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 /q what can you say about the prime factors of q?

(1) 43.123456789

(2) 0.120120012000120000. . .

(3) `43.bar(123456789)`

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#### Solution

(1) 43.123456789

It is a rational number and q will have factors of 2 and 5 only.

(2) 0.120120012000120000. . .

It is an irrational number as it is non-terminating and non-repeating.

(3) `43.bar(123456789)`

It is rational number, as it is non-terminating but repeating. The factors of q will be, apart from 2 and 5, some other factor also.

Concept: Revisiting Rational Numbers and Their Decimal Expansions

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