QUICK(ER) CALCULATIONS
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ToKath
Thesumthattwomarriedpeopleowetooneanotherdefiescalculation.Itisaninfinite debtthatcanonlybedischargedthroughalleternity.
JohannWolfgangvonGoethe, ElectiveAffinities Book1,Chapter9
ForMary,Ann,Clare,Therese,andPeter “DadwatchingyoudomathhomeworkislikeMichaelPhelpswatchingyoudrown.”
Preface
Multiplicationisvexation
Divisionisasbad
Theruleofthreedothpuzzleme Andpracticedrivesmemad.
FromanElizabethanmanuscriptof1570 (asquotedin Bartlett’sFamiliarQuotations 15thand125thAnniversaryedition (Boston,MA:LittleBrown,1980),p.917)
Coffeeshopscrowdontoeverycityblock,orsoitseems.Somediscerningsoulsrefusetodrinkatsuchplaces.Instead,theypurchasefair-trade beans,grindthemup,anduseaFrenchpresstoproducethelife-giving liquid.Othersforegothebriochesoldbysupermarkets,preferringto buytheirbakedgoodsfromsmallcraftbakeries.
Whatbetterwaytoaccompanyyourhand-craftedcoffeeandcroissantthanwithartisanalarithmetic?Stowyourcomputerandusethe mostaffordableandecofriendlycalculatoryoupossess—yourbrain. Thisbookshowsyouhowtocarryoutsimplearithmetic—addition, subtraction,multiplication,division—aswellashowtosquareorto findroots(square,cubic,andevenquintic)quickly.Faster,infact,than youcouldwalkacrosstheroom,findacalculator,putinthenumbers, andhitthebigredbuttonwithaConit.Oruseyourlandlinetocall yoursmartphone,findit,andgettothecalculatorfunction.
Whybother?Ioffertwodifferentanswers,neitherofwhichisoriginal. First,“Forthosewhobelieve,noexplanationisnecessary;forthose whodonotbelieve,noexplanationispossible,”whichisvariously attributed,inchronologicalorder,toSaintThomasAquinas,Saint IgnatiusofLoyola,andFranzWerfel—thescript-writerforthemovie TheSongofBernadette.Inthiscontext,thosewhohavefunfrolickingwith formulasneednoexplanationforwhytheyshouldattempttocarryout fantasticallyfastarithmetic.
ThesecondanswerbelongstoSirGeorgeMallory,whorepeatedly triedtoclimbMountEverest,andwhoeventuallydiedonitsslopes. Whenaskedwhyhemadesomanyattempts,Malloryreplied“Because it’sthere”—aphraseoftenmisattributedtoSirEdmundHilary,who, withTenzingNorgay,firstconqueredthemountainin1953.Inother
words,weshouldpracticequickcalculationssimplybecausewecan. Afterall,internationalgrandmastersplaychessnotonlyagainstother internationalgrandmasters,butalsoagainsttheclock.1 Likewise,inthe cutthroatworldofcrosswordcompetitions,it’snotsufficientmerely tocomplete,say, TheTimes or TheNewYorkTimes puzzle—youhaveto dosofasterthaneveryoneelsedoes.So,whynotcalculateagainst theclock?Ihopethisshortbookwillbeastepinthatdirection.As afurtherincentive,every2years,the WeltmeisterschaftenimKopfrechnen (looselytranslatedastheWorldCupinMentalArithmetic)takesplace inGermany.Togiveaflavorofthecompetition,eventsincludeadding 10ten-digitnumbers,multiplying2eight-digitnumbers,andcalendar calculations(inwhichcontestantsperusealistofdatesfrom1600–2100 andhave1minutetostatecorrectlyonwhichdayoftheweektheyfell). Toprovideabenchmark,thecurrentWorldRecordstandsat66correct answersintheallotted60seconds.Asforcalendricalcalculations,see “InterludeIII:Doomsday”lateroninthisbook.
