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Math43–PreparationforLiberalArtMajorsHandout1.5–Areyou“rational”or“irrational”?
FoxtrotByBillAmend
1).Groupwork:Inyourgroupof2or3students,makealistofalltheDIFFERENTnumbersyoucancomeupwith.
Inmathematics,thenumbersystemevolvedthroughtime:
• N:Atfirst,“numbers”meantsomethingyoucancount,likehowmanysheepafarmerowns.ThesearecalledNaturalNumbers(orcountingnumbers).
• W:Atsomepoint,theideaof“zero”cametobeconsideredasanumber.Ifthefarmerdoesnothaveanysheep,thenthenumberofsheepthatthefarmerownsiszero.NaturalNumberstogetherwith“zero”makeuptheWholeNumbers.
• Z:Evenmoreabstractthanzeroistheideaofnegativenumbers.If,inadditiontonothavinganysheep,thefarmerowessomeone3sheep,youcouldsaythatthenumberofsheepthatthefarmerownsisnegative3.TheexpandedsetofnumbersthatwegetbyincludingnegativeversionsofthecountingnumbersiscalledtheIntegers.
• Q:Thenextgeneralizationthatwecanmakeistoincludetheideaoffractions.Whileitisunlikelythatafarmerownsafractionalnumberofsheep,manyotherthingsinreallifearemeasuredinfractions,likeahalf-cupofsugar.Ifweaddfractionstothesetofintegers,wegetthesetofRationalNumbers.
• H:Therearenumbersthatcannotbeexpressedasafraction,andthesenumbersarecalledIrrationalbecausetheyarenotrational.
1 -1 13 if if
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a
IN
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= { O,±l
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2.Isthefraction!"equivalentto0.333… = 0. 3?Explainwhyorwhynot?
3.Convertthefollowingtoafraction:a).1.434343…. b).25. 84.Sortthenumbersinto2groups,rationalorirrational.Writethenumbersintheappropriatebubble. 0.8 64 0 32 -19 100- 2.343443444… 7
3 75
726 76.12 121 5
12
5.Plotandlabeleachnumberonthenumberlinebelow.
A 0.75 B 3
C 9
D 212-
E 1015
-
F 6.2
G 2- H
Rational Irrational
-5 -4 -3 -2 -1 0 1 2 3 4 5
3..
..
3*0. 9¥
.
-
= 1.43
use
1001004.4343... ) .
- 143.4343 . . |O( 25.5 )= 258.8,÷, /'s:* :
- / ( 1,4343 .D= -
.4343 ...
- 1 ( 25.5 )= - 25.8
9*343.1=14 965=2331.4343 . ... 25.5 = 2¥
06kg 0-19 ft 2.343443444 :
3 ftto T
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.
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6.WorkwithapartnertodrawaVennDiagramfortherealnumbers.Makesuretoincludethefollowing:naturalnumbers,wholenumbers,integers,rational,andirrationalnumbers.
DivisionTricks:Divisibleby2:Thenumberiseven–itendsin0,2,4,6,or8 Ex:Divisibleby3:Thesumofthedigitsisdivisibleby3 Ex:Divisibleby5:Thenumberendsin0or5 Ex:Divisibleby6:Thenumberisdivisibleby2and3 Ex:Divisibleby9:Thesumofthedigitsisdivisibleby9 Ex:Divisibleby10:Thenumberendsin0 Ex:
,@@@*f@7311177534
24 : 2+4=6
111 : 1+1+1=3
37936875.
222 : z€+Tf¥ 6
ys:ztI8Ihn=i8✓
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7.Primenumbersarenumbersdivisiblebyonly1andthemselves.Listafewprimenumbers.8.Forthenumbers1-100below,circlealltheprimenumbers.Doyouthinktherearemoreprimenumbers?Do
youthinktherearefinitenumberofprimesorinfinitelymany?Explainbriefly.
t2 3 57
13
101/00/0/0#9=3.3
0/01/-0-0-composite- onto.oxyoo.ox-onto,¥⇐t÷⇐⇒on.ro/~r/r- / - D
I t
7. 13 5 's
whidsijspid Riemann Hypothesis
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9.Whichofthefollowingnumbersisaprimenumber?Explainyourstrategytoyourpartner. 3931 5009 4100 8999
10.Thecastoftheschoolplayhadapartyatthedramateacher’shouse.Therewere20cookiesand40carrotsticksservedasrefreshment.Eachcastmemberhadthesamenumberofwholecookiesandthesamenumberofwholecarrotsticks,andnothingwasleftover.Thedramateacherdidnoteat.Howmanycastmembersmighthavebeenattheparty?Explainyouranswer.
11.Giventhefollowingclues,whatismynumber?
• Mynumberisaperfectsquare.• Theonlyprimenumberisinitsprimefactorizationis2.• Mynumberisafactorof80.• Thesumofitsdigitsisodd.
Prime # it 0end Prime
2/7#(31 5=961
31 #236## F99t95 !!* HK Lily
.
20 cookies
40 carrot sticks
# Cast Members : 20,10 , 5,4 ,
2,1
l6:l+6=@ ←
T3/6=3.3-2.24=2.24/9=7.791=3.364=2.2-2.2-2.216=2.2-2.28/1=3.3-3.32/5=5.51/00=5.5-2.2
Ex : Find the Prime Factorization
64 11}Fa¥Ee
. \20.4@ - 41 ' 1 '
�2�. @20 .@
64=2-2-2.21500= 5. 5. 5. 3. 2.2
n -
15 100 Multiply to
051.130 /\check
• . 10 .
10*5.0*0
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HOMEWORK:1).Describethedifferencebetweenrationalandirrationalnumbers.Youcanuseexamplestohelpsupportyour
answer.
2).Identifyeachnumberasrationalorirrational.
3.676776777…. 76.12 32 49 57-
3).Isitpossibletoconvert3.676776777….toafraction?Ifyes,findthefraction?Ifno,whynot?4).Isitpossibletoconvert 76.12 toafraction?Ifyes,findthefraction?Ifno,whynot?5).Statewhetherthefollowingstatementsaretrueorfalse.a).Arealnumberiseitherarationalorirrational.
b).Anintegerisalsoarationalnumber.
c).Zeroisnotaninteger.
d).1.625isanirrationalnumber.
e).0.333...isarationalnumber.
f).3.31662479...isairrationalnumber.
g).9/11isnotawholenumberbutitisarationalnumber.
h).20%isawholenumber.
1.5
