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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

p>

一一01

OneTwo02

OneThree03

OneFour04

OneFive05

OneLiu06

OneSeven07

OneEight08

一九09

二一02

TwoTwo04

TwoThree06

TwentyFour08

TwentyFive10

TwentySix12

TwoSeven14

TwoEight16

TwoNine18

三一03

三二06

ThreeThree09

ThreeFour12

ThreeFive15

March18

March21

March24

Sanjiu27

FourOne04

FourTwo08

FourThree12

FourFour16

FourFive20

FourSix24

FourSeven28

FourEight32

FourNine36

May105

May210

FiveThree15

FiveFour20

FiveFive25

FiveSix30

FiveSeven35

FiveEight40

FiveNine45

6106

SixTwo12

SixThree18

SixFour24

SixFive30

SixSix36

SixSeven42

Sixeight48

Sixnine54

SevenOne07

SevenTwo14

SevenThree21

SevenFour28

SevenFive35

SevenSix42

SevenSeven49

SevenEight56

SevenNine63

AugustOne08

AugustTwo16

AugustThree24

AugustFour32

EightFive40

EightSix48

EightSeven56

八八64

八九72

九一09

九二18

NineThree27

NineFour36

NineFive45

NineSix54

NineSeven63

Nine-Eight72

Nine-Eight81

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

p>

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

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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.

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