SirEdmundHillary,togetherwithCharlesLindbergh—thefirstpersontoflysoloacrosstheAtlantic—showthatrapidmentalmathematicscansaveyourlife.Hillary,ashereportsinhisbook HighAdventure, pushedontothesummitofMountEverest,butknewoxygenwasrunninglow.Everest(orSagarmathainNepalese),rises29,029feet(8,840m) abovesealevel—althoughChinaandNepaldisputetheactualheightof themountain,whosesummitformspartoftheirborder.Shouldheand Tenzingcontinue,orretracetheirstepsdownthemountain?Hillary’s quickestimateconvincedhimtogoon.Theydidso,andclaimedthe honorofbeingthefirsttwopeopletostandatoptheworld’stallest mountain.
Lindberghhadasimilarpredicament,requiringrapidaviationarithmetic:didhehaveenoughfueltomakeitovertheocean?Awrong answerwouldhavespelleddisaster.The450gallons(1,704liters)offuel inhisplane,hereckonedcorrectly,sufficedtopowerthe SpiritofSt.Louis fromRooseveltField,NewYork(nowthehomeofashoppingmall)to Paris,touchingdownattheLeBourgetAerodromeontheeveningof May21,1927.
1 There’saformulatoratechessprowess,theEloratingsystem.Ifyouplay N games,win W,lose L,andplayopponentswhoseratingssumto R,thenyourrating is [R + 400 (W L)] /N .Asyoucan’tlosemoregamesthanyouplay,yourscorewill increaseifyouplayonlythosepeoplewhoseratingsarehigherthan400.
Therearepracticalbenefits—lessdrasticthansavingalife—incalculatingmorequickly.Inaneraoftimedmultiple-choicetests,the quickeryouperformaparticularmathematicaloperation,themore timeyoucandevotetotheotherquestions.That’swhysomesections inthisbookaredevotednotto exact answers,butto approximate ones.If youcancalculateanapproximateanswerswiftlytowithinacouple ofpercentoftheactualanswer,that’softensufficienttodetermine whichmultiple-choiceanswertoselect.Oratleastthatpermitsyou toeliminateacoupleofanswers,therebyincreasingyourchancesof plumpingfortherightonepurelybyguesswork.
IsaacNewtonwrotethatifhehadseenfurtherthananyoneelse,it wasbecausehehadstoodontheshouldersofgiants.Likewise,mathematicaltechniquestocalculatequicklyarenotrecentinventions.They comefromavarietyofculturesandfromacrosstheglobe.Western scholarsonlyrecentlybegantopayseriousattentiontothemathematicsofothercultures,givingrisetothesubjectofethnomathematics. Likewise,thehistoryofmathematicsrevealsmanydifferentwaysthat non-Europeanculturesdevisedtomultiply,divide,andextractsquare roots,someofwhichareincludedinthisslimvolume.Iamprivileged togiveaglimpseintohowothercultureshavecalculated,andhope somereadersmaydelvefurtherintothetopic.Therearesomefresh techniquesoriginaltothisbook,however,suchasthesectiononmultiplyinganddividingbyirrationalnumbersandtheentriesonsquaring certainthree-digitnumbers.
Ihopethat Quick(er)Calculations achieveswhattheBBCsetsoutto do:inform,educate,andentertain.Bytheend,Itrustthatyouwillbe abletocomputeanswerstobasicarithmeticalproblemsfarfasterthan youhaveuptothispoint.Beingabletodosomaybeofgreathelpfor thosestillinhighschool,studyingmathematics,physics,orchemistry. Itshouldalsohelpthosewhohavealreadyenteredintoprofessions,who arecard-carryingscientists,engineers,accountants,andactuaries.Ialso hopethebookwillfamiliarizeyouwith,andgiveyoumoreappreciation for,thoseofdifferenttimesandplaceswhohavedevisedingeniousways tocalculateanswerstoquestionswestillposetoday.
TrevorLipscombe Baltimore,Maryland FeastoftheAssumptionoftheBlessedVirginMary,2020
Acknowledgments
Anotherdamnedthicksquarebook!Alwaysscribble,scribble, scribble!Eh!Mr.Gibbon.
WilliamHenry,DukeofGloucester, afterreceivingvolumeIIofGibbon’s DeclineandFalloftheRomanEmpire
Myfamilyisusuallypleasantandpeaceful.Exceptduringthecardgame “Pounce.”Intensecompetitiontakescontroland,withtheroundcompleted,I’minvariablyinlastplace.Somethingoddhappens,though, whenwetotupthescores.Someonesays,“Addon30,takeoff4,”or somethinglikeit,pokinggentlefunatthewayIdomath.Onenight, afterbeingdrubbedyetagain,theideaforthisbookpounced(badpun intended)uponme.
