Binary - techintroduce

Countingsystem

Base

Thecountingsystematthebaseb(wherebisApositivenaturalnumberiscalledabase),andbbasicsymbols(ornumbers)correspondtothesmallestbnaturalnumbersincluding0.Togenerateothernumbers,thepositionofthesymbolinthenumbermustbeused.Thesymbolofthelastdigitusesitsownvalue,andthevalueofonedigittotheleftismultipliedbyb.Generallyspeaking,ifbisthebase,weexpressthenumberinthebbasesystemintheformof,andwritethenumbersinordera₀a₁a₂a₃...a_k.Thesenumbersarenaturalnumbersfrom0tob-1.

Generallyspeaking,thenumberinthebhexadecimalsystemhasthefollowingform:

Number

Andaretheproportionsofthecorrespondingnumbers.

Binarycounting

TheGermanmathematicianLeibnizfromthe17thtothe18thcenturywasthefirstpersonintheworldtoproposethebinarynotation.Usebinarynotation,onlyusethetwosymbolsof0and1,noothersymbolsareneeded.

Binarydataalsoadoptsthepositioncountingmethod,anditspositionweightisapowerof2asthebase.Forexample,thebinarydata110.11,every2enters1,theweightorderis2²,2¹,2º,,.Forbinarydatawithn-digitintegersandm-digitdecimals,expressedasaweightedcoefficientexpansion,itcanbewrittenas:

Binarydatacangenerallybewrittenas：

[Example]:Writethebinarydata111.01intheformofweightingcoefficient.

Solution:

Binary,hexadecimal,andoctalarethesame,andtheyareallcarriedinthepoweroftwo.

Binary

Operation

Addition

Therearefourcasesofbinaryaddition:0+0=0,0+1=1,1+0=1,1+1=10(0isroundedto1).

Multiplication

Therearefourcasesofbinarymultiplication:0×0=0,1×0=0,0×1=0,1×1=1.

Subtraction

Therearefourcasesofbinarysubtraction:0-0=0,1-0=1,1-1=0,0-1=1.

Division

Therearetwocasesofbinarydivision(thedivisorcanonlybe1):0÷1=0,1÷1=1.

Example

Thearithmeticoperationoftwobinarynumbers1001and0101canbeexpressedas:

Binaryconversion

Binaryconversiontodecimal

Method:"Expandsumbyweight".Thespecificstepsofthismethodaretofirstwritebinarynumbersasweightedcoefficientexpansions,andthensumthemaccordingtotheadditionrulesofthedecimalsystem.

[Example]:

Rule:Thenumberofdigitsintheonesplaceis0,thenumberoftens'digitsis1,......,increasingsequentially,andthenumberofdigitsinthetenthplaceis-1,andthenumberofdigitsinthepercentileis-2,...,indescendingorder.

Convertingdecimaltobinary

Adecimalnumberisconvertedtoabinarynumbertobeconvertedintointegerpartanddecimalpartseparately,andfinallycombinedtogether.

Theintegerpartadoptsthemethodof"divideby2andtaketheremainder,arrangeinreverseorder".Thespecificmethodis:Dividethedecimalintegerby2togetaquotientandremainder;thendividethequotientby2togetaquotientandremainder.Continuethiswayuntilthequotientislessthan1,andthenusethefirstobtainedremainderasbinaryThelow-significantdigitsofthenumber,andtheremainderobtainedasthehigh-significantdigitsofthebinarynumber,arearrangedinsequence.Example:125.

Thedecimalpartshouldusethe"multiplyby2roundingmethod".Thatis,multiplythedecimalfractionby2andtakeawaytheintegeroftheresult(itmustbe0or1),andthenrepeatthepreviousstepswiththeremainingdecimalsuntiltheremainingdecimalsare0andstop,andfinallytheintegersobtainedeachtimeThepartsarearrangedinorderfromlefttorighttogetthecorrespondingbinarydecimal.Forexample,theprocessofconvertingadecimaldecimalnumber0.8125intoabinarydecimalnumberisasfollows:

Universalhexadecimalconversion

Theessenceofconversionbetweendifferenthexadecimalsistodeterminethedigitsatdifferentweightpositions.Thereisasimplealgorithmforconvertingthebaseofapositiveinteger,thatis,byusingthetargetbaseforlongdivision;theremaindergivesthe"number"startingfromthelowestdigit.Forexample,1020304fromdecimalto7:

Anotherexample,10110111from2to5:

Thereasonwhythecomputerusesbinary

Firstofall,thebinarysystemusesonlytwodigits.0and1,soanyelementwithtwodifferentstablestatescanbeusedtorepresentacertaindigitofanumber.Infact,therearemanycomponentswithtwoobviousstablestates.Forexample,the"on"and"off"oftheneonlamp;the"on"and"off"oftheswitch;the"high"and"low","positive"and"negative"ofthevoltage;the"holes"and"Nohole";"signal"and"nosignal"inthecircuit;thesouthandnorthpolesofmagneticmaterials,etc.,tonameafew.Itiseasytousethesedistinctstatestorepresentnumbers.Notonlythat,butmoreimportantly,thetwocompletelydifferentstatesarenotonlyquantitativelydifferent,butalsoqualitativelydifferent.Inthisway,theanti-interferenceabilityofthemachinecanbegreatlyimproved,andthereliabilitycanbeimproved.Itismuchmoredifficulttofindasimpleandreliabledevicethatcanexpressmorethantwostates.

Secondly,thefourarithmeticrulesofthebinarycountingsystemareverysimple.Inaddition,thefourarithmeticoperationscanbesummedupasadditionoperationsandshifts.Inthisway,thearithmeticcircuitintheelectroniccomputerbecomesverysimple.Notonlythat,thelineissimplified,andthespeedcanbeincreased.Thisisalsoincomparablewiththedecimalcountingsystem.

Third,theuseofbinaryrepresentationinelectroniccomputerscansaveequipment.Itcanbeprovedtheoreticallythatthethree-digitsystemsavesthemostequipment,followedbythebinarysystem.However,becausethebinarysystemhasadvantagesthatothercarrysystems,includingtheternarysystem,donothave,mostelectroniccomputersstillusebinarysystems.Inaddition,sinceonlytwosymbols"0"and"1"areusedinthebinarysystem,Booleanalgebracanbeusedtoanalyzeandsynthesizethelogiccircuitsinthemachine.Thisprovidesaveryusefultoolfordesigningelectroniccomputercircuits.

Fourth,thebinarysymbols"1"and"0"correspondtothe"true"and"false"(false)inlogicaloperations,whichisconvenientforcomputerstoperformlogicaloperations.