Question

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000. . .

Answer 1

Step-by-step explanation:

Given :-

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form, p/q what can you say about the prime factors of q?

To Find :-

(i) 43.123456789

(ii) 0.120120012000120000.

Solution :-

(i) 43.123456789

=> Method 1 :-

43.123456789 is terminating.

So, it would be a rational number.

$43.123456789 = \frac{43123456789}{1000000000} \\ \\ = \frac{43123456789}{ {(10)}^{9} } \\ \\ = \frac{43123456789}{ {(2 \: \times \: 5)}^{9} } \\ \\ = \frac{43123456789}{ {2}^{9} \times \: {5}^{9} }$

Hence, 43.123456789 is now in the form of p/q and the prime factors of q are in the terms of

2 and 5.

=> Method 2 :-

43.123456789 is terminating.

So, it would be a rational number.

In a terminating expansion of p/q, q is the form of

${2}^{n} \: \: {5}^{m}$

So, the prime factors of q will be 2 or 5 or both only.

(ii) 0.120120012000120000

0.120120012000120000 is not terminating and non-repeting.

So, it is not a rational number.

Answer 2

$\large{ \mathbb{ \colorbox{blac} { \boxed{ \boxed{ \colorbox{g} {-:Answer:-}}}}}}$

$\large{ \pmb{ \underline{ \underline{\frak{ \color{red}{Given::}}}}}}$

$\pink{➠}{ \sf{Real \: number}}$

$: : \implies{43.123456789}$

$: : \implies{120120012000120000...}$

$\large{ \pmb{ \underline{ \underline{\frak{ \color{bl}{To \: find::}}}}}}$

$\pink{➠}{ \sf{Which \: type \: of \: Real \: number \: is \: this?? }}$

$\pink{➠}{ \sf{If \: any \: of \: this \: is \: rational \: number \: then,what\:is\:prime \: factor \: }}$

$\sf{of \: q \: in \: \frac{p}{q} ?? \: \: \: \: \: \: }$

$\large{ \pmb{ \underline{ \underline{\frak{ \color{blue}{Conceptual \: point::}}}}}}$

$\pink{➠}{ \sf{Rational \: no. \: is \: written \: in \: \frac{p}{q} \: form\:and \: it \: is \:either \:}}$

$\sf{ terminating - non \: repeating\:or \: non \: terminating \: - repeating.}$

$\pink{➠}{ \sf{Irrational \: no. \: is \: not\: written \: in \: \frac{p}{q} \: form\:and \: it \: is }}$

$\sf{non \: terminating - non \: repeating.}$

$\large{ \pmb{ \underline{ \underline{\frak{ \color{green}{According \: to \: Question::}}}}}}$

$\pmb{ \bf{Let's \: start \: with \: concepts!!! }}$

$\pink{➠}{ \sf{(i) \rm43.123456789}}$

$\sf{Here \: it \: is \: terminating \: decimal \: expansion. }$

$\sf{So, it \: would \: be \: rational \: number. }$

$\bf{Then, we \: change \: the \: decimal \: expansion \: to \: fraction. }$

${: : \implies{ \sf{ \rm43.123456789 = \frac{43123456789}{1000000000} }}}$

${: : \implies{ \sf{ \rm43.123456789 = \frac{43123456789}{ { {(10)}^{9} } } }}}$

${: : \implies{ \sf{ \rm43.123456789 = \frac{43123456789}{ { {(2 \times 5)}^{9} } } }}}$

${: : \implies{ \sf{ \rm43.123456789 = \frac{43123456789}{ { {2}^{9} \times {5}^{9} } } }}}$

$\bf{After \: that, it \: is \: in \: the \: form \: of \: \frac{p}{q}\:and \: the \: prime \: factor }$

$\bf{ \: is \: 2 \: and \: 5.}$

$\pink{➠}{ \sf{(ii) \rm \: 0.120012000120000...}}$

$\sf{Here \: it \: is \: non \: terminating - non \: repeating \: decimal \: expansion. }$

$\sf{So, it \: would \: be \: irrational \: number. }$