Abacus formula

Abacus

ThewordappearedinLiuYin[1248-1293]"Mr.JingxiuCollection"inYuanDynasty

Thetitleofafive-characterquatrain;

TheYuanDynastypainterWangZhenpeng's"ApictureoftheUniverseandOneDan"[1310]ThereisanabacusinthegoodsoftheZhonghuLangDan;

AttheendoftheYuanDynasty,TaoZongyi's"SouthVillageStopsFarmingRecord"[1366]VolumeThereisametaphorfor"abacus"inthetwenty-nine"wellbead"article;

The"abacus"isalsomentionedintheYuantune,whichshowsthattheabacushasbeenusedintheYuanDynasty.

Theabacusfirstappearedinthefamouswork"ShangHeTuduringtheQingMingDynasty"duringtheNorthernSongDynastybyZhangZeduan.Inthepicturescroll,theabacusplacedonthecounteroftheshoponChangjieinthecapitalcityofBianjingisexactlythesameasthemodernabacus.Theprofessionaldocumentcontainingtheabacusdiagramisthebook"QueiBen'sCorrespondencewithFourWordsandMiscellaneousCharacters"engravedinthefourthyearofHongwu[1371]oftheMingDynasty.TheearliestsurvivingabacusbookisXuXinlu'srevised"PanzhuAlgorithm"[1573].Themostpopularandmosteffectiveabacusbookinhistory

isthe"StraightFingerAlgorithms"[1592]compiledbyChengDaweiintheMingDynasty.

Theadditionandsubtractionformulaisuniquetoabacus,andwasfirstseeninWuJing's"NineChaptersofAlgorithmComparison"[1450].

Theformulaofmultiplicationanddivisionistheformulaofcalculation.

The"Nine-Nine"formulaofmultiplicationhasbeenusedincalculationsduringtheSpringandAutumnPeriodandtheWarringStatesPeriod;

TheformulafordivisionisthefirsttoseeYangHui's"MultiplicationandDividingTongbianbao"[1274],

ZhuShijie’s"MathematicsEnlightenment"[1299]containedthenine-guidanceformulabasicallythesameasthemodernone.

Withfourformulas,theabacusalgorithmhasformedasystem,whichhasbeenusedforalongtime.

NineNineFormulas

Additionformula

Number

Addwithoutcarry

Addwithcarry

Directadd

Fullfiveadd

Tenplusplus

Breakfiveplustenplus

One

Oneup,one

Onegotofivetofour

Onegotonineentersone

Two

Twoontwo

Twounderfivetothree

Twotoeightinone

Three

ThreeonThree

Threeunderfivetotwo

Threetoseveninone

four

Fouronfour

Fourunderfivetoone

Fourtosixinone

Five

FiveonFive

Fivetofiveinone

Six

Sixonsix

Sixtofourinone

Sixononetofiveinone

Seven

Sevenonseven

Seventothreeinone

Sevenuptotwotofiveinone

eight

eightoneight

Eightgotwoinone

eightonthreeGotofiveinone

nine

Nineonnine

Ninetooneentersone

Nineonfourtofiveintoone

Subtractionformula

Decrease

Decreasewithoutabdication

AbdicatedMinus

Directreduction

Brokenfiveminus

Retreatminus

Returntentocomplementfiveminus

One

Oneclick

Oneupfourtofive

Onerefund,onereturnnine

Two

Twodowntwo

Twoupthreetofive

Tworefunds,onerefund,eight

Three

Threeunderthree

Threeup,twotofive

Threerefundsandonereturnseven

Four

Fourunderfour

Fourup,onegotofive

Fourrefundsandonereturnsix

Five

Fiveunderfive

Fiverefundsandonerefund

Six

Sixundersix

Sixrefunds,onerefund,four

Sixrefunds,onerefund,fivetoone

Seven

Sevenunderseven

Sevenrefundsandonerefund

SevenrefundsandonerefundFivetotwo

Eight

EightNexteight

Eightreturn,onereturn,two

Eightrefunds,onereturn,fivetothree

Nine

Nineundernine

Ninerefunds,onerefund

Ninerefunds,onereturn,fivetofour

ExceptSongJue

ZhuShijie's"MathematicsEnlightenment"(1299)volume"returningtoremovethesongtactics"...

