Lect3 Number Representation

7/29/2019 Lect3 Number Representation

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TA C162

Lecture 3 Representation of Binary Numbers


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Todays Agenda Representation of Binary Numbers

Unsigned, Signed Magnitude, 1s Compliment

Done !!! 2s Complement

Base Conversion Binary to Decimal and Decimal to Binary

Decimal to Octal and Octal to Decimal

Binary to Hexadecimal and Hexadecimal to Binary

Tuesday, January 12, 2010 2Biju K Raveendran@BITS Pilani.


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How to represent Signed Integers

2s complement representation



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Representation of Signed Integers Cont

Ex: n = 3

Signed

magnitude

000 +0

010 +2

100 -0

101 -1

110 -2

111 -3



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1s Complement Representation Positive numbers representation is same as signed

integers.

Negative numbers represented by flippingall the bits ofcorresponding positive numbers

For Example:

+5 is re resented as 00101

-5 is represented by 11010

This representation was used in some early computers



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Question Given n bit string, What is the Maximum number we can representin 1s complement form?

he Maximum +ve value is : 2n-1 -1

The Maximum -ve value is : 2n-1 -1

Ex: Given 5 bits,

. - =

Min. number in signed magnitude form is: -24 -1= -15



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Representation of Signed Integers Cont

Ex: n = 3

Signed 1s

magnitude complement

000 + 0 + 0

010 + 2 + 2

100 0 3

101 1 2

110 2 1

111 3 0



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Example: 1s ComplementExample 1:How will we represent-12in 1s complement form in 5 digits?

Step1: Take +12 in binary representation

Step2: Flip all the bits of the above

10011

Inference

-12 representation in 1s complement form: 10011

+12 representation in 1s complement form: 01100



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Representation of Signed IntegersEx: n = 4 Signed magnitude 1s Complement

0000 + 0 + 0

0001 + 1 + 1

0010 + 2 + 2

0011 + 3 + 3

0100 + 4 + 4

0101 + 5 + 5

0110 + 6 + 6

0111 + 7 + 7

1000 0 7

1001 1 6

1010 2 5

1011 3 4

1100 4 31101 5 2

1110 6 1

1111 7 0



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Limitations of Signed magnitude

Problems with sign-magnitude and 1s complement!

Two representations of zero (+0 and 0)

Arithmetic circuits are complex to implement aboverepresentation (hardware complexity is more)

How to add two sign-magnitude numbers?

e.g., try 2 + (-3)00010

10011

10101 => -5 ??

How to add two ones complement numbers?

e.g., ry -



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A Good Representation

Representation of Negative Numbers should makeArithmetic & Logic Unit (ALU) simple

Addition of an arbitrary integer with same magnitude butopposite sign should return 0

.

A + (-A) = 0

From (2n-1-1) to +(2n-1-1) ALU adds 1 to successive representations

Mathematically we can represent as:

REPRESENTATION(value+1) =REPRESENTATION(value) + REPRESENTATION(1)



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Twos Complement Representation If number is positive or zero Normal binary representation

Start with positive number Flip every bit (i.e., take the ones complement)

Then add one

Example:

11010 (1s comp) 10110 (1s comp)

+ 1 + 1

11011 (-5) 10111 (-9)



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Example Represent -12 in 2s complement form (5

Step 1:Represent +12 in binary

Step 2: Find 1s complement of that

Step 3: Add 00001 to 1s complement

Result (-12 in 2s complement form) is



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Twos Complement Shortcut To take the twos complement of a number: Copy bits from right to left until (and including) the

first 1 Flip remaining bits to the left

011010000 011010000

+ 1

100110000 100110000

(copy)(flip)



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Advantage of 2s complement Twos complementrepresentation developed to makecircuits easy for arithmetic.

-, ,

such that X + (-X) = 0 with normal addition, ignoring carry out

00101 (5) 01001 (9)

+ 11011 (-5) +10111 (-9)00000 (0) 00000 (0)



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Exam le:2+(-3) =?? (in 5 digits)

Ex ress -3 in 2s com lement form:

+3 000111s Complement of 3 11100

s omp emen o .e. -+2 00010+ +

-3 1110111111

Look at MSB 1 So take 2s complement again 00001



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Twos Complement Signed Integers MS bit is sign bit: it has weight(2n-1)

Range of an n-bit number: (2n-1)through 2n-1 1.-

- .

23 22 21 20 23 22 21 20

-

0 0 0 1 1

0 0 1 0 2

1 0 0 1 -7

1 0 1 0 -6

0 0 1 1 3

0 1 0 0 4

1 0 1 1 -5

1 1 0 0 -4

0 1 0 1 5

0 1 1 0 6

1 1 0 1 -3

1 1 1 0 -2

0 1 1 1 7 -



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What is Base? Decimal Number System

Base is 10

All numbers are represented by 0 to 9

Binary Number System

ase s

All numbers are represented by 0 and 1

Base is 8

Inference

-



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Converting Binary (2s C) to Decimal8-bit 2s complement Number is of the form

a7 a6 a5 a4 a3 a2 a1 a0n 2n

0 1

ea ng a7 s one; en e num er s

negative take twos complement to get apositive number; Otherwise number is positive.

1 2

2 4

3 8

The Magnitude is:

a6.26

+a5.25

+a4.24

+a3.23

+a2.22

+a1.21

+a0.20

4 16

5 32

Add powers of 2 that have 1 in the7 128

8 256

.

If original number was negative, affix a minussign in front.

9 51210 1024

ssumng - s comp emen num ers.



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Example: Binary to Decimal Conversion

X = 01101000two

Step 1: Leading bit is 0 no need to take 2s complementX = 01101000

Step 2: Add powers of 2 that have 1 in the correspondingt pos tons.

= 26+25+23 = 64+32+8

ten



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More Examples

n 2nX =00100111two1 2

2 4

3 8

= + + + = + + +X =39ten

4 16

5 32

6 64

X =11100110two- = 7 128

8 256

9 512

=24+23+21=16+8+2=26ten

X =-26ten

Assuming 8-bit 2s complement numbers.



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Examples

= wo

X=10001100two



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Algorithm

-

Step 1:

Step 2:

,

If N is negative, then form negative of this 2s



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Addition & SubtractionExample 1: 5 + 4 ? (in 5 bit notation)

5 001014 00100

Example 2: 9 + - 12? (in 5 bit notation)9 01001

-12 101009 + (-12) 11101 (-3)Example 3: -3 + -7? (in 5 bit notation)

3 00011- 7 00111-7 11001

- + - -Tuesday, January 12, 2010 24Biju K Raveendran@BITS Pilani.

Lect3 Number Representation

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