Lect2 Number Representation

7/29/2019 Lect2 Number Representation

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TA C162

LECTURE 2 REPRESENTATION OF BINARY NUMBERS


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Todays Agenda

Binary Numbers

Non Positional

Positional

Signed Magnitude

1s Complement

Saturday, January 9, 2010 2Biju K Raveendran@BITS Pilani.


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Representation of Information

One, Two, Three, .., Dozen, Gross

One + Two = Three

Dozen x Dozen = Gross

Problem:

Cumbersome when working with large Numbers

pera ons are no sys ema c



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Representation of Information

I, II, III, IV, V,, IX, X, XI,

X * X = C

XX * XX = CCCC

Problems

Operations are slightly better still not a good



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Representation of Information Simplest way

Unary Representation

Example 5 is represented as 11111

Problems

Cumbersome when working with large numbers



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Representation of Information Decimal Numbers

, , , , ,

Operations are systematic i.e. algorithmic

At the lowest level, a computer is an electronic machine.

Com uter is workin with devices which react topresence or absence of voltage (controlling the flow ofelectrons).

1. Presence of a voltage well call this state 1

2. Absence of a voltage well call this state 0



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Computer is a binary digital system

Di ital s stem:

Finite number of symbols

Binary (base two) system:

Has two states: 0 and 1



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Computer is a binary digital system Cont

Basic unit of information is the binary digit, or bit.

Values with more than two states re uire

multiple bits (wires). A se uence of two bits has four ossible states:

00, 01, 10, 11

A sequence of three bits has eight possible states:, , , , , , ,

Inference: A sequence ofn bits has 2n possible states.



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What kinds of data do we need torepresent? , , ,

point, complex, rational, irrational, , ,

Images pixels, colors, shapes,

oun

Logical true, false

Instructions



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Representation of a Number um er can e represente as Five

5

V 11111

A representation is a data type if there are operations in thecompu er a can opera e on n orma on enco e n a

representation

In this course we mainly use 2 data types 2s complement integers for representing +ve and ve integers

(to perform arithmetic operations)

co es or representng c aracters



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Unsigned Integers - No weightage for the position {0th, 1st, etc.. }

Problems?

We g te pos tona notaton

like decimal numbers: 329

3 is worth 300, because of its position, while 9 is only worth 9

329 101most

significantleast

significantmost

significantleast

significant

102 101 100 22 21 20

= =



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Unsigned Integers (cont.)An n-bit unsigned integer represents 2n values:from 0 to 2n-1.

22

21

20

0 0 0 0

0 1 0 2

0 1 1 3

1 0 0 4

1 0 1 5

1 1 0 61 1 1 7



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Unsigned Binary Arithmetic Base-2 addition just like base-10! Add from right to left, propagating carry

10010 10010 1111

carry

+ 1001 + 1011 + 1

11011 11101 10000

10111

11110



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Signed Integers With n bits, we have 2n distinct values. Assign about half to positive integers (1 through 2n-1 - 1)

- n-1 - -

That leaves two values: one for 0, and one extra

Positive integers

J ust like unsigned : zero in most significant (MS) bit00101 = 5

Ne ative inte ers

Set MS bit to show negative, other bits are the same as unsigned10101 = -5

MS bit indicates sign: 0=positive, 1=negative



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Question Given n bit string, What is the Maximum number we can representin Signed Magnitude form?

. . . . . .

2n-1 2n-2 . . . . . 22 21 20

The 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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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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Limitations! Problems with sign-magnitude representation!

Two representations of zero (+0 and 0)

complexity is more)Because:

How to add two sign-magnitude numbers?

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

10011

10101 => -5 ??



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1s Complement Representation Positive numbers representation is same as signed

integers.

Negative numbers represented by flipping all 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 -12 in 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 71001 1 6

1010 2 5

1011 3 4

1100 4 31101 5 2

1110 6 1

1111 7 0



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Limitations! Problems with sign-magnitude and 1s complement!

Two representations of zero (+0 and 0)

complexity is more)Because:

How to add two sign-magnitude numbers?

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

10011

10101 => -5 ??

How to add two ones complement numbers?

e. ., tr 4 + -3


Lect2 Number Representation

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