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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