NumberSystems-BinarySystem
NumberSystem• Number• Itisasymbolrepresen2ngaunitorquan2ty.
• NumberSystem• Definesasetofsymbolsusedtorepresentquan2ty
• Radix• Thebaseorradixofnumbersystemdetermineshowmanynumericaldigitsthenumbersystemuses.
TypesofNumberSystem• DecimalSystem• BinaryNumberSystem• OctalNumberSystem• HexadecimalNumberSystem
DecimalNumberSystem• Ingeniousmethodofexpressingallnumbersbymeansoftenssymbolsoriginated from India. It iswidelyusedand isbasedonthetenfingersofahumanbeing.
• Itmakesuseoftennumericsymbols• 0,1,2,3,4,5,6,7,8,9
InherentValueandPositionalValue
• Theinherentvalueofasymbolisthevalueofthatsymbolstandingalone.• Example6innumber256,165,698• Thesymbolisrelatedtothequan2tysix,evenifitisusedindifferentnumberposi2ons
• Theposi2onalvalueofanumericsymbolisdirectlyrelatedtothebaseofasystem.• Inthecaseofdecimalsystem,eachposi2onhasavalueof102mesgreaterthattheposi2ontoitsright.Example:423,thesymbol3representstheones(units),thesymbol2representsthetensposi2on(10x1),andthesymbol4representsthehundredsposi2on(10x10).Inotherwords,eachsymbolmovetotheleVrepresentsanincreaseinthevalueoftheposi2onbyafactoroften.
InherentandPositionalValuecont.
2539=2X1000+5X100+3X10+9X1=2X103+5X102+3X101+9x100
Thismeansthatposi2onalvalueofsymbol2is1000orusingthebase10itis103
BinaryNumberSystem• Usesonlytwonumericsymbols1and0• Underthebinarysystem,eachposi2onhasavalue22mesgreaterthantheposi2ontotheright.
OctalNumberSystem• Octalnumber system isusing8digits to representnumbers.Thehighestvalue=7.Eachcolumnrepresentsapowerof8.Octalnumbersarerepresentedwiththesuffix8.
HexadecimalNumberSystem• Providesanotherconvenientandsimplemethodforexpressingvaluesrepresentedbybinarynumerals.
• Itusesabase,orradix,of16andtheplacevaluesarethepowersof16.
Decimal Binary Hexadecimal Decimal Binary Hexadecimal
0 0000 0 8 1000 8
1 0001 1 9 1001 9
2 0010 2 10 1010 A
3 0011 3 11 1011 B
4 0100 4 12 1100 C
5 0101 5 13 1101 D
6 0110 6 14 1110 E
7 0111 7 15 1111 F
RadixConversion• Theprocessofconver2ngabasetoanother.• Toconvertadecimalnumbertoanyothernumbersystem,dividethedecimalnumberbythebaseofthedes2na2onnumbersystem.Repeattheprocessun2lthequo2entbecomeszero.Andnotedowntheremaindersinthereverseorder.
• Toconvertfromanyothernumbersystemtodecimal,taketheposi2onalvalue,mul2plybythedigitandadd.
RadixConversion
RadixConversion
DecimaltoBinaryConversionofFractions• Division–Mul2plica2onMethod• Stepstobefollowed
• Mul2plythedecimalfrac2onby2andno2ngtheintegralpartoftheproduct
• Con2nuetomul2plyby2aslongastheresul2ngproductisnotequaltozero.
• Whentheobtainedproductisequaltozero,thebinaryofthenumberconsistsoftheintegralpartlistedfromtoptobocomintheordertheywererecorded.
• Example1:Convert0.375toitsbinaryequivalent
Multiplication Product Integral part 0.375 x 2 0.75 0 0.75 x 2 1.5 1 0.5 x 2 1.0 1
0.37510 is equivalent to 0.0112
Exercises• Convertthefollowingdecimalnumbersintobinaryandhexadecimalnumbers:1. 1282. 207
• Convertthefollowingbinarynumbersintodecimalandhexadecimalnumbers:1. 111110002. 1110110
Exercises• Convertthenumberinbinary(110110)intooctalandhexformat.• Inoctal(base8)• InHexadecimal(base16)
• Convertthenumberinbinary(1110110)intooctalandhexformat.• Inoctal(base8)• InHexadecimal(base16)
Exercises• Convertdecimal12.75tobinaryrepresenta2on
• Convertbinarynumber1010.0011intodecimalrepresenta2on
FastConversionBinarytoPowerof2Base• Ifyouhaveabinarynumbertobeconvertedintobasewhichispowerof2,• Splitthenumberinagroupbeginningfromtherightbythefactorofpower«n»(2n)
• Thenconvertthebinarygroupdirectlytothepowerof2base
• Example
• (100110010)2=(……)8• (1100110)2=(……)8
FastConversionBinarytoPowerof2Base
• Examples
• (10110010)2=(……)16• (1100110)2=(……)16
FastConversionPowerof2BasetoBinary• Ifyouhaveanumber,whichisapowerof2,tobeconvertedintobasetwo,• Spliteachdigitofthenumber,• Thenconverteahdigitdirectlytobinarynumberwithndigits
• Wherenisthepowerfactor
• Examples
• (53227)8=(……)2• (125)8=(……)2• (AD2)16=(……)2• (C3)16=(……)2•
WhataboutOctaltoHexConversion• Examples• (125)8=(……)16• (125)16=(……)8
THANKSFORYOURATTENTION!