Iowethanks,then,toQuickHandsMcGee,theEvens,andtheOdds (youknowwhoyouare)forservingasmyinspiration.Ialsothank themforallthetimeswe’vespentovertheyearsenduringthatmost unenjoyableofthings,mathematicshomework.It’stherethatmykids sufferedwhileIdidthewhole“Takeoff60,addon3”routine,and theyputupwithmyoft-ventedfrustrationthatno-oneteacheselegant mathematicsingradeschool,middleschool,highschool. ...
Myownhigh-schoolmathematicsteachersdeserveagreatdealof thanks.JohnEvansandKenThomasoftheHughChristieSchoolin Tonbridge,Kent,werealwaysencouragingandhelpedsteermesuccessfullydowntheroadtouniversity.Ihopethisbookdoesthem, andtheschool,proud,andhelpstheircurrentandfuturestudentsdo arithmeticmorequickly.
Myschoolreport,fromwhenIwas10yearsold,asserts“Trevormust bemuchmorecarefulinMechanicalArithmetic.Hemakesmorecarelesserrorsthanheshould.”ItturnsoutthatwhatMissWatsonwrote atIcknieldPrimarySchoolinLutonseveraldecadesagoremainstrue today.Iamespeciallythankful,then,forCamilleBramall’sattentive copyediting;shehasgraciouslyrootedoutmorecarelesserrorsthan Ishouldeverhavecommitted.Thosethatremainaremyfaultalone, servingassuresignsIshouldhavepaidmoreattentiontowhatMiss Watsonwastryingtoteachme.
MygratitudealsoextendstothoselaboringinthevineyardsofOxford UniversityPress,inparticularSonkeAdlungandKatherineWard,for seeingsomethingofmeritinmyefforts.Ihopeyouwill,too.
Multiplyordivideby647
Multiplyordivideby748
Multiplyordivideby851
Multiplyby952
Multiplyby1154
Multiplyordivideby1256
Multiplyby1357
Multiplyby1457
Multiplyordivideby1558
Multiplyordivideby15,25,35,ornumbersendingin559
Multiplyby16,26,36,ornumbersendingin662
Multiplyby18,27,36,ormultiplesof962
Multiplyby19,29,39,ornumbersendingin963
Trythese 65
Multiplyby21,31,41,ornumbersendingin166
Multiplyby32,42,52,ornumbersendingina267
Multiplyordivideby7568
Trythese 69
Multiplyby11171
Multiplyordivideby12573
Multiplyby316 2 3 ,633 1 3 ,and95073
Multiplyby999or1,00174
5.CalculationswithConstraints:MultiplyandDivide byNumberswithSpecificProperties81
Multiplytwonumbers,10apart,endingin581
Multiplytwonumbers,20apart,endingin582
Multiplyaone-ortwo-digitnumber,lessthan50,by9883
Multiplyaone-ortwo-digitnumberby9985
Multiplyaone-ortwo-digitnumberby10185
Multiplyaone-ortwo-digitnumber,lessthan50,by10286
Trythese
86
Multiplytwonumbersthatdifferby2,4,6,or2087
Multiplytwo“kindred”numbers89
Multiplyby23,34,45,ornumbers“remainder1” whendividedby1190
Multiplyby24,36,42,48,ornumberswhereone digitistwicetheother90
Multiplytwonumbers(orsquareanumber) justunder10092
Multiplytwonumbers(orsquareanumber) justover10092
Multiplytwonumberseithersideof10093
Multiplytogethertwotwo-digitnumbers94
InterludeIV:MulticulturalMultiplication
6.SuperPowers:CalculateSquares,SquareRoots, CubeRoots,andMore103
Squareanytwo-digitnumberendingin1or9105
Squareanynumberendingin5106
Squareanynumberendingin6or4107
Squareanytwo-digitnumber108
Squareanythree-digitnumberwithmiddledigit0or9109
Squareanythree-digitnumberwithmiddledigit4or5110
Squaring,sortof
Squareroots,Babylonianstyle113
Squareroots,Chinesestyle115
Squareroots,Indianstyle116
Squareroots,Frenchstyle118
Findthesquarerootofamysteryperfectsquare121
Findthecuberootofamysteryperfectcube122
Findtherootofamysteryfifthpower122
Trythese 123
7.Close-EnoughCalculations:QuickandAccurate Approximations125 Divideby9125 Divideby11127
Introduction
Thedifferentbranchesofarithmetic—ambition,distraction, uglification,andderision.
