### Chapter 1: Real Numbers

Q
##### Real Numbers Solutions

Question:

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion:

(i)              (ii)                  (iii)            (iv)           (v)

(vi)           (vii)          (viii)            (ix)                (x)

Solutions: We know that   x =   be a rational number, such that the prime factorization of q is of the form  , where n, m are non-negative integers. Then, x  has a decimal expansion which terminates.

And   x=    be a rational number, such that the prime factorization of q is not of the form    , where n, m are non-negative integers. Then, x  has a decimal expansion which non-terminating repeating.

(i)

If we factorize the denominator, we get
3125 = 5 × 5 × 5 × 5 × 5 =

So, the denominator is in the form of   so, ​    is terminating decimal expansion.

(ii)

If we factorize the denominator, we get
8 = 2 × 2 × 2 =
So, the denominator is in the form of     so,     is terminating decimal expansion.

(iii)

If we factorize the denominator, we get

455 = 5 × 7 × 13

So, the denominator is not in the form of   so,

is non-terminating repeating decimal expansion.

(iv)

If we factorize the denominator, we get

1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 =

So, the denominator is in the form of   . So,    is terminating decimal expansion.

(v)

If we factorize the denominator, we get

343 = 7 × 7 × 7 =

So, the denominator is not in the form of   . So,   is non-terminating repeating decimal expansion.

(vi)

Here, the denominator is in the form of     . So,    is terminating decimal expansion.

(vii)

Here, the denominator is not in the form of    only. So,   is non-terminating repeating decimal expansion.

(viii)

If we divide nominator and denominator both by 3 we get
So, the denominator is in the form of .   So,   is terminating decimal expansion.

(ix)

If we divide nominator and denominator both by 5, we get
If we factorize the denominator, we get

10 = 2 × 5

So, the denominator is in the form of   .  So,         is terminating decimal expansion.

(x)

If we divide nominator and denominator both by 7, we get
If we factorize the denominator, we get

30 = 2 × 3 × 5

So, the denominator is not in the form of  .  So,     is non-terminating repeating decimal expansion.

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