The math hacker book

Contents
Name/Copyright
SeriesContents
Proverb
HowToBestUseThisBook
IntroductiontoMultiplication
WhatMultiplicationIs
NewMethodforTimesTables
WhatBracketsMean
2x2DigitNumbers
3x3DigitNumbers
BiggerNumbers
WhatMultiplicationIsFor
WhatIsDivision
WhatIsDivision-Part2
ThreeTypesofDivision
Less-StraightforwardDivision
TheThirdRuleofMaths
So-CalledLongDivision
IntroductionToFractions
AdditionOfFractions
SubtractionofFractions
MultiplicationofFractions
DividingFractions
"MixedNumbers"
DivisionofMixedNumbers
AdditionofMixedNumbers
Ultra-FastWaysofAddingandSubtractingFractions
Decimals
KeyChangeEffect

ReverseSituation-takingtheKeyChangeEffectfurther
DivisionofDecimals
Percentages-InAMinute
FindingaPercentageofAnyNumber
CalculatePercentagesMentally
Usefulpercentages
CalculatingDiscounts

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RealLifeexample-usingtheMultiplierinreality
ReversePercentage
CompoundInterest
TheFlipSide
Fastcalculationofcompoundinterest
Depreciation-thereverseofCompoundInterest
WritingADivisionAsAPercentage
IntroductiontoNegativeNumbers
Everydaylife
MultiplyingNegativeNumbers
DivisionofNegativeNumbers
FinalProblem
IntroductiontoSquaring&Area
Squaringiscalled‘Squaring'
TheSquaringSystem
CalculationofAreas
ANewAngle
IntroductiontoCubing
HowEnginesWork

UseofLetters
VolumesofOtherShapes
IntroductiontoIndices
Indices-InAMinute
SecondRule
ThirdRule
FourthRule
FifthRuleofIndices
TheSixthRule
Rule7ofIndices
AlgebraicIndices
Surds
RationalisingTheDenominator
IntroductiontoStandardForm
StandardForm-TheShortWayofWritingLargeNumbers
StandardFormforSmallNumbers
MultiplyingNumbersInStandardForm
DividingNumbersInStandardForm
TypicalExampleQuestion
FurtherExamplesofuseofStandardForminScience

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IntroductiontoLogarithms
WhatAreLogarithms?
TouchBase
Logarithmsineverydaylife
TheRulesofLogarithms
Introduction&Puzzles

Hints
Solutions
IntroductiontoSequences
Sequences-InAMinute
Non-linearSequences
AManCalledAl
Child,12,AmazesTeacher
ExampleQuestions
IntroductiontoGradient
WhatIsGradient?
TheEquationofAStraightLine
CalculatingtheGradientand‘Cut
OtherStraightLineGraphs
IntroductiontoSimultaneousEquations
SimultaneousEquations
Type2Algebra
Type3Algebra
WhatWe’reDoingThisFor
Real-LifeExampleofSimultaneousEquations
SimultaneousEquations-SchoolStyle
IntroductiontoQuadraticEquations
MultiplyingBrackets
Factorising
Multiplying-InAMinute
WhatWe’reDoingThisFor
SolvingAQuadratic
Non-factorisableQuadratics
AnyTypeofQuadratic
AnotherExample
CompletingtheSquare

ACloserLookAtTheQuadraticFormula
WhatMultiplyingTwoExpressionsTogetherWillGive
WhyIsitCalled‘CompletingTheSquare'?

