Theorem-1.3 : Let x be a rational number whose decimal form terminates. Then x can
be expressed in the form of
p
q
, where p and q are coprime, and the p
Write the following terminating decimals in the form of
p
q
, q¹ 0 and p, q are
co-primes
(i) 15.265
(ii) 0.1255
(iii) 0.4
(iv) 23.34
(v) 1215.8
Write the denominators in 2n5
m
form.
You are probably wondering what happens the other way round. That is, if we have
a rational number in the form of
p
q
and the prime factorization of q is of the form 2n5
m
,
where n, m are non-negative integers, then does
p
/
q
have a terminating decimal expansion?
So, it seems to make sense to convert a rational number of the form
p
/
q
, where q is of the
form 2n5
m
, to an equivalent rational number of the form
a
b
, where b is a power of 10. Let us go
back to our examples above and work backwards.
(i)
3/8=3/3= 3×5 3
/2 3×5 3=375/103
(ii)
26/25 = 26/52 =13/23 /22×
52=104/102=1.04
(iii) 7/80=7/24×5=/7×53/24×54=875/104=0.0875
25/2= 53/5×2
=125/10=12.5
So, these examples show us how we can convert a rational number of the form
p
/
q
,
where q is of the form 2n5
m
, to an equivalent rational number of the form
a
/
b
, where b is a
power of 10. Therefore, the decimal forms of such a rational number terminate. We find
that a rational number of the form
p
/
q
, where q is a power of 10, is a terminating decimal.
So, we conclude that the converse of theorem 1.3 is also true which can be formally