Number System111

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Number SystemNumber SystemA number system defines a set of values used toA number system defines a set of values used to

define define quantity .quantity .withwith the advancement ofthe advancement of

machines different number systems were formed tomachines different number systems were formed tomake the task simple, accurate & fast. Thesemake the task simple, accurate & fast. These

number systems worked on the principle ofnumber systems worked on the principle of

DLDP(digital logic design presentation).DLDP(digital logic design presentation).


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Number System(contd)Number System(contd)

Number are of two types:Number are of two types:--

1)1)NonNon--positionalpositional : Had no symbol for zero, clumsy &: Had no symbol for zero, clumsy &difficult to do large calculation because of limiteddifficult to do large calculation because of limitedSymbols. e.g. Roman Number SystemSymbols. e.g. Roman Number System

2)2)PositionalPositional : They have a finite numbers of: They have a finite numbers ofsymbols/digits to represent large numbers. the value ofsymbols/digits to represent large numbers. the value of

each digit in a number is defined not only by a symboleach digit in a number is defined not only by a symbolbut also by the symbols position. e.g. Decimal Numberbut also by the symbols position. e.g. Decimal NumberSystem.System.


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Base or Radix of No. systemBase or Radix of No. system

The word Base or Radix means the quantity ofThe word Base or Radix means the quantity of

admissible marks used in a given number system.admissible marks used in a given number system.

these admissible marks are the characters such asthese admissible marks are the characters such asArabic numerals, Latin letters or otherArabic numerals, Latin letters or other

recognizable marks, which are used to present therecognizable marks, which are used to present the

numerical magnitude of a quantity.numerical magnitude of a quantity.

e.g. (2239)e.g. (2239)10, (10, (10101)10101)22 etc.etc.


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Types of Number systemTypes of Number system

1.1. Decimal number system. (base=10 )Decimal number system. (base=10 )

(0,1,2,3,4,5,6,7,8,9)(0,1,2,3,4,5,6,7,8,9)

2. Binary number system. (base=2)2. Binary number system. (base=2)

(0,1)(0,1)

3. Octal number system. (base=8 )3. Octal number system. (base=8 )

(0,1,2,3,4,5,6,7)(0,1,2,3,4,5,6,7)

4. Hexadecimal number system. (base=16)4. Hexadecimal number system. (base=16)

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


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Decimal number systemDecimal number system : has a base value 10 and having: has a base value 10 and having10 admissible marks.It is a positional number system.10 admissible marks.It is a positional number system.they uses powers of 10 determine their number position.they uses powers of 10 determine their number position.

PositionPosition fifthfifth fourthfourth thirdthird secondsecond firstfirst

w

eightw

eight 101044

(10,000)(10,000)

101033

(1000)(1000)

101022

(100)(100)

101011

(10)(10)

101000

(1)(1)


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Binary number systemBinary number system :this number system provides basis:this number system provides basisfor all computer operations. They has a base value 2for all computer operations. They has a base value 2and having 2 admissible marks(1,0).It is a positionaland having 2 admissible marks(1,0).It is a positional

number system. they uses powers of 2 determine theirnumber system. they uses powers of 2 determine theirnumber position.number position.


weightweight 2244

(16)(16)

2233

(8)(8)

2222

(4)(4)

2211

(2)(2)

2200

(1)(1)


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Octal number systemOctal number system : The octal number system is a base 8: The octal number system is a base 8system, having 8 admissible marks(0,1,2,3,4,5,6,7). It is asystem, having 8 admissible marks(0,1,2,3,4,5,6,7). It is apositional number system. they uses powers of 8positional number system. they uses powers of 8

determine their number position.determine their number position.


weightweight 8844

(4096)(4096)

8833

(512)(512)

8822

(64)(64)

8811

(8)(8)

8800

(1)(1)


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Hexadecimal number systemHexadecimal number system :The hexadecimal number:The hexadecimal number

system is a base 16 system,having 16 admissiblesystem is a base 16 system,having 16 admissiblemarks(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). It is a positionalmarks(0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F). It is a positionalnumber system. they uses powers of 16 determine theirnumber system. they uses powers of 16 determine theirnumber position. This number system uses 0 to 9 numbersnumber position. This number system uses 0 to 9 numbers

and A to F characters to represent 10 to 15 respectivelyand A to F characters to represent 10 to 15 respectively


weightweight 161644

(65,536)(65,536)