TheMockTurtle, Alice’sAdventuresinWonderland,byLewisCarroll.
Mymother—aformidableLondonerwholivedthroughtheBlitzand whowoulddelightinbeingdescribedasaNoNonsensePerson—hadan unusualwayoftellingtime.Iftheclockshowed6:25,sheproclaimedit “Fiveandtwentypastsix.”Tenminuteslater,naturally,shereported thetimeas“Fiveandtwentytoseven.”
Mymum’swaytoreporttimeshowsaninsight.Namely,sometimes it’seasiertobreakacalculationupintotwosteps,ratherthanone. Ratherthanadding25,insteadadd20,thenadd5.Dissolvingadifficult problemintotwosimpleronesliesatthecoreofacceleratingyour arithmetic.Twootherkeyingredients—which,dependingonyour viewpoint,areeithertrivialandobviousordeeplyprofound—isthat youcanadd0toanysumwithoutdoinganyharm,ormultiplyany expressionby1withoutchangingtheoutcome.Enigmatically,much dependsonwhatthe0andthe1happentobe.Whenpresentedwitha complicatedsum,itmaybeeasiertoadd3toit,tocreateafarsimpler addition,andthensubtract3attheend.As3 3 = 0,youhaveadded 0—andthushavethesameanswer—buthavemadethecalculation quicker.Likewise,forsomedivisions,multiplyingthetopby4may makelifeeasierand,ifyoudivideby4lateron,youwillhavemultiplied by4/4,whichis1,leavingtheanswerunchanged.Allshallberevealed.
Thisbookbeginswithabriefglanceathowyoumaycalculate currently,anddispensesadviceonhowtoimprovebothspeedand accuracy.Thencomesachapterdevotedtoadditionsandsubtractions, beforeturningtotheaccountingchallengeofsumminguplong columnsofnumbersatspeed.Fromthisfollowsabird-by-birdguide, sotospeak,tomultiplicationanddivisionbyspecificnumbers.Someof thetipsandtechniquesholdformorethanonenumberso,ifyouread straightthroughthatsection,therewillbesomerepetition.Thisisgood formakingapoint,butitalsomeansthatatanytimeyouwishtolearna techniqueforaspecificnumber,youcansimplygotothenumberitself
andseewhattodo.Thenextdivision(punintended),dealswithnonspecificnumbers.Thingssuchas“multiplyingtwonumbers,bothof whicharejustover100.”Theselendthemselvestoimpressingfriends, relatives,students,orevenanaudience(“givemetwonumbers,any twonumbers,between100and110...”).
Practicalapplicationsforsquarerootsabound,especiallyifyouknow ofPythagoras’theorem.Thisgivesrise,then,toachapterdevotedto findingthesquareroot,approximately,ofagivennumber.Thissection alsodealswithanotherwaytoimpresspeople,inthattheycanuse acalculatortosquare,cube,orraisetothefifthpoweranytwo-digit numbertheychoose,andyoucandivinethatoriginalnumberina matterofsecondsusingonlyyourbrain—oftensimplybyinspection.
Thebookcloseswithsomeapproximatewaystomultiplyanddivide. Forexample,todivideby17,youcangetfairlyclose(within2percent) bymultiplyingthenumberby6anddividingby100.Therearealsoa numberofwaystocalculatemultiplesof √2,theGoldenRatio, e,and π .Formoredetails,readon!
Scatteredwithinthesesectionsareinterludes.Thesearedigressions, hopefullyenjoyable,thattouchlightlyonthemoretechnicalmaterialinthebook.Theydealwithsupermarketcheckouts,determiningwhatdayoftheweekaparticulardatefellon(usefulforthose mullingarunattheWorldCupinMentalArithmetic),andalook atcalculationsthroughnon-modernornon-Westerneyes.Theinterludesalsopresentamathematicaltrickinvolvingthenumber111,111 and—alwaysimportant—thewaystoestimatethenumberofextraterrestriallifeforms.
