Question

What is the condition for the decimal expansion of rational number to terminate ? Explain

Answer 1

The condition is the denominator q should be in $\bold{2^{a} \times 2^{b}}$  form.

Given:

$\mathrm{p} / \mathrm{q}$ contains a terminating expansion in decimal.

To find:

The necessary condition to be satisfied by q.

Solution:

The necessary condition which is to be satisfied by q such that the rational number p/q contains a terminating expansion in decimal is:

The denominator, that is, q should be in the form of $\left(2^{2} \times 5^{b}\right),$

Where, a, b denotes some non-negative integer, that is, a, b are positive integers.

Answer 2

Answer:

Let $x=\frac{p}{q}$ be a rational number, such that the prime factorisation of q is of the form $2^{n}5^{m}$, where n and m are non-negative integers . Then x has a decimal expansion which terminates.

Example:

1) Rational number x = 3/20

Denominator q = 20

q = 2×2×5 = 2²×5¹

q is of the form $2^{n}5^{m}$

Therefore,

3/20 is a terminating decimal.

$\frac{3}{20}$

= $\frac{3}{2^{2}\times 5}$

= $\frac{3×5}{2^{2}\times 5^{2}}$

= $\frac{15}{(10)^{2}}$

= $0.15$ [ Terminating decimal ]

Therefore,

$\frac{3}{20} = 0.15$

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