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 from , what can you say about the prime factors of q?i) 43.123456789

Foundation

Mathematics Foundation

HCF and LCM

Properties of Real Number

Rational Numbers

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 from $\frac{p}{q}$ , what can you say about the prime factors of q?
i) 43.123456789

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43.123456789

is terminating

So, it would be a rational number

43.123456789 = \frac{43.123456789}{1000000000}

= \frac{43.123456789}{10^{9}}

= \frac{43.123456789}{(2 \times 5)^{9}}

= \frac{43.123456789}{2^{9} \times 5^{9}}

Hence 43.123456789 is now in the form

\frac{p}{q}

And the prime factors of q are in terms of 2 and 5