Thechallengeinwritingabookonhowtoadd,subtract,multiply, divide,andfindsquarerootsistodosoinawaythatitdoesn’tendup withthethrillingprosefoundinrentalagreementsortheinstructions thatcomewithdo-it-yourselffurniture.Pepperedwithineachchapter, then,areanumberoffanciful“applications.”Theseincludethenumber ofpeoplewhoaccompaniedKingHenryVIIItoFranceandtookpart inoneofhistory’sgreatestbinges,theFieldoftheClothofGold;the amountofbreadconsumedannuallybythecourtofKingCharlesI; thenumberofgoalsscoredbyCharltonAthleticduringtheiryearsin thePremierLeague;andthenumberoflinesofpoetryinShakespeare’s sonnets.Whilethesearefar-fetched,youcanapplythemethodsinthe booktocalculatetheanswers.Arguablymoreimportant,theyshow thatyoucanfindopportunities,orexcuses,tousequick(er)calculations almosteverywhere.
Challenge
Theaimofthisbookistohelpyoucalculatemorequickly.Tobegin,I inviteyoutorisetothischallenge.Trythe25questionsbelow(26ifyou feeluptoit),beforereadingthebook,andseehowlongittakes.Once you’vereadthebookandlearnedthestreamlinedwaystocalculate, trythemagaintoseehowmanyseconds,orminutes,youhaveshaved fromyourtime.Tohelp,sectionsofthebookthatmaylaterassistyou aregiveninparentheses.
Pencilandbrainsharp?Go!
1.171–46(Subtractbyadding)
2.1.27 + 0.98(Two-stepsubtraction)
3.Thebirthdaysofmyfamilyfallonthefollowingdaysofthe month.Addthem:14,17,2,23,13,20,27(Tallyingintens)
4.Thefirst10Fibonaccinumbersare1,1,2,3,5,8,13,21,34,55. Addthem(Addingnumbersmystically)
5.Oneofmykids,preparingtorunamarathon,kepta“joglog” totrackhermiles.Overa10-dayperiod,sheranthefollowing numberofmileseachday:8,3,2,3,9.5,4,4,3,2,3.Howmany milesdidsherunintotal?(Cancelasyoucalculate)
6.87 × 1.1(Multiplyby11)
7.4.1 × 1,800(Multiplyby18)
8.53 × 2.9(Multiplyby19,29,39...)
9.2,100 × 4.3(Multiplyby21,31,41...)
10.3,018/7.5(Multiplyordivideby75)
11.620 × 11.1(Multiplyby111)
12.9.5 × 130(Multiplyby316 2 3 ,633 1 3 ,and950)
13.8.5 × 75(Multiplytwonumbers,10apart,endingin5)
14.430 × 0.98(Multiplyaone-ortwo-digitnumber, lessthan50,by98)
15.27 × 2.9(Multiplytwonumbersthatdifferby2,4,6,or20)
16.2,300 × 2.7(Multiplytwo“kindred”numbers)
17.940 × 9.7(Multiplytwonumbersjustunder100)
18.1072 (Multiplytwonumbers(orsquareanumber) justover100)
19.792 (Squareanytwo-digitnumberendingin1or9)
20.4.52 (Squareanynumberendingin5)
21.39.12 (Squareanythree-digitnumberwithmiddledigit0or9)
22.What,approximately,is √7.9?(Squareroots—anystyle!)
23.Whatisthecuberootof175,616?
(Findthecuberootofamysteryperfectcube)
24.What,roughly,is130/1.7?(Multiplyordivideby17)
25.What,approximately,is8π ?(Multiplyby π )
26.(Bonus)January3,1957wasaThursday.Thespaceracebegan onOctober4,1957,whentheSovietUnionlaunchedthesatellite Sputnik.Whichdayoftheweekwasthat?
(AppendixICalculatingDoomsday)
1 ArithmeticalAdvice
2 + 2in#Theologycanmake5
AtweetonJanuary5,2017byFr.AntonioSpadaro, SJ,aclosefriendofPopeFrancis,causingmuchhead scratchingamongmathematiciansandtheologians
IntheendthePartywouldannouncethattwoandtwomade five,andyouwouldhavetobelieveit.Itwasinevitablethatthey shouldmakethatclaimsoonerorlater:thelogicoftheirposition demandedit.
