The following real number has decimal expansion as given below. Decide whether they are rational or not. If they are rational and of the form $\frac{p}{q}$ , what is the index of prime factors of $q$ ?
$43. ¯ ¯¯¯¯¯¯¯¯¯ ¯ 1234$

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Solution

let us assume $x = 43.1234$
$x = 43.12341234 = 43. ¯ ¯¯¯¯¯¯¯¯¯ ¯ 1234$ .... eq1
$x \times 10^{4} = 431234.1234...$ .... eq2
Subtract eq(1) from eq(2), we get
$9999 x = 431234$
$x = \frac{431234}{9999} = \frac{p}{q}$
Hence, $q = 9999 = 9^{1} \times 11^{1} \times 101^{1}$ .

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The following real number has decimal expansion as given below. Decide whether they are rational or not. If they are rational and of the form pq, what is the index of prime factors of q?43.¯¯¯¯¯¯¯¯¯¯¯1234

let us assume x=43.1234x=43.12341234=43.¯¯¯¯¯¯¯¯¯¯¯1234 .... eq1x×104=431234.1234... .... eq2Subtract eq(1) from eq(2), we get 9999x=431234x=4312349999=pqHence, q=9999=91×111×1011.

The following real number has decimal expansion as given below. Decide whether they are rational or not. If they are rational and of the form $\frac{p}{q}$ , what is the index of prime factors of $q$ ?
$43. ¯ ¯¯¯¯¯¯¯¯¯ ¯ 1234$

let us assume $x = 43.1234$
$x = 43.12341234 = 43. ¯ ¯¯¯¯¯¯¯¯¯ ¯ 1234$ .... eq1
$x \times 10^{4} = 431234.1234...$ .... eq2
Subtract eq(1) from eq(2), we get
$9999 x = 431234$
$x = \frac{431234}{9999} = \frac{p}{q}$
Hence, $q = 9999 = 9^{1} \times 11^{1} \times 101^{1}$ .