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TA C162

Lecture 5 Binary Arithmetic

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Todays Agenda

Binary Arithmetic

Addition & Subtraction

Sign Extension

Representation of Fractions

Floating Point Next Class IEEE-754 Standard

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2s C Addition & Subtractionxampe : + n t notaton11 01011

3

0001111 + 3 01110 (14)

Example 2:14 - 9 ? (in 5 bit notation)14 011109 01001-9 1011114 + -9 00101 5Tuesday, January 19, 2010 3Biju K Raveendran@BITS Pilani.

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Signed Extension Useful to represent a small number withfewer bits

Example: 6

00000110 can be represented as0110.

A ng ea ng s oes not a ect t e va ue o

a Number. Question

How about Negative Numbers ??

Try this out: 13 + (-5) where 13 with 8 bits and-5 with 5 bits

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Sign Extension (SEXT)To add two numbers, we must represent them with the samenumber of bits.

4-bit 8-bit

Instead, replicate the MS bit i.e. the sign bit:

1100 (-4) 00001100 (12, not -4)

- -

0100 (4) 00000100 (still 4)

Leading 0s will not affect the value of positive numbers &1100 (-4) 11111100 (still -4)

Tuesday, January 19, 2010 5Biju K Raveendran@BITS Pilani.

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Overflow operan s are too g, t en sum cannot e representeas an n-bit 2s compliment number.

+ 01001 (9)

Above result is Incorrect !!!

10001 (-15)

We have overflow if:

Signs of both operands are the same, and Sign of sum is

. Another test -- easy for hardware:

Carr into MS bit does not e ual carr out

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Example:

Carry in for MS BitCarry out for MS Bit1 0

-+ 10111 (-9)

sign of result is positive

Look at both carry!!!

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Overflow

Overflow

Overflow

negative number is added?

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Addition & Subtraction Adding a Number to itself (Multiply by 2)

X + X = 2X

4 00100 8 01000 7 00111 14 01110

-211110 -4 11100

Equivalent to shifting each bit one position to the left

Dividing a number by 2

Equivalent to shifting all the bits right one position

Tuesday, January 19, 2010 9Biju K Raveendran@BITS Pilani.

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Fractions: Fixed-Point How can we represent fractions?

negative powers of two -- just like decimal point.

2-1 = 0.5

2-2 = 0.25xampe:

11111110.110

2- = 0.125

Tuesday, January 19, 2010 10Biju K Raveendran@BITS Pilani.

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Conversion from Binary to Decimal(Fractions)Convert to decimal: 00101000.101

ep : a e n eger par an conver o ec ma

23 + 25 = 40

Step2: Take fraction part and convert to decimal

2-1+2-3 = .625

Step3: Add result of Step1 and Step2

40.625

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Arithmetic on Fractionss comp a ton an su tracton st wor .

If binary points are aligned 2-1 = 0.5

2-2 =0.25

00101000.101 (40.625)

2-3

= 0.125

+ 11111110.110 (-1.25)00100111.011 (39.375)

=-

.110 = 0.75

. - . = - .

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Precision and Range Precision: Difference between successive values

for a given data type

Precision of the number, is determined by the number offractional bits.

-,to the right of the binary point has a precision of 2-3

i.e. value of least significant bit

Range

The range of numbers

For example, for 8 bit 2s compliment data type therange is 128.

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Representations of very Big and very

ma um ersHow many Bits required to represent 6.023 x 1023 ?

How many Bits required to represent 6.626 x 10-34 ?

Can we represent above numbers by using 32 Bits? IfYES, HOW?

By Floating Point Representation

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Floating-Point Representation Use equivalent of scientific notation: F x 2E

Need to represent F (fraction), E (exponent), andsign.

IEEE 754 Floating-Point Standard (32-bits):

1b 8b 23b

S exponent fraction

0exponent,2fraction.0)1(

254exponent1,2fraction.1)1(126

127exponent

==

=

S

S

N

N

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Floating Point Example Single-precision IEEE floating point number

1 01111110 10000000000000000000000

sign exponent fraction

Sign is 1number is negative.

= .

Fraction is 0.100000000000 = 0.5 (decimal).

Value = -1.1 x 2(126-127) = -1.1x 2-1 = -0.11

- . .

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Floating Point ExampleRepresent 1/8 (0.125) in IEEE 754 format?

Binary equivalent of 0.125 is 0.001 or 1.0 x 2-3

orma ze

N = (-1)s X 1.fraction X 2exponent-127

Sign bit = 0 (Number is positive)

exponent - 127 = -3 i.e. exponent = 124Binary equivalent of 124 = 01111100

Fraction = 00000000000000000000000

Final representation of 1/8 in IEEE 754 format is

Tuesday, January 19, 2010 17Biju K Raveendran@BITS Pilani.