Chapter 1: Real Numbers

Q
Real Numbers Solutions

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  what can you say about the prime factors of q?

(i) 43.123456789

(ii) 0.120120012000120000. . .

(iii) $43.\overline{123456789}$

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

And   $x = \frac{p}{q}$   be a rational number, such that the prime factorization of q is not of the form $2^{n}5^{m}$ , where n, m are non-negative integers. Then, $x$  has a decimal expansion which non-terminating repeating.

(i)

$43.123456789 = \frac{43123456789}{100000000} = \frac{43123456789}{10^{9}} = \frac{43123456789}{2^{9} \times 5^{9}}$

Hence, 43.123456789 is now in the form of   $\frac{p}{q}$  and prime factors of q are in terms of 2 and 5.

(ii) 0.120120012000120000. . .

It is non-terminating non-repeating decimal expansions, so 0.120120012000120000. . . is an irrational number.

(iii) $43.\overline{123456789}$

It is non-terminating repeating decimal expansions, so $43.\overline{123456789}$  is a rational number of the form of  $\frac{p}{q}$   and prime factors of q are not in terms of 2 and 5.

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