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Q1.
Ans
Exercise1.4
Withoutactuallyperformingthelongdivision,statewhetherthe
followingrationalnumberswillhaveaterminatingdecimal
expansionoranon-terminatingrepeatingdecimalexpansion:
(i)13/3125
3125=55
Thedenominatorisoftheform5m.
Hence,thedecimalexpansionof13/3125isterminating.
(ii)17/8
8=23
Thedenominatorisoftheform2m.
Hence,thedecimalexpansionof17/8isterminating.
(iii)64/455
455=5x7x13
Sincethedenominatorisnotintheform2m×5n,anditalso
contains7and13asitsfactors,itsdecimalexpansionwillbenonterminatingrepeating.
(iv)15/1600
1600=26x52
Thedenominatorisoftheform2m×5n.
Hence,thedecimalexpansionof15/1600isterminating.
(v)29/343
343=73
Sincethedenominatorisnotintheform2m×5n,andithas7asits
factor,thedecimalexpansionof29/343isnon-terminating
repeating.
(vi)23/23x52
Thedenominatorisoftheform2m×5n.
Hence,thedecimalexpansionof23/23x52isterminating.
(vii)129/22x57x75
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Sincethedenominatorisnotoftheform2m×5n,anditalsohas7
asitsfactor,thedecimalexpansion129/22x57x75isnonterminatingrepeating.
(viii)
Thedenominatorisoftheform5n.
Hence,thedecimalexpansionof6/15isterminating.
(ix)
(x)
10=2x5
Thedenominatorisoftheform2m×5n.
Hence,thedecimalexpansionof35/50isterminating.
30=2x3x5
Sincethedenominatorisnotoftheform2m×5n,anditalso
has3asitsfactors,thedecimalexpansionof77/210isnonterminatingrepeating.
Q2. Writedownthedecimalexpansionsofthoserationalnumbersin
Question1abovewhichhaveterminatingdecimalexpansions.
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(viii)
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Q3.
Ans
Thefollowingrealnumbershavedecimalexpansionsasgiven
below.Ineachcase,decidewhethertheyarerationalornot.Ifthey
arerational,andoftheformp/qwhatcanyousayabouttheprime
factorofq?
(i)43.123456789(ii)0.120120012000120000…(iii)
(i)43.123456789
Sincethisnumberhasaterminatingdecimalexpansion,itisa
rationalnumberoftheformp/qandqisoftheform2m×5n
i.e.,theprimefactorsofqwillbeeither2or5orboth.
(ii)0.120120012000120000…
Thedecimalexpansionisneitherterminatingnorrecurring.
Therefore,thegivennumberisanirrationalnumber.
(iii)
Sincethedecimalexpansionisnon-terminatingrecurring,the
givennumberisarationalnumberoftheformp/qandqisnotof
theform2m×5ni.e.,theprimefactorsofqwillalsohaveafactor
otherthan2or5.
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