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)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
(ix)
(x)
Solution:
According to Theorem, any given rational number of the form
where p and q are co-prime, has a terminating decimal expansion if q is of the form
, where m and n are non-negative integers.
(i)
Therefore, denominator is of the form
, where m=5 and n=0.
It means rational number
has a terminating decimal expansion.
(ii)
Therefore, denominator is of the form
, where m=0 and n=3.
It means rational number
has a terminating decimal expansion.
(iii)
Therefore, denominator is not of the form
, where m and n are non-negative integers.
It means rational number
has a non-terminating repeating decimal expansion.
(iv)
Therefore, denominator is of the form
, where m=1 and n=6.
It means rational number
has a terminating decimal expansion.
(v)
Therefore, denominator is not of the form
, where m and n are non-negative integers.
It means rational number
has non-terminating repeating decimal expansion.
(vi)
Therefore, denominator is of the form
, where m=2 and n=3 are non-negative integers.
It means rational number
has terminating decimal expansion.
(vii)
Therefore, denominator is not of the form
, where m and n are non-negative integers.
It means rational number
has non-terminating repeating decimal expansion.
(viii)
Therefore, denominator is of the form
, where m=1 and n=0.
It means rational number
has terminating decimal expansion.
(ix)
Therefore, denominator is of the form
, where m=1 and n=1..
It means rational number
has terminating decimal expansion.
(x)
Therefore, denominator is not of the form
, where m and n are non-negative integers.
It means rational number
has non-terminating repeating decimal expansion.
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