GeorgeOrwell, 1984
Beforedelvingintothedelightsofquickercalculations,thereareafew pointersworthfollowing.Thesewillhelpspeedthingsupandreduce mistakes.
Takelesstimeonyourtimestable
Consider7 × 9 = 63.
Howdidyoureadit?Asyoungsters,weoftenlearntoreciteour multiplicationtablesalongwithotherkidsintheclass.Ifso,youmay havechanted“seventimesnineequalssixty-three.”True—butslow.1 Shredsecondsfromthetimeittakestotellyourtimestablebyreplacing thislaboriousprocesswithaquestion.“Sevennines?Sixty-three.”This providesagoldenopportunitytorefreshyouracquaintancewiththe multiplicationtables,butinthisnewformat.There’sanaddedbonus: Notonlydoesthenewwayshavetwosyllablesoffthephrase,it’salso, well,anewwaytosayit.Inotherwords,youdon’thavetorepeat yourmultiplicationtablesintheslowsing-songwaythatyoumayhave
1 AnearlyEnglisharithmeticbookfrom1552,theanonymous AnIntroductionforto LernetoreckenwiththepenorwyththecountersaccordyngetothetrewecastofAlgorisme tellsus“7tyme 9maketh63,”soslow-pacedmultiplicationisnothingnew.Thebook,though,went througheighteditions,makingitasixteenth-centurymathematicalbestseller.
2Quick(er)Calculations
learned.Reciteyourseventimestableyourusualway,buttimeyourself. Thentrythecompactway,sayingasswiftlyasyoucan,“Oneseven? Seven.Twosevens?Fourteen...”Betteryet,timehowlongittakesyou togoallthewayfrom“Onetimesoneisone”upto“Twelvetimes twelveisonehundredandforty-four”andcomparewiththetimeit takestogofrom“Onceone?One”upto“Twelvetwelves?Onehundred andforty-four!”Seehowmuchtimeyoucansave!
Thecaseofthecrookedcolumns
Yourhandwritingispreciselythat:yours.Evenifitisunintelligible tootherreaders,aproblemthatplaguesmedicaldoctors,thatwon’t matterwhenyoutrytodoquickcalculations,unlessyouhavesome numbersthatcanbeeasilymistakenforothers.
Abiggerpoint,though,iskeepingthetensandtheunitscolumns aligned.Itisalltooeasytowritedownasum,say,as:
34+ 17
Inwhichthe4driftsdangerouslytotheright,or,vieweddifferently,the 7crowdsoutthe1.Overthecourseofthecalculation,especiallyifdone atspeed,thingscangetmessedupandsuddenlyyouarecalculating:
304+ 170 Puttingzeroswherethesort-ofblankspaceswere.Insteadofgettingthe result51,theansweronofferis474.So,please,keepcolumnsaligned. Thismayseemlikenobigdeal,andwhenyouaddjusttwonumbers oftwodigits,itprobablyisn’t.Butifyouhavetoadduplonglistsof numbersthathavemultipledigits,mistakescanrackuprapidly.
WelshmathematicianRobertRecorde(ca.1512–1558)wrotesomeof thefirstbooksinEnglishonarithmetic.In TheGroundofArtes,published in1543,Recordeadvisesstudentsengagedincurrencycalculations: “Ifyourdenominationsbepounds,shillings,&pennies,writepounds underpounds,shillingsundershillings,andpenniesunderpennies: Andnotshillingsunderpennies,norpenniesunderpounds.”2 Authors
2 “Yfyourdenominationsbepoundes,shyllyns,&pennes,wrytepoundesunder po[ndes],shyllyngesundershyllynges,andpennysunderpe[n]nys:Andnotshyllynges underpennys,norpe[n]nysunderpoundes.”
ofarithmeticbookshavebeenurgingreaderstokeepcolumnsaligned foratleast550years!