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SimultaneousEquations-Extended
Epilogue
IntroductiontoInequalities
ThreeTypesofAlgebraforInequalities
Type2Algebra
Type3Algebra
WhentheInequalitySignFlips
InequalitiesofQuadratics
QuadraticsandStraightLines
IntroductiontoChangingTheSubject
UsingBIDMAStoChangetheSubject
Examples
HowWe'veUsedThisBefore
ExceptionsToTheRule
FamousScienceFormulaeRearranged
IntroductiontoCubics
MultiplyingThreeBrackets
FactorisingaCubic
FactorisingaCubic2
RemainsofTheDay
Findingthemaximumorminimumvalues
IntroductiontoAdvancedMentalMultiplication
TheTimesTables

HowtoMultiplyTwoDigitNumbersInYourHead
SquaringLargeNumbers
FindingSquareRootsfromSquareNumbers
TheChristmasPartyWhereIWasCalledAWizard
HowToCheckMultiplicationsAreCorrect-InSeconds
TheChristmasPartyMagicTrick
AlgebraBehindtheMultiplicationMethod
AlgebraBehindTheSquaringSystem
AlgebraBehindAdvancedMultiplicationUsingSquares
IntroductiontoGradient/Tangent
ConceptConnectedToGradient
PuttingOurValuesToUse
SecondarySolutions
IntroductiontoSine&Cosine
NormalisingtheHypotenuse
PuttingOurValuesToUse1

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SecondarySolutions
UsingSineToFindtheAreaofATriangle
TheOtherTwoSidesofTheTriangle
PuttingOurValuesToUse2
TheRelationshipBetweenSine,Cosine,Tangent,Gradient
ReverseofSine
ANewAngle
Pythagoras’theorem
SpecialSituation
ReverseSituation

ThePythagoreanTheoremIsNotJustForTrianglesOnce
IntroductiontoSine&CosineRules
TheSineRule
Right-AngledTriangles
OneSituationRemains-TheCosineRule
DerivationofCosineRule
ExamTechnique
THEEND

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Name/Copyright
Math-Hacker

©PaulCarson2015

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Dedication
ToTymoteusz,Ihopetheworldyougrow
upinteachesmathseasily.

Blog

Website

Twitter
@In_A_Min

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Email

YouTube
ChannelContainingInstructionalVideos
/>
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SeriesContents
Seriesofbooksandtitles
NakedNumbers:The3RulesToMakeYourLifeAddUp,HodderEducation,
theMichelThomasMethod(2010)

MathsinaMinuteSeries
Contentsandnumbering2ndSeptember2013
1. Multiplication
2. Division
3. Fractions
4. Decimals
5. Percentages
6. NegativeNumbers
7. Squaring&Area
8. Cubing&Volume
9. Indices
10. StandardForm
11. Logarithms

12. Sequences
13. Gradient/EquationofAStraightLine
14. SimultaneousEquations
15. Quadratics
16. Inequalities
17. ChangingtheSubject
18. Cubics
19. AlgebraicFractions
20. AlgebraofArithmetic
21. AnglesandRadians
22. Gradient/Tangent
23. SineandCosine
24. SineandCosineRules

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25. Pythagoras’Theorem
26. Gradient/Differentiation1
27. Integration1
28. Polynomials
29. TypesofFunctions
30. Co-ordinateGeometry
31. Differentiation2
32. Integration2
33. BinomialTheorem
34. Trigonometry
35. NumericalMethods
36. DifferentialEquations
37. Mechanics

38. Vectors
39. EngineeringFundamentals
40. CodingFundamentals
41. Statistics
42. Matrices
43. TheSlideRule
44. RichardFeynman
45. TheCurrentMathematicsTeachingSystem
46. HowtheMinAMinMethodWorks
47. ComplexNumbers
48. PartialDifferentiation
49. Series
50. Integration3
51. MultipleIntegrals
52. DifferentialEquations
53. FourierSeries
54. PartialDifferentialEquations
55. VectorCalculus

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Proverb

‘NilSatisNisiOptimum’
(Nothingbutthebestisgoodenough-EvertonFCproverb)

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HowToBestUseThisBook
HowToBestUseThisBook
Thisbookshouldbereadinchapterorder.Startatthebeginningandwork
throughtotheend!Attheendofsomechaptersthereisatypicalexamquestion
foryoutotry.Makesureyoudoit.Thenpracticequestionsliketheseasmuch
asyoucan.
AlongsidethebookthereisaYouTubechannelwhichdemonstratesmanyofthe
techniques.

