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
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 | p>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
p>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.