Asoneentersintoone,oneentersintoten

TwooneaddsfivetimesTwointofourteenintotwenty-sixintothirty-eightintoforty

Threeonethirtyonethreetwosixtytwoeverythreeintosixteenintotwenty-nineintothirty

FourOneTwentyTwoFourTwoAddsFiveFourThreeSeventyTwoEveryFourAdvancesEighteenIn20>SixoneclickaddfoursixtwothirtytwosixthreeaddfivesixfoursixtyfoursixfiveeightytwoeverysixintotenFortytwosevenfourfiftyfivesevenfiveseventyonesevensixeightyfoureverysevencomesintotenEightFiveSixTwoEightSixSeventyFourEightSevenEightSixp>Inthebook"EnlightenmentofMathematicalStudies"(1299)byZhuShijie,thegreatmathematicianoftheYuanDynasty,itisfurtheradvancedintothefollowing15sentences:

Oneask,sixandtwenty-fiveeveryotherposition;(1/16=0.0625)

Tworequests,oneabdicationTwenty-five;(2/16=0.125)

Threeseeking,eightseven-fivenotes;(3/16=0.1875)

Fourseeking,changetotwenty-five;(4/16=0.25)

Fiveseeking,threeonetwofiveis;(5/16=0.3125)

Sixseeking,twopricethreesevenfive;(6/16=0.375)

Sevenasks,four,three,sevenandfivesets;(7/16=0.4375)

Eightasks,turnsintofive;(8/16=0.5)

Nineseeking,fivesixtwofive;(9/16=0.5625)

Tenseeking,sixtwofive;(10/16=0.625)

11seeking,Six,eight,seven,five;(11/16=0.6875)

12,seven,five;(12/16=0.75)

13,eightone,twofive;(13/16=0.8125)

14,eight,seven,five;(14/16=0.875)

15,nine,three,seven,five;(15/16=0.9375)abacus,division,abacusMantratable

Abacusdivision

Abacusdivisionhastwotypes:subtractionandquotientdivision.

Forsubtraction,usemantrasforcalculation.Therearenineformulasandreturnquotient.MantraandShangJiuMantra.

Thereare61sentencesinJiuguiMantra:

Yigui(divideby1):enteroneeveryone,entertwoeverytwo,enterthreeeverythree,Everyfourentersfour,everyfiveentersfive,everysixenterssix,everysevenentersseven,everyeightenterseight,andeverynineentersnine.

Tworeturn(divideby2):everytwoentersone,Everyfourenterstwo,everysixentersthree,everyeightentersfour,andtwooneaddsfive.

Threereturn(divideby3):everythreeentersone,everysixenterstwo,andeverynineentersthree,Threeonethreemorethanone,threetwosixmorethantwo.

Sigui(divideby4):enteroneeveryfour,entertwoeveryeight,addfourtwotofive,addfouronetwototwo,Four,three,sevenandmorethantwo.

FiveGui(divideby5):everyfiveentersone,fivetimesmakestwotimes,fivetimesmakesfour,fivetimesthreetimesmakessix,fivetimesfourtimesmakeseight.

Liugui(divideby6):Everysixentersone,sixthreeaddsfive,sixoneaddsfour,sixtwothreemorethantwo,sixfoursixmorethanfour,sixfiveeightmorethantwo.

Qigui(divideby7):everysevenentersone,sevenaddsthree,seventwoaddssix,seventhreefourmorethantwo,sevenfourfivemorethanfive,sevenfivesevenmorethanone,Seven,six,eightandmorethanfour.

