GREMathConventions18point12.docx
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GREMathConventions18point12.docx
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GREMathConventions18point12
GRADUATERECORDEXAMINATIONS®
MathematicalConventions
LargePrint(18point)Edition
Copyright©2010byEducationalTestingService.All
rightsreserved.ETS,theETSlogo,GRADUATERECORD
EXAMINATIONS,andGREareregisteredtrademarks
ofEducationalTestingService(ETS)intheUnitedStates
andothercountries.
Overview
Note:
SomeoftheMathematicalconventionsdiscussedin
thisdocumentareconventionsusedinprinteditionsoftests
andpracticematerial.BrailleeditionsusetheNemethCode
ofMathematicsandScienceNotation.
Themathematicalsymbolsandterminologyusedinthe
QuantitativeReasoningmeasureofthetestareconventionalat
thehighschoollevel,andmostoftheseappearintheMathReview.
Whenevernonstandardorspecialnotationorterminologyisusedin
atestquestion,itisexplicitlyintroducedinthequestion.However,
therearesomeassumptionsaboutnumbersandgeometricfigures
thatareparticulartothetest.Theseassumptionsappearinthetest
atthebeginningoftheQuantitativeReasoningsections,andthey
areelaboratedbelow.
Also,somenotationandterminology,whilestandardatthehigh
schoollevelinmanycountries,maybedifferentfromthoseused
inothercountriesorfromthoseusedathigherorlowerlevels
ofmathematics.Suchnotationandterminologyareclarified
below.Becauseitisimpossibletoascertainwhichnotationand
terminologyshouldbeclarifiedforanindividualtesttaker,more
materialthannecessarymaybeincluded.
Finally,therearesomeguidelinesforhowcertaininformation
givenintestquestionsshouldbeinterpretedandusedinthecontext
ofansweringthequestions—informationsuchascertainwords,
phrases,quantities,mathematicalexpressions,anddisplaysofdata.
Theseguidelinesappearattheend.
-2-
Numbersandquantities
1.Allnumbersusedinthetestquestionsarerealnumbers.
Inparticular,integersandbothrationalandirrationalnumbers
aretobeconsidered,butimaginarynumbersarenot.Thisisthe
mainassumptionregardingnumbers.Also,allquantitiesarereal
numbers,althoughquantitiesmayinvolveunitsofmeasurement.
2.Numbersareexpressedinbase10unlessotherwisenoted,using
the10digits0through9andaperiodtotherightoftheonesdigit,
orunitsdigit,forthedecimalpoint.Also,innumbersthatare1,000
orgreater,commasareusedtoseparategroupsofthreedigitstothe
leftofthedecimalpoint.
3.Whenapositiveintegerisdescribedbythenumberofitsdigits,
e.g.,atwo-digitinteger,thedigitsthatarecountedincludetheones
digitandallthedigitsfurthertotheleft,wheretheleft-mostdigitis
not0.Forexample,5,000isafour-digitinteger,whereas031isnot
consideredtobeathree-digitinteger.
4.Someotherconventionsinvolvingnumbers:
912
countries);
onedozenmeans12;
theGreekletterπrepresentstheratioofthecircumferenceof
acircletoitsdiameterandisapproximately3.14.
-3-
onebillionmeans1,000,000,000,or10(not10,asinsome
5.Whenapositivenumberistoberoundedtoacertain
decimalplaceandthenumberishalfwaybetweenthetwo
nearestpossibilities,thenumbershouldberoundedtothe
greaterpossibility.
Example:
23.5roundedtothenearestintegeris24,
and123.985roundedtothenearest0.01is123.99.
Whenthenumbertoberoundedisnegative,thenumbershould
beroundedtothelesserpossibility.
Example:
-36.5roundedtothenearestintegeris-37.
6.Repeatingdecimalsaresometimeswrittenwithabarover
thedigitsthatrepeat,asin
25
12
=2.083and
1
7
=0.142857.
7.Ifr,s,andtareintegersandrs=t,thenrandsarefactors,or
divisors,oft;also,tisamultipleofr(andofs)andtisdivisible
byr(andbys).Thefactorsofanintegerincludepositiveand
negativeintegers.
Example1:
-7isafactorof35.
Example2:
8isafactorof-40.
Example3:
Theinteger4hassixfactors:
-4,-2,-1,1,2,
and4.
-4-
Thetermsfactor,divisor,anddivisibleareusedonlywhenr,s,
andtareintegers.However,thetermmultiplecanbeusedwith
anyrealnumberssandtprovidedrisaninteger.
Example1:
1.2isamultipleof0.4.
Example2:
-2πisamultipleofπ.
8.Theleastcommonmultipleoftwononzerointegersaandb
istheleastpositiveintegerthatisamultipleofbothaandb.The
greatestcommondivisor(orgreatestcommonfactor)ofaand
bisthegreatestpositiveintegerthatisadivisorofbothaandb.
