Real Numbers

$\displaystyle \textbf{Question 1. }\text{Without actually performing the long division, state whether the following rational numbers}$
$\displaystyle \text{will have a terminating decimal expansion or a non-terminating repeating decimal expansion.}$
$\displaystyle (i)\ \frac{23}{8}\qquad (ii)\ \frac{125}{441}\qquad (iii)\ \frac{35}{50}\qquad (iv)\ \frac{77}{210}\qquad (v)\ \frac{129}{2^{2}\times5^{7}\times7^{17}}\qquad (vi)\ \frac{987}{10500}$
$\displaystyle \text{Answer:}$
$\displaystyle (i)\ \frac{23}{8}=\frac{23}{2^{3}}$
$\displaystyle \text{The denominator is of the form }2^{m}\times5^{n}.$
$\displaystyle \therefore \frac{23}{8}\text{ has a terminating decimal expansion.}$
$\displaystyle \text{It terminates after }3\text{ places of decimals.}$
$\\$
$\displaystyle (ii)\ \frac{125}{441}=\frac{125}{3^{2}\times7^{2}}$
$\displaystyle \text{The denominator is not of the form }2^{m}\times5^{n}.$
$\displaystyle \therefore \frac{125}{441}\text{ has a non-terminating repeating decimal expansion.}$
$\\$
$\displaystyle (iii)\ \frac{35}{50}=\frac{7}{10}=\frac{7}{2\times5}$
$\displaystyle \text{The denominator is of the form }2^{m}\times5^{n}.$
$\displaystyle \therefore \frac{35}{50}\text{ has a terminating decimal expansion.}$
$\displaystyle \text{It terminates after }1\text{ place of decimal.}$
$\\$
$\displaystyle (iv)\ \frac{77}{210}=\frac{11}{30}=\frac{11}{2\times3\times5}$
$\displaystyle \text{The denominator is not of the form }2^{m}\times5^{n}.$
$\displaystyle \therefore \frac{77}{210}\text{ has a non-terminating repeating decimal expansion.}$
$\\$
$\displaystyle (v)\ \frac{129}{2^{2}\times5^{7}\times7^{17}}$
$\displaystyle \text{The denominator contains the factor }7.$
$\displaystyle \text{So, it is not of the form }2^{m}\times5^{n}.$
$\displaystyle \therefore \frac{129}{2^{2}\times5^{7}\times7^{17}}\text{ has a non-terminating repeating decimal expansion.}$
$\\$
$\displaystyle (vi)\ \frac{987}{10500}=\frac{47}{500}$
$\displaystyle =\frac{47}{2^{2}\times5^{3}}$
$\displaystyle \text{The denominator is of the form }2^{m}\times5^{n}.$
$\displaystyle \therefore \frac{987}{10500}\text{ has a terminating decimal expansion.}$
$\displaystyle \text{It terminates after }3\text{ places of decimals.}$
$\\$

$\displaystyle \textbf{Question 2. }\text{Write down the decimal expansions of the following rational numbers by writing their}$
$\displaystyle \text{denominators in the form }2^{m}\times5^{n},\text{ where }m,n\text{ are non-negative integers.}$
$\displaystyle (i)\ \frac{3}{8}\qquad (ii)\ \frac{13}{125}\qquad (iii)\ \frac{7}{80}\qquad (iv)\ \frac{14588}{625}\qquad (v)\ \frac{129}{2^{2}\times5^{7}}$
$\displaystyle \text{Answer:}$
$\displaystyle (i)\ \frac{3}{8}=\frac{3}{2^{3}}$
$\displaystyle =\frac{3\times5^{3}}{2^{3}\times5^{3}}$
$\displaystyle =\frac{375}{1000}$
$\displaystyle =0.375$
$\\$
$\displaystyle (ii)\ \frac{13}{125}=\frac{13}{5^{3}}$
$\displaystyle =\frac{13\times2^{3}}{2^{3}\times5^{3}}$
$\displaystyle =\frac{104}{1000}$
$\displaystyle =0.104$
$\\$
$\displaystyle (iii)\ \frac{7}{80}=\frac{7}{2^{4}\times5}$
$\displaystyle =\frac{7\times5^{3}}{2^{4}\times5^{4}}$
$\displaystyle =\frac{875}{10000}$
$\displaystyle =0.0875$
$\\$
$\displaystyle (iv)\ \frac{14588}{625}=\frac{14588}{5^{4}}$
$\displaystyle =\frac{14588\times2^{4}}{2^{4}\times5^{4}}$
$\displaystyle =\frac{233408}{10000}$
$\displaystyle =23.3408$
$\\$
$\displaystyle (v)\ \frac{129}{2^{2}\times5^{7}}$
$\displaystyle =\frac{129\times2^{5}}{2^{7}\times5^{7}}$
$\displaystyle =\frac{4128}{10000000}$
$\displaystyle =0.0004128$
$\\$

$\displaystyle \textbf{Question 3. }\text{Write the denominator of the rational number }\frac{257}{5000}\text{ in the form }$
$\displaystyle 2^{m}\times5^{n},\text{ where }m,n\text{ are non-negative integers. Hence, write the decimal expansion,} \\ \text{without actual division.}$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{257}{5000}=\frac{257}{2^{3}\times5^{4}}$
$\displaystyle =\frac{257\times2}{2^{4}\times5^{4}}$
$\displaystyle =\frac{514}{10000}$
$\displaystyle \therefore \frac{257}{5000}=0.0514$
$\\$

$\displaystyle \textbf{Question 4. }\text{What can you say about the prime factorisations of the denominators }$
$\displaystyle \text{of the following rationals:}$
$\displaystyle (i)\ 42.123456789\qquad (ii)\ 43.\overline{123456789}\qquad \\ (iii)\ 27.142857\qquad [\mathrm{CBSE}\ 2010]\qquad (iv)\ 0.120120012000120000\ldots$
$\displaystyle \text{Answer:}$
$\displaystyle (i)\ 42.123456789\text{ is a terminating decimal expansion.}$
$\displaystyle \therefore \text{The denominator has only the prime factors }2\text{ and }5.$
$\\$
$\displaystyle (ii)\ 43.\overline{123456789}\text{ is a non-terminating repeating decimal expansion.}$
$\displaystyle \therefore \text{The denominator has at least one prime factor other than }2\text{ or }5.$
$\\$
$\displaystyle (iii)\ 27.142857\text{ is a terminating decimal expansion.}$
$\displaystyle \therefore \text{The denominator has only the prime factors }2\text{ and }5.$
$\\$
$\displaystyle (iv)\ 0.120120012000120000\ldots\text{ is a non-terminating non-repeating decimal expansion.}$
$\displaystyle \therefore \text{It is an irrational number and cannot be expressed in the form }\frac{p}{q}.$
$\displaystyle \therefore \text{Its denominator has no prime factorisation as a rational number.}$
$\\$

$\displaystyle \textbf{Question 5. }\text{A rational number in its decimal expansion is }327.7081.\text{ What can you say about}$
$\displaystyle \text{the prime factors of }q,\text{ when this number is expressed in the form }\frac{p}{q}\text{? Give reasons.}$
$\displaystyle \text{Answer:}$
$\displaystyle 327.7081\text{ is a terminating decimal expansion.}$
$\displaystyle \therefore \text{When expressed in the form }\frac{p}{q}\text{ in lowest terms,}$
$\displaystyle \text{the prime factorisation of }q\text{ contains only the prime factors }2\text{ and }5.$
$\\$