Guesswhat?Makeestimates
Asyourmentalmathheatsup,thetimetocalculatequantitiesdrops impressively.Theabilitytomessup,however,increases—unlessyou makeestimates.Guessingananswertakesafractionofasecondbut cancatchembarrassingmistakes.Inthepreviousexample,ifyoustart byobservingthat34isroughly30and17isalmost20,you’dexpectan answerofabout30 + 20 = 50.Ifyougettheanswer51,thenyou’dfeel happyandconfident.Mixupyourcolumnsandget474andyourrough estimateof50screamsatyoutogobackandcheck.
Thereisasubtleartinestimation,whichisperhapsnotquiteso subtle.Inanaddition,suchas34 + 17,ithelpstooverestimateone number(say20for17)andcompensateforthatbyunderestimatingthe othernumber(30for34).Thatway,youendupwitharapidestimate that’slikelytobefairlyclosetotherealanswer.Forsomethingmore complicated,suchas513/3.78,youmightbrackettheanswerwithtwo estimates.Forafraction,youcanoverestimatethetop(thenumerator) andunderestimatethebottom(thedenominator),bothofwhichoverestimatethefraction.Thus,513/3.78 < 540/3andso513/3.78 < 180. Thendothereverse—underestimatethenumerator,overestimatethe denominator,andsounderestimatetheanswer.Herewecouldset 513/3.78 > 500/4 > 125.Whateveranswerwegetisonlyreasonableif itliesbetween125and180.
What’sthepoint?Extrazerosanddecimals
Anothervitaltiptobeefupyourcalculationistoignoreinconvenience— suchaszerosattheendofnumbers,ordecimalsinthemiddle.These canbestrippedout,sothattheydon’tgetinthewayofcalculation, andthenreinsertedlateron.Afterall,youhavealreadyestimated theanswer—soyouknowhowmanyzerostoaddorwheretoinsert thefinaldecimalpoint.There’snoneedtoslowyourselfdownby keepingtrackofallthosedecimalpointsorextrazerosateachstepof thejourney.
Consider,forexample,0.9 × 1.2.Inregularmode,youwouldwrite thesenumbersoneabovetheother,startmultiplyingwiththedecimals
inplacetocomeupwithananswer.Don’t.Letcriticsturnredwith anger,whitewithfear,orwatchyouandgogreenwithenvy.
First,makeanestimate:0.9isroughly1.So,too,is1.2.Weexpectour answertobeabout1 × 1 = 1.Removethedecimalpointstoleave9 × 12, whoseansweryou’veknownsinceyourearlydays,since9 × 12 = 108. Astheanswerisabout1,stickthedecimalpointintogettheactual answer,namely0.9 × 1.2 = 1.08.
Asasecondexample,thinkof1,300 × 40.Asbefore,estimate.We know1,300ismorethan1,000,sotheanswerisgoingtobeabitmore than1,000 × 40 = 40,000.Now13 × 4—aseverycardplayerknows— is52.Andso,astheanswerslightlyexceeds40,000,wemusthave 1,300 × 40 = 52,000.
Letmemakeanappeal:clothethenakeddecimalpoint!It’sbadstyle towriteonetenthas.1,thereasonbeingthatitistooeasytooverlook thedecimalpoint.No-oneeverwrites01,soifyoualwayswrite0.1,you knowit’sonetenth,andyouareunlikelytomistakeitfor01.
And,nowourdecimalpointsareclothedinalltheirsplendor,here’s areminderaboutscientificnotation,whichwon’tbeusedmuchinthis book,Iadmit.Ratherthanwrite12,345,forexample,youcanwriteitas 1.2345 × 104 .Ifyoustruggletorememberthepowerof10that’sused, imagineadecimalpointbeinginserted,whichwouldgive12,345.0.How manynumbersaretherebeforethedecimalpoint?Five.Subtract1,to get4,so12,345 = 1.2345 × 104 .Butnowlet’ssupposewehavethe number0.054321.Thefirstnon-zeronumberafterthedecimalpointis a5,whichoccurstwoplacesin.Hence,wewritethisas5.4321 × 10 2 Ifyoufeeltheneed,12,345 × 0.054321 = 1.2345 × 5.4321 × 102 ,for whenyoumultiplypowersof10together,youaddtheexponents.As 12 × 5.5 = 66,wecanestimatetheanswerasabout660.