Bagui(divideby8):everyeightentersone,eightfouraddsfive,eightaddstwo,eighttwoaddsfour,eightthreeaddssix,Eightyfivesixmorethantwo,eightsixsevenmorethanfour,eightseveneightmorethansix.

Ninereturn(divideby9):everynineentersone,nineaddsoneatatime,ninetwoaddstwo,Addthreeunderninetythree,addfourunderninefour,addfiveundernine-five,addsixundernine-six,addsevenundernine-seven,addeightundernine-eight.

Quitbusinessformula

Returnonewithoutdeduction,returntwowithoutdeduction,returnthreewithoutdeduction,

Returnfourwithoutdeduction,fivewithoutdeduction,sixwithoutdeduction,returnsevenwithoutdeduction,returneightwithoutdeduction,andninewithoutdeduction.

Thereare9sentencesintheShangjiuformula:

Seeingonewithoutdividingintoninetyone,seeingtwowithoutdividingintoninetytwo,seeingthreewithoutdividingintoninetythree,

Seefourwithoutdivisionforninetyfour,fivewithoutdivisionforninefive,seesixwithoutdivisionforninety-six,seesevenwithoutdivisionforninety-seven,seeeightwithoutdivisionforninety-eight,seeNinenodivisionisninety-nine.

Thedivisionofasingledigitiscalled"singlereturn";thedivisionoftwoormoredivisorsiscalled"division",andthefirstdigitofthedivisoriscalled"gui.",Thefollowingpeoplearecalled"division".Forexample,thedivisionofthedivisoris534,whichiscalled"fivereturnthreeandfourdivision".Thatistosay,usethefivereturnformulatofindthequotient,andthendivideby34.

Abacusterm

Term1

Neutral:Whentheupperandlowerpartsofacertaingeararebothawayfromthebeam,itiscalledneutral.Neutralmeansthatthisgearisnotcounted,oritmeans0.

Emptyplate:Wheneachgearoftheabacusisinaneutralposition,itmeansthatthereisnocountinthewholeplate,whichiscalledanemptyplate.

Innerbeads:Theabacusbeadsthatcountbybeamsarecalledinnerbeads.

Outerbeads:Abacusbeadsthatdonotcountfromthebeamarecalledouterbeads.

Dialup:referstothelowerbeadagainstthebeam.

Pulldown:Referstopressingtheupperbeadagainstthebeam.

Pullingout:referstopullingtheupperorlowerbeadsawayfromthebeam.

Thisfile:referstothefilethatisabouttobedialedandcounted.

Frontgear:referstothepreviousgearofthisfile,alsocalledtheleftgear(position).

Backgear:Referstothenextgearofthisfile,alsocalledthefirstgear(position)ontheright.

Term2

Floatingbead:Theabacusbeadfloatinginthemiddleofthegearwithoutusingtoomuchforcewhendialingthebead.

Beads:Whendialingbeads,bringinoroutthebeadsthatshouldnotbedialedinoroutofthisfileorneighboringfiles.

Realbeads:abacusbeadswithabeamthatrepresentsapositivenumber.

Virtualbead:alsocallednegativebead,itreferstothehangingbeadthatissettoneitherthebeamnortheframe,indicatinganegativenumber.

Setthenumber:alsocalledthenumberofcloth,accordingtotherequirementsofthecalculation,dialthenumberintotheabacustoprepareforthecalculation.

Gearposition:Alsocalledgrade,itreferstothepositionofthegear.

Wronggear:Alsocalledmisalignment,itmeansthatthebeadisnotdialedintothegearthatshouldbedialedduringtheoperation.

Interval:Itisalsocalledtheinterval,whichreferstothesecondgear(position)withonegearleftandright.Forexample,whentwonumbersaremultipliedtogetherinbit-by-bitmultiplication,theonesdigitoftheproductisplacedontherighttwodigitsofthemultiplicand;inbit-by-bitdivision,thequotientofthebitisthefirsttwodigitsfromtheleftofthedividend.