SUBSCRIBE
Andyoucanseethetechniquescarriedoutinrealtime,asdescribedbythe
book.
Finally,attheendisaguidetothetrickythingoftakinganexamitself!
Read,understand,workthroughandmastertheskills.AnA*willbecome
incrediblyeasytoyouandyou’llbeamazedathoweasyitcanbe!

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IntroductiontoMultiplication
Introduction
Theideaofoptimisedmathematics.
Thisbookistheculminationofyearsofresearch,experience,studentfeedback,
atirelesssearchforbettermethodsandadesiretomakemathseasy.Thishas
takenfifteenyears.Inthattime,mystudentshavea95%successrate,agreater
understandingofmaths,nomiseryorpainbecausemathsis‘boring’,theirselfrespectandpridehasreturnedandofcourse,theyseetheworldinanew,
mathematicalway.
Theyalsorealisethatlearningdoesn’thavetobepainfulanditispossibleto
learnsomethingthatmayappeartobedifficultwitheaseandinrelativelylittle
time.

ThemethodIhavecreatedisonethatisholistic,containingasfewmethodsas
possible,subliminal,sothatyoulearnadvancedmathsasyouaredoingthe
fundamentals,algebraic,meaningyouthinkalongabstractlineswithoutinitially
realising,andjoyful,becauseyoucandothingsyouhithertothoughtimpossible.
Whilereadingthisbook,onequestionwillconstantlyreturntoyourmind.
‘WHYDON’TTHEYTEACHTHISATSCHOOL?’
I’vebeenaskedthismany,manytimes,andthereareavarietyofanswers.
Fornow,foryou,youhaveinyourhandtheguideandpassporttounlockingthe
secretsofmathsandbecomingoneofitsbetterusers.Youwillbeabletodo
thingsthatwillimpressyourfriends,family,teachersandmostofallyourself.
Takeiton,beinspired,usethemethodforyouandbeasuccess!
Thefirstchapterisallaboutmultiplication.
Goodluck.Havefun.

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>>>
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Chapter1
Thereareloadsofmultiplicationtechniquesoutthere.
Egyptian
Babylonian
Chinese
Russian
Napier’sbones

Gridmethod
‘Long’multiplication
Thelistgoeson.
Youcouldtryeachoftheseyourselfanddecidewhichisbest.
WhatIwantyoutodoisoptimise.Iwantyoutodomultiplicationinthemost
efficient,intuitiveandmathematicallyadvantageousway.Amethodthatallows
youtodofiveotherthings.Andahalf.ButI’llgettothat.
Schoolteachesallsortsofwayswiththehopethatonewillsnagandyou’llbe
abletodoit.Themostpopularisthegridmethod.
Beforewediscussthese…
Whyschooltechniquesdon’twork
Someoftheirmethodswork.Ofcoursetheydo.Butwhydopeoplestruggle
so?Becausethemethodsrequirememorisationofanumberofsteps,which,if
anyarewrong,makestheanswerincorrect.Worse,itisnotpossibletoknow
yourselfifitisincorrect.Youhavetoasksomeoneelse.Whatkindofsystemis
that?
Studentsfindthemselvesasking:
‘Isthisright,Miss?’
Plus,becauseyou’rememorisingsteps,youdon’treallyunderstandwhatis

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happening.Andthisleadstouncertainty.Thismeansthatyou’rebeingtrained
toactlikearobot.Androbotsdon’tthinkforthemselves.Sohowwouldyou
knowifitwasrightorwrong?Onlyifyourcontrollertellsyouso!
Itisvital,reallyvital,thatweknowwhywearedoingthings.Thismakesiteasy
toremember,moreinterestingtolearn,andallowscreativityofthought.And
evenimprovementsofexistingsystems.
Ishalluseananalogytoillustrate.