161633

(4096)(4096)

161622

(256)(256)

161611

(16)(16)

161600

(1)(1)


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Comparing Decimal,Binary, Octal & HexadecimalComparing Decimal,Binary, Octal & Hexadecimal

DecimalDecimal BinaryBinary OctalOctal HexadecimalHexadecimal

00 00000000 000000 00

11 00010001 001001 11

22 00100010 002002 22

33 00110011 003003 33

44 01000100 004004 44

55 01010101 005005 55

66 01100110 006006 66

77 01110111 007007 77

88 10001000 010010 88

99 10011001 011011 99

1010 10101010 012012 AA

1111 10111011 013013 BB

1212 11001100 014014 CC

1313 11011101 015015 DD

1414 11101110 016016 EE

1515 11111111 017017 FF


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What is the decimal equivalent of thebinary number 1101110?

1 x 26 = 1 x 64 = 64+ 1 x 25 = 1 x 32 = 32

+ 0 x 24 = 0 x 16 = 0

+ 1 x 23 = 1 x 8 = 8

+ 1 x 22 = 1 x 4 = 4

+ 1 x 21 = 1 x 2 = 2

+ 0 x 2 = 0 x 1 = 0

= 110 in base 10

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Converting Binary to DecimalConverting Binary to Decimal


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What is the decimal equivalent of the octal

number 642?

6 x 82 = 6 x 64 = 384

+ 4 x 81 = 4 x 8= 32

+ 2 x 8 = 2 x 1 = 2

= 418 in base 10

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Converting Octal to DecimalConverting Octal to Decimal


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What is the decimal equivalent of the

hexadecimal number DEF?

D x 162 = 13 x 256 = 3328

+ E x 161 = 14 x 16= 224

+ F x 16 = 15 x 1 = 15

= 3567 in base 10

Remember, the digits in base 16 are

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

Converting Hexadecimal to DecimalConverting Hexadecimal to Decimal


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Converting Decimal to Binary

(3567)(3567)1010 = (110111101111)= (110111101111) 22

2 3567 Remainder Remainder

2 1783 1

2 891 1

2 445 1

2 222 1

2 111 02 55 1

2 27 1

2 13 1

2 6 1

2 3 0

2 1 1

0 1


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Converting Decimal to OctalConverting Decimal to Octal

(1988)(1988)1010 = (3704)= (3704)88

8 1988 R emainder

8 248 4

8 31 0

8 3 7

33


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(3567)(3567)1010 = (DEF)= (DEF)1616

Converting Decimal to HexadecimalConverting Decimal to Hexadecimal

16 3567 R emainder

16 222 15 = F

16 13 14 = E

13 =D


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Mark groups ofthree (from right)

Convert each group

10101011 10 101 0112 5 3

10101011 is 253 in base 8

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Converting Binary to OctalConverting Binary to Octal


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Mark groups offour(from right)

Convert each group

10101011 1010 1011

A B

10101011 is AB in base 16

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Converting Binary to HexadecimalConverting Binary to Hexadecimal


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Converting Octal to HexadecimalConverting Octal to Hexadecimal

OctalN

umberOctalN

umber 22 33 22 77Binary valueBinary value 010010 011011 010010 111111

Combine the 3-bit binary blocks, we have

010011010111.

Determine the hexadecimal equivalent of (2327)8


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Separate the group of binary numbers from leftside into the 4-bit binary number & convert

these blocks into their respective hexadecimal

symbols.

0100 1101 0111

4 D

SoSo

(2327)(2327)88 = (4D7)= (4D7)1616


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Converting Hexadecimal to OctalConverting Hexadecimal to Octal

Determine the Octal equivalent of (2B6)16

Hexadecimal no.Hexadecimal no. 22 BB 66

Binary valueBinary value 00100010 10111011 01100110

Combine all the 4-bit binary blocks, we have

001010110110


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Separate the group of binary numbers into 3-bit

binary blocks and convert these blocks into their

corresponding octal symbols

001 010 110 110

1 2 6

So

(2B6)16 = (1266)8


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Number System111

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