Prove that the following are irrationals:
(i)1/✓2
(ii) 7✓5
(ii) 6 + ✓2
Answers
Answer:
3r d answer for this question
Answer:
i)
2
1
Let us assume
2
1
is rational.
So we can write this number as
2
1
=
b
a
---- (1)
Here, a and b are two co-prime numbers and b is not equal to zero.
Simplify the equation (1) multiply by
2
both sides, we get
1=
b
a
2
Now, divide by b, we get
b=a
2
or
a
b
=
2
Here, a and b are integers so,
a
b
is a rational number,
so
2
should be a rational number.
But
2
is a irrational number, so it is contradictory.
Therefore,
2
1
is irrational number.
(ii) 7
5
Let us assume 7
5
is rational.
So, we can write this number as
7
5
=
b
a
---- (1)
Here, a and b are two co-prime numbers and b is not equal to zero.
Simplify the equation (1) divide by 7 both sides, we get
5
=
7b
a
Here, a and b are integers, so
7b
a
is a rational
number, so
5
should be a rational number.
But
5
is a irrational number, so it is contradictory.
Therefore, 7
5
is irrational number.
(iii) 6+
2
Let us assume 6+
2
is rational.
So we can write this number as
6+
2
=
b
a
---- (1)
Here, a and b are two co-prime number and b is not equal to zero.
Simplify the equation (1) subtract 6 on both sides, we get
2
=
b
a
−6
2
=
b
a−6b
Here, a and b are integers so,
b
a−6b
is a rational
number, so
2
should be a rational number.
But
2
is a irrational number, so it is contradictory.
Therefore, 6+
2
is irrational number.
Step-by-step explanation: