# Real Numbers : Exercise 1.4 (Mathematics NCERT Class 10th)

Q.1     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) ${{13} \over {3125}}$                                   (ii) ${{17} \over 8}$
(iii) ${{64} \over {455}}$                                   (iv) ${{15} \over {1600}}$
(v) ${{29} \over {343}}$                                    (iv) ${{23} \over {{2^3}{5^2}}}$
(vii) ${{129} \over {{2^2}{5^7}{7^5}}}$             (viii) ${6 \over {15}}$
(ix) ${{35} \over {50}}$                                      (x) ${{77} \over {210}}$

Sol.     We know that if the denominator of a rational number has no prime factors other than 2 or 5, then it is expressible as a terminating, otherwise it has non - terminating  repeating decimal representation. Thus, we will have to check the prime factors of the denominators of each of the given rational numbers.

(i) In ${{13} \over {3125}}$, the denominator is 3125. We have, 3125 = 5 × 5 × 5 × 5 × 5.
Thus, 3125 has 5 as the only prime factor.
Hence, ${{13} \over {3125}}$ must have a terminating decimal representation.

(ii) In ${{17} \over 8}$, the denominator is 8. We have, 8 = 2 × 2 × 2
Thus, 8 has 2 as the only prime factor.
Hence, ${{17} \over 8}$ must have a terminating decimal representation.

(iii) In ${{64} \over {455}}$, denominator is 455. We have, 455 = 5 × 7 × 13
Clearly, 455 had prime factors other than 2 and 5. So, it will not have a terminating decimal representation.

(iv) In ${{15} \over {1600}}$, the denominator is 1600. We have, 1600
= 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5
Thus, 1600 has only 2 and 5 as prime factors.
Hence, ${{15} \over {1600}}$ must have a terminating decimal representation.

(v) In ${{29} \over {343}}$, the denominator is 343. We have, 343 = 7 × 7 × 7
Clearly, 343 has prime factors other than 2 and 5.
So, it will not have terminating decimal representation.

(vi) In ${{23} \over {{2^3}{{.5}^2}}}$ Clearly, the denominator ${2^3}{.5^2}$ has only 2 and 5 as prime factors.
Hence, ${{23} \over {{2^3}{{.5}^2}}}$ must have a terminating decimal representation.
(vii) In ${{129} \over {{2^2}{{.5}^7}{{.7}^5}}}$ Clearly, the denominator ${{2^2}{{.5}^7}{{.7}^5}}$ has prime factors other than 2 and 5.So, it will not have terminating decimal representation.

(viii) In ${6 \over {15}}$, we have 15 = 3 × 5
Clearly, 15 has prime factors other than 2 and 5. So, it will not have terminating decimal representation.

(ix) In ${{35} \over {50}}$ , we have 50 = 2 × 5 × 5 The denominator has only 2 and 5 as prime factors. Hence, ${{35} \over {50}}$ must have a terminating decimal representation.

(x) In ${{77} \over {210}}$, the denominator is 210. We have, 210 = 2 × 3 × 5 × 7
Clearly, 210 has prime factors other than 2 and 5.
So, it will not have terminating decimal representation.

Q.2     Write down the decimal expansion of those rational numbers in Question 1 above which have terminating decimal expansions.
Sol.      (i) ${{13} \over {3125}} = {{13} \over {5 \times 5 \times 5 \times 5 \times 5}}$
$= {{13 \times 2 \times 2 \times 2 \times 2 \times 2} \over {5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 2}}$
$= {{13 \times 32} \over {10 \times 10 \times 10 \times 10 \times 10}} = {{416} \over {100000}}$ = 0.00416

(ii) ${{17} \over 8} = {{17 \times {5^3}} \over {{2^3} \times {5^3}}} = {{17 \times {5^3}} \over {{{10}^3}}} = {{17 \times 125} \over {{{10}^3}}}$
$= {{2125} \over {1000}} = 2.125$

(iii) Non - terminating.

(iv) ${{15} \over {1600}} = {{15} \over {{2^6} \times {5^2}}} = {{15} \over {{2^4} \times {2^2} \times {5^2}}}$
$= {{15} \over {{2^4} \times {{10}^2}}} = {{15 \times {5^4}} \over {{2^4} \times {5^4} \times {{10}^2}}}$
$= {{15 \times 625} \over {{{10}^4} \times {{10}^2}}} = {{9375} \over {1000000}} = 0.009375$

(v) Non - terminating.

(vi) ${{23} \over {{2^3}{{.5}^2}}} = {{23} \over {{{2.2}^2}{{.5}^2}}} = {{23} \over {{{2.10}^2}}} = {{23 \times 5} \over {2 \times 5 \times {{10}^2}}}$
$= {{115} \over {10 \times {{10}^2}}} = {{115} \over {1000}} = 0.115$

(vii) Non - terminating.

(viii) ${6 \over {15}} = {2 \over 5} = {4 \over {10}} = 0.4$

(ix) ${{35} \over {50}} = {{35 \times 2} \over {50 \times 2}} = {{70} \over {100}} = 0.70$

(x) Non - terminating.

Q.3     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 ${p \over q}$, what can you say about the prime factors of q ?
(i) 43.123456789      (ii) 0.120120012000120000.......      (iii)
$43.\overline {123456789}$

Sol.     (i) 43.123456789 is terminating.
So, it represents a rational number.
Thus, 43.123456789 = ${p \over q}$, where $q = {10^9}$.

(ii) 0.12012001200012000... is non - terminating and non-repeating. So, it is irrational.

(iii) $43.\overline {123456789}$ is non - terminating but repeating. So, it is rational.
Thus, $43.\overline {123456789}$ $= {p \over q}$, where q = 999999999.