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) 59.120120012000…
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 = 22 × 53
Which is of the form 2m × 5n 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. 32 × 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 = 25 × 56
Is of the form 2m × 5n 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:
- Choose the correct statement:
- Between two rational numbers:
- The product of any two irrational numbers is:
- Which of the following is an irrational number?
- The following is an irrational number?
- A rational number between √2 and √3 is:
- The decimal expansion of the rational number:
- 2√3 + √3 is equal to:
- The number (2 – √3)2 is: