#### In each case, state whether the following numbers are rational or irrational. If they are rational and expressed in the form p/q, where p and q are coprime integers, then what can you say about the prime factors of q?

(i) 279.034

(iii) 3.010010001…

(iv) 39.546782

(v) 2.3476817681…

(vi) 59.1201200**12000…**

**Solution:**

(i) 279.034 is a rational number because it has terminating decimals

279.034 = 279034/1000 (in p/q form)

= 139517/500 (Dividing by 2)

We know that

Factors of 500 = 2 × 2 × 5 × 5 × 5 = 2^{2} × 5^{3}

Which is of the form 2^{m} × 5^{n} where m and n are positive integers.

It is a rational number as it has recurring or repeating decimals

Consider

x = 76.17893 17893 17893 …..

100000x = 7617893.178931789317893…..

By subtraction

99999x = 7617817

x = 7617817/99999 which is of p/q form

We know that

Prime factor of 99999 = 3 × 3 × 11111

q has factors other than 2 or 5 i.e. 3^{2} × 11111

(iii) 3.010010001….

It is neither terminating decimal nor repeating

Therefore, it is an irrational number.

(iv) 39.546782

It is terminating decimal and is a rational number

39.546782 = 39546782/1000000 (in p/q form)

= 19773391/500000

We know that p and q are coprime

Prime factors of q = 2^{5} × 5^{6}

Is of the form 2^{m} × 5^{n} where m and n are positive integers

(v) 2.3476817681…

Is neither terminating nor repeated decimal

Therefore, it is an irrational number.

(vi) 59.120120012000….

It is neither terminating decimal nor repeated

Therefore, it is an irrational number.

**More Solutions:**

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