The × factor
Supposeyoufeelasuddenurgetomultiply78 × 37.Thislooksnasty. Thevitalpointtorememberisthatallnumbers,withtheexceptionof primenumbers,havesmallerfactors.Forexample,111 = 37 × 3,so that37 = 111/3.Tocalculate78 × 37,itmaybequickertocomputeit as78 × 37 = 78 × 111/3 = (78/3) × 111 = 26 × 111,andthenusethe swiftmethodeitherformultiplyingby13anddoubling(tomultiply by26),ormultiplyby111swiftlybyshunting(seethenextsection!). Choosewhicheverfactorizationyouprefer!
Oneofthegloriesofpre-decimalcurrencyinEnglandandtheImperialsystemofmeasurementmaywellhavetodowithfactors.Twelve canbedividedby1,2,3,4,and6,whichmaybethebasisforhaving12 inchesinafootand12penniesina(pre-decimal)shilling.Likewisethe number60canbedividedby1,2,3,4,5,and6.(Itcanbedividedexactly byothernumbersaswell,butweonlyneedtocheckuptothesquare rootof60.Theotherfactorswillbethecomplementsofthese;thatis, thenumbersyouneedtomultiplybytheseinordertoget60.Theother divisorsare60,30,20,15,12,and10.)Thenumber60havingsomany integersasdivisorsmayexplainwhytheAncientBabyloniansusedthe sexagesimalsystem,countingin60s.Weacknowledgethisinthemetric system,wherearightangleremains90◦ .Thegradian,with100gradians inarightangle,nevercaughton.Sure,the45 45 90 ◦ triangle becomesa50–50–100gradiantriangle(anglesinatriangleaddupto 200gradians),buttheotherstandardtrianglegoesfrom30 60 90 ◦ to33.333 66.666 100gradians,whichisuser unfriendly.
Shuntingforshow
It’sjustajumptotheleft,andthenasteptotheright.
“TheTimeWarp”
Tobebrief—whichhelpssavespaperandthereforetrees—I’llusethe wordshunting.Toshunt,Imeanshiftanumberovertotheleftorto theright.Thenumber1,shuntedtwoplacestotheleft,is100,andtwo placestotherightitbecomes0.01.
Asimpleexampleofshuntingisifyouwanttomultiply47 × 111. Simplywrite47downthreetimes,butshuntedoveroneplaceeach time,toget:
Thisyouaddupswiftlytoget5,217.
How?
Thereareseveralbooksavailableonthesubjectofquickcalculations— nottomentionsomewebsitesandYouTubevideos.Thesecanbegoodat
teachingthetricksandtechniquesincarryingoutacalculation.Edward H.Julius,forexample,haswrittenseveralbooksonRapidMath.Ashout outshouldalsogotoJacowTrachtenberg,whodevelopedanentire systemforrapidcalculationswhiledetainedinaNaziconcentration camp. 3
Few,ifany,otherbooksexplain why or how aparticularmethodworks. Thisbookdoes:manysectionshaveshortexplanationsofwhythings work.Thesenearlyalwaystieintosomesimpleformulasfromhighschoolmathematics,onesyoumayneverhaveusedtoaccelerateyour arithmetic.
Trythese
Estimatethreeways:firstoverestimate,thenunderestimate,andfinally, constructa“bestestimate.”Do not workouttheanswer—theidea hereistolearntoestimaterapidly.Sometimes,rapidestimatesform partofoffbeatjobinterviews.“Howmanypingpongballscanfitina 747airplane”isastandardnon-standardquestion(goahead,findthe data,andcomeupwithyourownestimate!)orhowmanybabies areborneachyearintheUK.(Ifthereare66millionBritswholive onaverageto75,andthepopulationisn’tgrowing,that’sabout66/75 millionbabies,or880,000littleoneseachyear.)Herearesomefarmore straightforward“plugandchug”questionstogetyoustarted.
1.4π
2.1.87 × 2.34
3.24,000 × 26
4.0.007 × 265
5.0.05/547
6.547/0.051
7.456 4.63
8.0.93 0.296
9.412 + 891
10.1,075 + 2,342
3 Forhismethod,seeJacowTrachtenberg, SpeedSystemofBasicMathematics (NewYork: SouvenirPress,1989).