9.Whenanintegernisdividedbyanonzerointegerdresultingin
aquotientqwithremainderr,thenn=qd+r,where0≤r Furthermore,r=0ifandonlyifnisamultipleofd. Example1: When20isdividedby7,thequotientis2and theremainderis6. Example2: When21isdividedby7,thequotientis3and theremainderis0. Example3: When-17isdividedby7,thequotientis-3 andtheremainderis4. 10.Aprimenumberisanintegergreaterthan1thathasonly twopositivedivisors: 1anditself.Thefirstfiveprimenumbers are2,3,5,7,and11.Acompositenumberisanintegergreater than1thatisnotaprimenumber.Thefirstfivecompositenumbers are4,6,8,9,and10. -5- 11.Oddandevenintegersarenotnecessarilypositive. Example1: -7isodd, Example2: -18and0areeven. 12.Theinteger0isneitherpositivenornegative. Mathematicalexpressions,symbols,andvariables 1.Asiscommoninalgebra,italicletterslikexareusedtodenote numbers,constants,andvariables.Lettersarealsousedtolabel variousobjects,suchasline,pointP,functionf,setS,listT, eventE,randomvariableX,BrandX,CityY,andCompanyZ. Themeaningofaletterisdeterminedbythecontext. 2.Whennumbers,constants,orvariablesaregiven,their possiblevaluesareallrealnumbersunlessotherwiserestricted. Itiscommontorestrictthepossiblevaluesinvariousways. Herearethreeexamples. Example1: nisanonzerointeger. Example2: 1≤x<π Example3: Tisthetensdigitsofatwo-digitpositiveinteger, soTisanintegerfrom1to9. -6- 3.Standardmathematicalsymbolsatthehighschoollevel areused.Theseincludethestandardsymbolsforthearithmetic operationsofaddition,subtraction,multiplicationanddivision (+,-,∞,and∏),thoughmultiplicationisusuallydenotedby juxtaposition,oftenwithparentheses,e.g.,2yand(3)(4.5);and divisionisusuallydenotedwithahorizontalfractionbar,e.g., isequaltothefraction; 121 22 1 100 andx0=1forallnonzeronumbersx. 3 8 . 4.Mathematicalexpressionsaretobeinterpretedwithrespect toorderofoperations,whichestablisheswhichoperationsare performedbeforeothersinanexpression.Theorderisasfollows: parentheses;exponentiation;negation;multiplicationanddivision (fromlefttoright);additionandsubtraction(fromlefttoright). Example1: Thevalueoftheexpression1+2∞4is9,because theexpressionisevaluatedbyfirstmultiplying2and4andthen adding1totheresult. -7- w 3 Sometimesmixednumbers,ormixedfractions,areused,like4 and-10.(Themixednumber4 1335 2 8 8 andthemixednumber-10 isequaltothefraction- .) Exponentsarealsoused,e.g.,210=1,024,10-2= Example2: -32means“thenegativeof‘3squared’” becauseexponentiationtakesprecedenceovernegation. Therefore,-32=-9,but(-3)2=9becauseparentheses takeprecedenceoverexponentiation. 5.Herearenineexamplesofotherstandardsymbolswiththeir meanings: Example1: x≤y Example2: xðy Example3: x♠y Example4: x xislessthanorequaltoy xandyarenotequal xandyareapproximatelyequal theabsolutevalueofx Example5: x thenonnegativesquarerootofx, wherex≥0 Example6: -x Example7: n! thenonpositivesquarerootofx, wherex≥0 theproductofallpositiveintegers lessthanorequalton,wherenis anypositiveintegerand,asaspecial definition,0! =1. Example8: Example9: ||m ⊥m lines lines andmareparallel andmareperpendicular -8- 6.Becauseallnumbersareassumedtobereal,someexpressions arenotdefined.Herearethreeexamples: Example1: Foreverynumberx,theexpression isnotdefined. x 0 Example2: Ifx<0,thenxisnotdefined. 0 7.Sometimesspecialsymbolsornotationareintroducedin aquestion.Herearetwoexamples: Example1: Theoperationisdefinedforallintegersr rs 2 Example2: Theoperationisdefinedforallnonzero numbersxby x=- 1 x . 8.Sometimesjuxtapositionoflettersdoesnotdenote multiplication,asin“considerathree-digitintegerdenotedby XYZ,whereX,Y,andZaredigits.”Whetherornotjuxtaposition oflettersdenotesmultiplicationdependsonthecontextinwhich thejuxtapositionoccurs. -9- Example3: 0isnotdefined. andsbyrs= . 1+r 9.Standardfunctionnotationisusedinthetest,asshowninthe followingthreeexamples. Example1: Thefunctiongisdefinedforallx≥0by g(x)=2x+x. Example2: Ifthedomainofafunctionfisnotgivenexplicitly, itisassumedtobethesetofallrealnumbersxforwhichf(x) isarealnumber. Example3: Iffandgaretwofunctions,thenthecomposition ofgwithfisdenotedbyg(f(x)). Geometry 1.Inquestionsinvolvinggeometry,theconventionsofplane (orEuclidean)geometryarefollowed,includingtheassumption thatthesumofthemeasureso
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