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1、NumberSystemsPositionalnumbersystemsWholenumbersFractionsAccuracyCopyright1993-95,ThomasP.SturmFundamentalsofComputerScience16NumberSystemsConceptofaPositionalNumberSystemWhatdoes4,752(base10)represent(inbase10)?4x100040004x103+7x1007007x102+5x10505x101+2x122x100
2、Sowhatdoes1011(base2)represent(inbase10)?1x881x230x400x221x221x211x111x20Whilethismakesanicedefinition,itisfartoomuchworkfor"practical"use-binarynumbersgettobelongFundamentalsofComputerScience16NumberSystemsCountingRevisitedDecimalBinaryOctalHexadecimal0000111121
3、02231133410044510155611066711177810001089100111910101012A11101113B12110014C13110115D14111016E15111117F161000020101710001211118100102212191001123132010100241421101012515221011026162310111271724110003018251100131192611010321A2711011331B2811100341C2911101351D3011110
4、361E3111111371F321000004020FundamentalsofComputerScience16NumberSystemsFasterConversionofBinarytoDecimalDibble-Dabble,Horner'sMethod,Algorithm1.11.Startattheleft,startwith02.Addinleadingdigit3.Ifthere'sanotherdigit,multiplytotalby2,addinnextdigit,repeatthisstepun
5、tilyoureachtheradix(binary)point.Ex:11010(binary)convertedtobase10is:0+11x22+13x26+06x212+113x226+026FundamentalsofComputerScience16NumberSystemsEvenFasterinYourHeadEx:11010010110(binary)convertedtobase10is:(inyourhead1,3,6,13,26,52,105,210,421,843,1686)Whichhasj
6、ustgottobealotquickerthan1024+512+128+16+4+2(notevencountingthetimeittakestogeneratethepowersof2)FundamentalsofComputerScience16NumberSystemsFastDecimaltoBinaryConversion(Dibble-Dabble,Horner'sMethod,Algorithm1.2)Sinceallevennumbersendin0(binary),andalloddnumbers
7、endin1(binary),andsincedivisionby2yieldsa0remainderforevennumbersanda1remainderforoddnumbers,theremainderafterdivisionby2givesyoutheLSB(leastsignificantbit)ofthecorrespondingbinarynumber.Justrepeatthisprocessuntilyougeta0quotientEx:Convert75(decimal)tobinary2
8、752
9、
10、37+12
11、18+12
12、9+02
13、4+12
14、2+02
15、1+00+1Now,thetricktoremember,theLSBisONTOP,soreadtheanswerfromthebottomup:75(decimal)=1001011(binary)FundamentalsofComputerScience1
