Exercise
1.4
1.
Without actually performing the long division, state whether the
following rational numbers will have a terminating decimal expansion or a
non-terminating decimal expansion.
(i) 13
3125
Solution
: q = 3125 = 5 x 5
x 5 x
5 x 5 = 55
Therefore,
denominator is of the form 2n5m, where
m = 5 and n = 0.
It
means rational number 13 has a terminating decimal
expansion.
3125
(ii)
17
8
Solution
: q = 8 = 2 x 2 x 2 = 23
Therefore,
denominator is of the form 2n5m, where
m = 0 and n = 3.
It
means rational number 17 has a terminating decimal
expansion.
8
(iii) 64
455
Solution
: q = 455 = 5 x 91
Therefore,
denominator is not of the form 2n5m. It means rational number 64
has 455
a non-terminating
repeating decimal expansion.
(iv) 15 = 3
1600
320
Solution
: q = 320 = 2 x 2
x 2 x
2 x 2
x 2 x 5
Therefore,
denominator is of the form 2n5m, where
m = 1 and n = 6.
It
means rational number 15 has
a terminating decimal expansion.
1600
(v) 29
343
Solution
: q = 343 = 7
x 7 x 7
Therefore,
denominator is not of the form 2n5m. It means
rational number 29
343
has non-terminating
repeating decimal expansion.
(vi) 23
2352
Solution
: q = 23 x 52
Therefore,
denominator is of the form 2n5m, where
m = 2 and n = 3.
It
means rational number 23 has terminating decimal
expansion.
2352
(vii) 129
225775
Solution
: q = 22
x 57 x 75
Therefore,
denominator is not of the form 2n5m.It
means rational number 129 225775
has non-terminating repeating decimal expansion.
(viii) 6 = 2
15
5
Solution
: q = 5 = 51
Therefore,
denominator is of the form 2n5m, where
m = 1 and n = 0.
It
means rational number 6 has terminating decimal
expansion.
15
(ix) 35
= 7
50 10
Solution
: q = 10 = 2
x 5 = 2151
Therefore,
denominator is of the form 2n5m, where
m = 1 and n = 1..
It
means rational number 35 has terminating decimal expansion.
50
(x) 77 = 11
210
30
Solution
: q = 30 = 5
x 3 x 2
Therefore,
denominator is not of the form 2n5m .It
means rational number 77
210
has non-terminating repeating decimal expansion.
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2.
Write down the decimal expansions of those rational numbers in question 1 above
which have terminating decimal expansions.
(i) 13 = 0.00416
3125
(ii)
17 =
2.125
8
(iv) 15 = 3 = 0.009375
600
320
(vi) 23 = 0.115
2352
(viii) 6 = 2 = 0.4
15
5
(ix) 35
= 7 = 0.7
50 10
3. The
following real numbers have decimal expansions as given below.
In each case, decide whether they are rational or not.
If they are rational and are of the form p ,
q
what can you say about the prime factors of q?
In each case, decide whether they are rational or not.
If they are rational and are of the form p ,
q
what can you say about the prime factors of q?
(i) 43.123456789
It is a rational number because decimal expansion is
terminating
and it can be expressed in p form, where the factors of
q
q are of the form 2n5m, where n and m are non-negative integers.
and it can be expressed in p form, where the factors of
q
q are of the form 2n5m, where n and m are non-negative integers.
(ii) 0.120120012000120000...
It is
irrational as the decimal expansion is neither terminating nor non-terminating
repeating.
(iii) 43.̅1̅2̅3̅4̅5̅6̅7̅8̅9̅
It is a rational number because decimal expansion is
non terminating and repeating and it can be expressed in p form,
non terminating and repeating and it can be expressed in p form,
q
where the factors of q are not of the form 2n5m,
where n and m are non-negative integers.
where the factors of q are not of the form 2n5m,
where n and m are non-negative integers.
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