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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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Deimal to &inaryDeimal to &inary
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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Deimal to &inaryDeimal to &inary
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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Hexadeimal NumberHexadeimal Number
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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Hexadeimal NumberHexadeimal Number
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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Hexadeimal NumberHexadeimal Number
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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Hexadeimal NumberHexadeimal Number
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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Hexadeimal NumberHexadeimal Number
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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)e!resenting Signed)e!resenting Signed
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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)e!resenting Signed)e!resenting Signed
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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arity Method for 9rrorarity Method for 9rror
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"
7/21/2019 Number Systems Lecture 2
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arity Method for 9rrorarity Method for 9rror
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 ' ' $ould
re-uire the addition of a !arity bit 1' 5 ' ' Note that the !arity bit may be added at
either end of a grou!.
Ronald Tocci/Neal Widmer/Gregory
MossDigital Systems: Principles and
Applications, 10e
Copyright 2007 by Pearson Education !nc"
Co#u$bus %& 43235
'## rights resered"
7/21/2019 Number Systems Lecture 2
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arity Method for 9rrorarity Method for 9rror
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! ' ' ' ' $ould
re-uire the addition of a !arity bit 1' ' ' '
Ronald Tocci/Neal Widmer/Gregory
MossDigital Systems: Principles and
Applications, 10e
Copyright 2007 by Pearson Education !nc"
Co#u$bus %& 43235
'## rights resered"
7/21/2019 Number Systems Lecture 2
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arity Method for 9rrorarity Method for 9rror
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$ing E bit 7SC::odes(
I '55'5'5
I 5'5''5'
I 5''5'5'
Determine the e%en !arity bit re-uiredfor eah E bit 7SC:: ode listed abo%e.
Ronald Tocci/Neal Widmer/Gregory
MossDi i l S P i i l d
Copyright 2007 by Pearson Education !nc"
Co#u$bus %& 43235