Doyouhavetoremindyourselfnottoputyourhandonahotstove?
No?
Whynot?
Becauseyouunderstandtheconsequencesifyouweretodoso.
Youdon’thavetomemorisearule‘Nevertouchahotstove’andnotunderstand
why.Youalreadyunderstandwhy.Andsoitiseasytorememberand
impossibletoforget!
Recently,mycuriouscatdecidedtowalkonmykitchentopsandoverthe
cooker.UnfortunatelythestoveplatewasstillhotasIhadjustusedit.There
wouldhavebeennopoint‘teaching’himaboutthisbeforehand.Butnowhewill
neverforget.Ilearntmathsinthesameway...viapainfulfailure!Soyoucan
avoidmymistakesfromreadingthiscourse.
Understandingwhysomethingismakesiteasytorememberandimpossibleto
forget.Thatishowanythingshouldbelearned.Anditworksespeciallywell
withmaths.

Let’sstart.
Soinabookaboutmultiplication,thefirstquestionhastobe….
Whatismultiplication?

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BeforeIjustgiveyoutheanswer,Iwantyoutothinkaboutitalittlebit.Like
you’veprobablythoughtabouttheconsequencesofputtingyourhandonahot
stove,whatyouknow,youwon’tforget.Andifyoucomeupwithityourself,
you’lldefinitelyrememberit!
So,tomakeitalittlemorechallengingandpointyouintherightdirectionas
well,thinkforamoment,whatismultiplication?Butinyouranswer,you
cannot,CANNOT,usethewords

Multiply
Times
Product
By
Thinkaboutit.Thinkaboutit.
Abitmore.
Whathaveyougot?
Multiplicationis…..
Nowin13yearsoftutoring,Ihaveheardsomevariedanswers.AndI’mnot
goingtoembarrassmystudentsbyreferringtothemhere.Theoverridingresult,
althoughnotforeveryone,is…actually,Idon’tknow.
Isn’tthatsomething?
10yearsineducation(ormore)andtheydon’tknow.
Isthattheirfault,orschool’s?
Inmyopinionitisschool’s.WhenIshowyouhoweasythiscanbe,youwillbe
amazedthatschoolcanmakeitsohard.

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WhatMultiplicationIs

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Chapter1

Whatmultiplicationis
So,whatdidyoucomeupwith?Here’stheanswer:
Multiplicationisjust…..repeatedaddition.

That’sit.
(Isthatwhatyoucameupwith?Ifnot,don’tworry.)
Let’slookatthis.Multiplicationoftwonumberstendstobethoughtofas
‘timesingtwonumberstogether’.Theword‘times’here,whichhasmorphed
intoaverbovertheyears,actuallyreferstothenumberoftimesweadd.Thisis
veryimportant.Itishowmanytimesweadd.
Forexample,
3x5=15
Becauseweadd5…3times.
Soabove,youprobablyreadthatas3times5.Nowreaditas
3times(weadd)5.
3x5=3times(weadd)5=5+5+5=15.
Anotherexample
4x6=4times(weadd)6=6+6+6+6=24
Andsoon!
5x7=5times(weadd)7=7+7+7+7+7=35
Howexciting.Thisishoweasyitis.

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Multiplicationisjustaddition!Sothe‘times’isnotanotherwordformultiply,it
isactuallythenumberoftimesweadd.Youcanprobablyseethatthisis
Victoriansortoflanguage,whichhasgotdroppedovertheyears.Itsoundslike
somekindofproverb–5timesweadd7.Obviouslythe‘weadd’parthas
erodedawayandnoweveryonethinkstimes=multiply,butitdoesn’t.
Again,the‘times’isnotanotherwordformultiply,itisactuallythenumberof
timesweadd.
Sowhat?
Nowweknowthatwecannevergetamultiplicationwrong.Ifyoucanadd,you

canmultiply.Youdon’tactuallyneedtomemorisetimestablesanymore.You
couldworkeachoneouteverytimeifyouwanted!Thememorisationoftimes
tablesisokaywhenyouunderstandwhy5x7=35,butit’sbasicallyuselessif
youdon’t.
Nowyoudo.
Nowyoucandoanymultiplication.
Any.
Becauseyouknowthatyoucouldjustaddoverandover.
Ithoughtyousaidthiswasgoingtobeoptimised?
Itis!Butweneedtounderstandwhatwe’redoingfirst.
Let’slookatsomemoreexamples.
7x9=7times(weadd)9=9+9+9+9+9+9+9=63
8x12=8times(weadd)12=12+12+12+12+12+12+12+12=96
14x7=14times(weadd)7
Butwait.Here’sanotherconcept.Itdoesn’tmatterwhichorderwedoitin.