Carry:Itmeansthatafteraddinganumbertothisfile,itisgreaterthanorequalto10,and1mustbeaddedtotheprecedingposition,whichiscalledcarry.

Termthree

Rebate:Itmeansthatwhenthecurrentfileisnotenoughwhenonenumberissubtractedfromthecurrentfile,itiscalledabdicationbysubtracting1fromthepreviousdigit.

Firstplace:Alsocalledthehighestplace,itmeansthatthefirstnon-zerodigitofamulti-digitnumberisthefirstplace.Suchas3in3284and7in0.0726.

Lastdigit:Alsocalledthelowestdigit,itreferstothelastnumberofamulti-digitnumber.Forexample,5in3275,0,481.29in120

9.

Secondplace:Thesecondnumberthatisessentiallyamulti-digitnumber.Suchas8in3865and1in0.4178.

Realnumbers:Inancientarithmeticbooks,themultiplicandandthedividendarecommonlyreferredtoasrealnumbers,referredtoasrealnumbers.

Lawnumber:Inancientarithmeticbooks,themultiplieranddivisorarecommonlyreferredtoasthelawnumber,whichisreferredtoasthemethod.

Multiplicationandaddition:referstothemultiplicationofeachdigitofthemultiplicandbythemultiplier,andthemultiplicationontheabacuswhileaddingthenumberofproducts.

Multiplicationandsubtraction:Alsocalledsubtraction,itmeansthateachquotientismultipliedbythedivisor,andtheproductissubtractedfromthedividend.

Exceptforthefirst:referstothehighestdigitofthedivisor.

Thefirstdigitoftheproduct:referstothefirstdigitoftheproductnumber.

Term4

Thefirstquotient:referstothefirstdigitofthequotient.

Estimator:Indivision,everyquotientneedstobecalculatedmentallytoestimatehowmanytimesthedividendisthedivisor.Thismentalarithmeticprocessiscalledquotient.

TrialQuotient:Alsocalledpreliminaryquotient,itmeansthatthequotientthatistoolargeorsmallisinitiallyobtainedwhenestimatingthequotient,whichiscalledtrialquotient.

Buybusiness:alsocalledLishang,itmeanstoputtrialbusinessintotheabacus.

Adjustthequotient:Afterbuyingthequotient,itisprovedbymultiplicationandsubtractionthatthetrialquotationisincorrectandtheinitialquotationneedstobeadjusted.

Confirmquotient:Afterbuyingaquotient,itisprovedbymultiplicationandsubtractionthatthetrialquotientisneitherbignorsmall.

Division:referstothedivisionofthedividendbythedivisor,andthedivisiontoacertainplace,whichhappenstohavenoremainder,iscalleddivision.

Inexhaustibledivision:Referstothedivisioncalculationthatcannotbedividedwhenthereisaninfinitecycleornocycleofdecimals.Forexample:1÷3=0.333……;1÷7=0.142857142857…….

TermFive

Remainder:Adivisionthatisnotevenlydivisible.Whenthequotientreacheseachbitorapredetermineddigit,theremainingnumberinthedividendiscalledtheremainder.Intheprocessofcalculation,everytimetheproductofthequotientandthedivisorinthesuburbofthedividendhasaremainingnumber,itisusuallycalledaremainder.

Refundofbusiness:Theinitialbusinessistoolarge,sochangeitto"Refundofbusiness".

Supplementaryquotient:Theinitialquotientistoosmall,soIchangeditto"makeupquotient".

Falsequotient:Inthedivisionoperation,fortheconvenienceofcalculation,aquotientisestablishedfirst,andthenthecorrectquotientisobtainedafteradjustment.Thequotientestablishedfirstiscalledafalsequotient.

Liquidation:Turnofftheabacusbeadsineachstalltomakethewholemarketempty,whichiscalledliquidation.

Comprehensiveexercises:practiceonallthefilesoftheabacus,oronmostfiles,andperformcomprehensiveexercisesaccordingtothebasicalgorithm,whichiscalledoverallexercises.