number is 0000 0000Largest 8-bit number is 1111 1111
Equivalent to a decimal 0 to 255.
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Data is called unsigned binarybecauseall bits are used to represent magnitudeof the corresponding decimal only
Limitations: With 8-bit unsigned arithmetic, all
magnitudes must fall in the range 0 to255
Addition and subtraction must also fallbetween 0 and 255.
If greater than 255, use 16-bit
arithmetic
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OVERFLOW In 8-bit arithmetic , addition of 2
unsigned numbers whose sum is
greater than 255 , causes overflow.
a carry into 9th column
Microprocessors have carry flag to
detect a carry
which warns , answer is invalid.
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Example Addition of 175 10 and 118 10
175 1010 1111118 0111 0110293 1 0010 0101
In case of overflow, programmer has tocarry instructions for carry flag and use16-bit arithmetic.
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Signed Integer Representation We have been ignoring signed
integers.
The PROBLEMwith signed integers ( -
45, + 27, -99) is the SIGN!
How do we encode the sign?
The sign is an extra piece of information
that has to be encoded in addition to the
magnitude.
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What can we do?
Following are the three possible
techniques
Signed Magnitude Representation
Diminished Radix-Complement Representation
Radix-Complement Representation
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Signed Magnitude
Representation
Signed Magnitude (SM) is a method forencoding signed integers.
The Most Significant Bit (MSB) is used
to represent the sign.
1 is used for a - (negative sign),
0 for a + (positive sign).
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Format of SM Representation The format of a SM number in 8 bits is:
smmmmmmm Where s is the sign bit
The other 7 bits represent the magnitude.
NOTE: for positive numbers, the result isthe same as the unsigned binary
representation.
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Signed Magnitude Ex. (8 bits) -5 = (1 0000101)2 = (85)16 +5 = (0 0000101)2 = (05)16
+127 = (0 1111111)2 = (7F)16
-127 = (1 1111111)2 = (FF)16
+ 0 = (0 0000000)2 = (00)16 - 0 = (1 0000000)2 = (80)16
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For N bits, can represent the signedintegers
- {2(N-1)
1} to {+ 2(N-1)
1} For 8 bits, can represent the signed integers
-127 to +127.
Signed magnitude easy to understand andencode.
Simple. Negative numbers are identical topositive numbers except the sign bit.
Used today in some applications.
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Problems with Signed Magnitude Representation
One problem is that it has two ways of
representing 0 (-0,and +0)
Another problem is that addition of
K + (-K) does not give Zero!
-5 + 5 = (85)16
+ (05)16
= (8A)16
Requires complicated arithmetic
circuits.
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Ones Complement
Representation
Ones complement is another way torepresent signed integers.
To encode a negative number, get thebinary representation of its magnitude,then COMPLEMENT each bit.
Complementing each bit mean that 1sare replaced with 0s, 0s are replacedwith 1s.
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Example: How is -5 represented in Ones
Complement (encoded in 8 bits) ?
The magnitude 5 in 8-bits is( 00000101)2 = (05)16
Now complement each bit:
(11111010)2 = (FA)16 (FA)16 is ones complement
representation of -5.
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Ones Complement Examples
+5 = (00000101)2 = ( 05 )16
-5 = (11111010) = ( FA )16
+127 = (01111111)2 = ( 7F )16
-127 = (10000000)2 = ( 80 )16
+ 0 = (00000000)2 = ( 00 )16 - 0 = (11111111)2 = ( FF )16
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For N bits, can represent the signedintegers
- {2(N-1) 1} to + {2(N-1) 1}
For 8 bits, can represent the signed
integers -127 to +127.
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Limitations of Ones
Complement Method
Still have the problem that there are twoways of representing 0 (-0, and +0)
Mathematically speaking, no such thing astwo representations for zeros.
However, addition of K + (-K) now gives
Zero!
-5 + 5 = ( FA )16 + ( 05 )16
= ( FF)16
= -0 !!!
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Unfortunately, K + 0 = K only works
if we use +0,
Does not work if we use -0.
5 + (+0) = (05)16 + (00)16 = (05)16 = 5 (ok)
5 + (-0) = (05)16 + (FF)16= (04)16 = 4(wrong)
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Twos Complement
Representation
Twos complement is another way torepresent signed integers.
To encode a negative number, get the binaryrepresentation of its magnitude,
COMPLEMENT each bit, then ADD 1.
or
See 1 from LSB , mark it and complement all
bits which are occurring before it.
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Example
What is -5 in Twos Complement, 8 bits?
The magnitude 5 in 8-bits is:
(00000101)2 = (05)16
Now complement each bit:
(11111010)2 = (FA)16
Now add one: (FA) 16 + 1 = (FB)16
(FB)16 is the 8-bit, twos complement
representation of -5.
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Twos Complement Examples
-5 = ( 11111011) = ( FB)16 +5 = ( 00000101 )= ( 05)16
+127 = ( 01111111) = (7F)16
-127 = ( 10000001) = ( 81)16
-128 = ( 10000000 )= ( 80 )16
+ 0 = ( 00000000) = (00)16
- 0 = ( 00000000) = (00 )16 (only 1 zero!!!)
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For N bits, can represent the signed
integers
-2(N-1) to + 2(N-1) - 1
Note that negative range extends one
more than positive range.
For 8 bits, can represent the signed
integers -128 to +127.
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Twos Complement
Comments
Twos complement is the method of choicefor representing signed integers.
It has none of the drawbacks of SignedMagnitude or Ones Complement.
There is only one zero, and K + (-K) = 0.
-5 + 5 = (FB)16 + ( 05)16 = (00)16 = 0 !!!
Normal binary addition is used for addingnumbers that represent twos complementintegers.
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Complements
Used in digital computer forsimplifying the subtraction operation
and for logic manipulations. There are two types of complements
for each base R system Diminished Radix-Complement Representation
[ (R-1)s Complement]
Radix-Complement Representation [ RsComplement ]
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(1) The rs complement. e.g. 2scomplement for binary and 10scomplement for decimal.
(2) The (r-1)s complement. e.g. 1scomplement for binary and 9scomplement for decimal.
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(R-1)'s Complement Represen tation
+ve numbers. : sign and magnitude
-ve numbers : -N represented by , the (R-1)'scomplement
where = ( Rn R-m ) N,
n = Total number of digits in integerpart of the number N
m = Total number of digits in fractionalpart of the number N
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Examples
9s complements of (52520)10(105-100-52520)= (105-1-
52520)=47479
9s complements of (0.3267)10(100-10-4-0.3267)= (0.9999- 0.3267)
=0.6732
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R's Complement Representat ion
+ve nos.: sign and magnitude
-ve nos. : -N is represented by N*, the
R's Complementwhere N* = Rn N
n = Total number of digits in integer
part of the number N
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Examples:
10s complements of (52520)10
(105
-52520) = (105
-52520)= 47480
10s complements of (0.3267)10
(100-0.3267) = (1.0- 0.3267)= 0.6733