Number Systems Lecture 2

7/21/2019 Number Systems Lecture 2

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MCT 2333MCT 2333

Digital Number System

Dr. Hazlina Md Yusof


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Digital Number SystemDigital Number System

The hexadeimal number system isintrodued in this ha!ter.

Sine di"erent number systems may

be used in a system# it is im!ortant fora tehniian to understand ho$ toon%ert bet$een them.

&inary odes that are used to

re!resent di"erent information arealso desribed in this ha!ter.


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&inary to Deimal&inary to Deimal

Con%ersionCon%ersionCon%ert binary to deimal by

summing the !ositions thatontain a '.

10012345 371432222222 =++=+++++

1 0 0 1 0 21


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Deimal to &inaryDeimal to &inary

Con%ersionCon%ersion T$o methods to on%ert deimal

to binary( )e%erse !roess desribed in 2*'

+se re!eated di%ision


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Con%ersionCon%ersion )e%erse !roess desribed in 2*

' Note that all !ositions must be

aounted for025

1020200237 +++++=

1 0 0 1 0 21


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Con%ersionCon%ersion)e!eated di%ision ste!s(

Di%ide the deimal number by 2

,rite the remainder after eah

di%ision until a -uotient of zero isobtained.

The rst remainder is the /S& andthe last is the MS& Note# $hen done on a alulator# a

frational ans$er indiates a remainderof '.


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Deimal to &inary Con%ersionDeimal to &inary Con%ersion

)e!eated di%ision0 This 1o$hart

desribes the!roess and anbe used to on%ertfrom deimal to

any other numbersystem.


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Hexadeimal NumberHexadeimal Number

SystemSystemMost digital systems deal $ith grou!sof bits in e%en !o$ers of 2 suh as #'# 32# and 4 bits.

Hexadeimal uses grou!s of 4 bits.&ase '

' !ossible symbols

5*6 and 7*8

7llo$s for on%enient handling of longbinary strings.


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SystemSystemCon%ert from hex to deimal by

multi!lying eah hex digit by its

!ositional $eight. 9xam!le(

)16(3)16(6)16(1163 01216

++=

131662561 ++=

10355=


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SystemSystem

Con%ert from deimal to hex by using there!eated di%ision method used for deimal

to binary and deimal to otal on%ersion.Di%ide the deimal number by 'The rst remainder is the /S& and the last

is the MS&.

Note# $hen done on a alulator a deimalremainder an be multi!lied by ' to get theresult. :f the remainder is greater than 6# theletters 7 through 8 are used.


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SystemSystem

9xam!le of hex to binaryon%ersion(

682'3 ; 6 8 2

'55' '''' 55'5 ;'55'''''55'52

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111


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SystemSystemCon%ert from binary to hex by

grou!ing bits in four starting $iththe /S&.

9ah grou! is then on%erted tothe hex e-ui%alent

/eading zeros an be added to

the left of the MS& to ll out thelast grou!.


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2*3 Hexadeimal Number2*3 Hexadeimal Number

SystemSystem

9xam!le of binary to hex on%ersion.

(Note the addition of leading zeroes)

'''5'55''52 ; 00'' '5'5 5''5

; 3 7

; 37'3Counting in hex re-uires a reset and arry

after reahing 8.

Decimal 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Hexadecimal 0 1 2 3 4 5 6 7 8 9 A B C D E F

Binary 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111


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SystemSystemHexadeimal is useful for

re!resenting long strings of bits.+nderstanding the on%ersion

!roess and memorizing the 4 bit!atterns for eah hexadeimaldigit $ill !ro%e %aluable later.


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&inary 7ddition&inary 7ddition

&inary numbers are added li


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)e!resenting Signed)e!resenting Signed

NumbersNumbersSine it is only !ossible to sho$

magnitude $ith a binary number# thesign => or ? is sho$n by adding anextra @signA bit.

7 sign bit of 5 indiates a !ositi%enumber.

7 sign bit of ' indiates a negati%e

number.The 2Bs om!lement system is themost ommonly used $ay tore!resent signed numbers.


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NumbersNumbers:n order to hange a binary number to 2Bs

om!lement it must rst be hanged to 'Bsom!lement. To on%ert to 'Bs om!lement# sim!ly hange eah bit to its

om!lement =o!!osite?. To on%ert 'Bs om!lement to 2Bs om!lement add ' to the

'Bs om!lement.7 !ositi%e number is true binary $ith 5 in the sign

bit.

7 negati%e number is in 2Bs om!lement form $ith 'in the sign bit.


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NumbersNumbers7 number is negated $hen on%ertedto the o!!osite sign.

7 binary number an be negated by

ta


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7ddition in the 2Bs Com!lement7ddition in the 2Bs Com!lement

SystemSystem

erform normal binary addition ofmagnitudes.

The sign bits are added $ith the magnitude

bits.:f addition results in a arry of the sign bit# the

arry bit is ignored.:f the result is !ositi%e it is in !ure binary

form.:f the result is negati%e it is in 2Bs om!lement

form.


