The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational or not. If they are rational, and of them , what can you say about the prime factors of q?i) 43.123456789ii) 0.1201120012000120000....iii) 43.123456789

Foundation

Mathematics Foundation

Properties of Real Number

Irrational 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 or not. If they are rational, and of them $\frac{p}{q}$ , what can you say about the prime factors of q?
i) 43.123456789

ii) 0.1201120012000120000....

iii) 43.123456789

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i) 43.123456789 is terminating or rational

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

This is in the form of

\frac{p}{q}

, and the prime factors of q are in terms of 2 and 5.

ii) 0.120120012000120000..... is non-terminating and non-repeating, it is irrational. Hence cannot be expressed in the form of

\frac{p}{q}

iii) 43.123456789 is non-terminating but repeating. So, it would be rational.

Let x = 43.123456789 ...(1)

1000000000 x = 43123456789,123456789 ...(2)

(2) - (1) \Rightarrow 999999999 x = 43123456746 \Rightarrow x = \frac{43123456746}{999999999} = \frac{43123456746}{3^{4} \times 37^{1} \times 333667^{1}}

In a non-termination repeating expansion of

\frac{p}{q}

, q will have factors 3,37,333667.