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Forexample,3x5=15.
And,5x3=5times(weadd)3=3+3+3+3+3=15.
Whichisquicker?Obviouslywealwayswanttoaddthelowestnumberof
times.Thisisalwaysquicker.
3x14ismuchquickerthan14x3!
Soalwaysaddthelowestnumberoftimes.Turn14x7into7x14,giving
7x14=7times(weadd)14=14+14+14+14+14+14+14=98
Isthisthebestwaytomultiply2numberstogether?No.Itdoesn’tfulfilallof
therequirements.Wecan’tevenbesureitiscorrect,becausewemighthave
madeamistakewhileadding.
Butwe’regettingthere.

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NewMethodforTimesTables
Chapter2
TheTimesTables
Thetimestablesarejustabunchofanswerstoquestionsthatyoucanalready
findtheanswertoyourself,nowyouknowthatmultiplicationisjustaddition.
Atschool,theydon’ttellyouthis,sotheymakeyoulearnthetablesbyheart,
whichisboring.Becauseyoudon’tknowwherethesenumberscamefrom,you
tendtoforgettheanswers.Remember,wheneverwelearnsomethingnew,
alwaysask‘Whyisitlikethat?’andthatquestioncanbemorevaluablethanthe
actualfactyou’relearning.Itcanleadtootherthingsandsparkaninterest.
Learningabunchofdry,boringfactsisaboutasmuchfunasreadinga
dictionaryinGreek.
So,asIsaidabove,youdon’tHAVEtoknowthetimestablesanymore.So,
don’tworryaboutthat.Ifyoudon’tknowone,justfigureitout.Addasmany
timesasyouneed.Isitquick?No,butitwillgetyouthere.
Anotherthingyoucandoisusereferencepointsfromonesthathavesnaggedin
yourmindandyoudoknow.
Forexample,let’ssayyouknow,fromrepetition,that7x8=56.Sowhen
someoneasks,what’s6x8?Youthink,well…
Iknowthatmultiplicationisjustaddition.So7x8=56means
7times(weadd)8.
IfIadd8sixtimesinstead,Ithereforeneedtotakeaway8from56,becauseitis
onetoomany!
Sothatleavesmewith7x8=56,6x8=56–8=48.
So6times(weadd)8mustbeequalto48.

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Sothat’sokifyouhavereferencepoints.
Ifyoudon’thaveanyatall,youcanusethefollowingsystem,whichhasthree
waysofdoingit.Someprefermethod1,othersmethod2…It’suptoyou.
Forthesakeofoptimisation,lownumbermultiplication,like4x5,couldbe
donebyaddition.4times(weadd)5=5+5+5+5=20.
Forhighernumbermultiplication,like7x9,say,additionhereisabitslow,so
it’dbebettertohaveaquicksystem.

TheNineTimesTablesystem.
Whatwecandohereisthis:
Writetwocolumnsofnumbersdown,from0-9andthen9-0.
Leftcolumn
0
1
2
3
4
5
6
7
8
9
Andrightcolumn
9
8
7
6

5
4

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3
2
1
0
Butwritethemnexttoeachother!
09
18
27
36
45
54
63
72
81
90
Andheypresto,wehavethenine-timestable.
Tofind7x9,wejustgotothe7thnumberfromthetop,(orthereverse,the
fourthfromthebottom),andthatgives
09
18
27
36
45
54

637(x9)
72
81
90
Anotherexamplewouldbe6x9.Gotothesixthentry:
09
18
27
36
45

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