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Subtration in the 2Bs Com!lementSubtration in the 2Bs Com!lement

SystemSystem

The number subtrated =subtrahend?is negated.

The result is added to the minuend.

The ans$er re!resents the di"erene.:f the ans$er exeeds the number of

magnitude bits an o%er1o$ results.


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Multi!liation of &inaryMulti!liation of &inary

NumbersNumbersThis is similar to multi!liation of

deimal numbers.9ah bit in the multi!lier is multi!lied

by the multi!liand.The results are shifted as $e mo%e

from /S& to MS& in the multi!lier.7ll of the results are added to obtain

the nal !rodut.


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&inary Di%ision&inary Di%ision

This is similar to deimal long di%ision.:t is sim!ler beause only ' or 5 are

!ossible.

The subtration !art of the o!eration isdone using 2Bs om!lement subtration.

:f the signs of the di%idend and di%isorare the same the ans$er $ill be !ositi%e.

:f the signs of the di%idend and di%isorare di"erent the ans$er $ill be negati%e.


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Hexadeimal 7rithmetiHexadeimal 7rithmeti

Hex addition( 7dd the hex digits in deimal.

:f the sum is ' or less ex!ress it diretly inhex digits.

:f the sum is greater than '# subtrat 'and arry ' to the next !osition.

Hex subtration 0 use the samemethod as for binary numbers.

,hen the MSD in a hex number is orgreater# the number is negati%e.,hen the MSD is E or less# the numberis !ositi%e.


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&CD&CD

&inary Coded Deimal =&CD? isanother $ay to !resent deimal

numbers in binary form.&CD is $idely used and ombines

features of both deimal and binary

systems.9ah digit is on%erted to a binary

e-ui%alent.


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&CD&CD

To on%ert the number E4'5to &CD(

E 4

5'55 5''' 5'55 ; 5'555'''5'55&CD

9ah deimal digit is re!resented using 4 bits.

9ah 4*bit grou! an ne%er be greater than 6.

)e%erse the !roess to on%ert &CD to deimal.


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&CD&CD

&CD is not a number system.&CD is a deimal number $ith

eah digit enoded to its binary

e-ui%alent.7 &CD number is not the same as

a straight binary number.

The !rimary ad%antage of &CD isthe relati%e ease of on%erting toand from deimal.


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Fray CodeFray Code

The gray ode is used ina!!liations $here numbershange ra!idly.

:n the gray ode# only one bithanges from eah %alue to thenext.


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Fray CodeFray Code

&inary Fray Code

555 55555' 55'5'5 5''

5'' 5'5'55 ''5'5' '''''5 '5'

''' '55


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utting :t 7ll Togetherutting :t 7ll Together


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The &yte# Nibble# andThe &yte# Nibble# and

,ord,ord' byte ; bits' nibble ; 4 bits' $ord ; size de!ends on data

!ath$ay size. ,ord size in a sim!le system may be

one byte = bits?

,ord size in a C is eight bytes =4bits?


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7l!hanumeri Codes7l!hanumeri Codes

)e!resents haraters and funtionsfound on a om!uter


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arity Method for 9rrorarity Method for 9rror

DetetionDetetion&inary data and odes are fre-uently

mo%ed bet$een loations. 8or exam!le( Digitized %oie o%er a miro$a%e lin


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DetetionDetetionThe !arity method of error detetion

re-uires the addition of an extra bitto a ode grou!.

This extra bit is alled the !arity bit.The bit an be either a 5 or '#

de!ending on the number of 's inthe ode grou!.

There are t$o methods# e%en andodd.

Ronald Tocci/Neal Widmer/Gregory

MossDigital Systems: Principles and

Applications, 10e

Copyright 2007 by Pearson Education !nc"

Co#u$bus %& 43235

'## rights resered"


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DetetionDetetion9%en !arity method 0 the total

number of bits in a grou!inluding the !arity bit must add

u! to an e%en number.The binary grou! ' 5 ' ' $ould

re-uire the addition of a !arity bit 1' 5 ' ' Note that the !arity bit may be added at

either end of a grou!.



Applications, 10e


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'## rights resered"


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DetetionDetetionGdd !arity method 0 the total

number of bits in a grou!inluding the !arity bit must add

u! to an odd number.The binary grou! ' ' ' ' $ould

re-uire the addition of a !arity bit 1' ' ' '



Applications, 10e


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'## rights resered"


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DetetionDetetionThe transmitter and reei%er

must @agreeA on the ty!e of!arity he


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2*'5 7!!liations2*'5 7!!liations

7 CD*)GM stores 5 megabytes ofdigital data. Ho$ many bits of data isthis

Determine the odd !arity bit re-uiredfor eah of the follo$ing E bit 7SC::odes(

I '55'5'5

I 5'5''5'

I 5''5'5'

Determine the e%en !arity bit re-uiredfor eah E bit 7SC:: ode listed abo%e.


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Number Systems Lecture 2

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