prove that 3a is an irrational number if a is an irrational nuber
Answers
Step-by-step explanation:
Since we know that the product of any number to the irrational number then we get irrational number.
So if 3 is a number and 'a' is any irrational number the its product will be irrational number.
Answer:
Let us assume on the contrary that
3
is a rational number.
Then, there exist positive integers a and b such that
3
=
b
a
where, a and b, are co-prime i.e. their HCF is 1
Now,
3
=
b
a
⇒3=
b
2
a
2
⇒3b
2
=a
2
⇒3 divides a
2
[∵3 divides 3b
2
]
⇒3 divides a...(i)
⇒a=3c for some integer c
⇒a
2
=9c
2
⇒3b
2
=9c
2
[∵a
2
=3b
2
]
⇒b
2
=3c
2
⇒3 divides b
2
[∵3 divides 3c
2
]
⇒3 divides b...(ii)
From (i) and (ii), we observe that a and b have at least 3 as a common factor. But, this contradicts the fact that a and b are co-prime. This means that our assumption is not correct.
Hence,
3
is an irrational number.