The math hacker book shortcut your way to maths success the only truly painless way to learn and unlock maths ( PDFDrive com )

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


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


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'?

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


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

Name/Copyright
Math-Hacker

©PaulCarson2015


Dedication
ToTymoteusz,Ihopetheworldyougrow
upinteachesmathseasily.

Blog

Website

Twitter
@In_A_Min


Email

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

MathsinaMinuteSeries
Contentsandnumbering2ndSeptember2013
1.
2.
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10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.

Multiplication
Division

Fractions
Decimals
Percentages
NegativeNumbers
Squaring&Area
Cubing&Volume
Indices
StandardForm
Logarithms
Sequences
Gradient/EquationofAStraightLine
SimultaneousEquations
Quadratics
Inequalities
ChangingtheSubject
Cubics
AlgebraicFractions
AlgebraofArithmetic
AnglesandRadians
Gradient/Tangent
SineandCosine
SineandCosineRules


25.
26.
27.
28.
29.
30.

31.
32.
33.
34.
35.
36.
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40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.

Pythagoras’Theorem
Gradient/Differentiation1
Integration1
Polynomials

TypesofFunctions
Co-ordinateGeometry
Differentiation2
Integration2
BinomialTheorem
Trigonometry
NumericalMethods
DifferentialEquations
Mechanics
Vectors
EngineeringFundamentals
CodingFundamentals
Statistics
Matrices
TheSlideRule
RichardFeynman
TheCurrentMathematicsTeachingSystem
HowtheMinAMinMethodWorks
ComplexNumbers
PartialDifferentiation
Series
Integration3
MultipleIntegrals
DifferentialEquations
FourierSeries
PartialDifferentialEquations
VectorCalculus


Proverb

‘NilSatisNisiOptimum’
(Nothingbutthebestisgoodenough-EvertonFCproverb)


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!


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.


>>>
>>>


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


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?


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.


WhatMultiplicationIs


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.


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.


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.


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.


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


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