Prove that the following are irrationals.
Q.3
Sol :
Let us assume that
is rational.
∴
There exist co-prime integers
a
and
b
(b ≠
0)
such that,
=
a
b
∴
=
7
a
Since
a
and
b
are integers,
∴
7b
a
is rational
it implies that
is also rational,
But this contradicts the fact that
∴
Our assumption that
Lets prove this by contradiction
It should be in form of a/b
it implies that 7b is also integer
Integer
Integer
∴
is irrational.
b
Exercise 1.3
Sol :
Let us assume that
is rational.
∴
There exist co-prime integers
a
and
b
(b ≠
0)
such that,
=
a
b
∴
=
b
a – 6b
Since
a
and
b
are integers,
∴
b
a – 6b
is rational
which implies that
is also rational,
But this contradicts the fact that
∴
Our assumption that
Lets prove this by contradiction
It should be in form of a/b
it implies that a – 6b is also integer
Integer
Integer
∴
is irrational.
∴
=
a
b
–
6
Prove that the following are irrationals.
Q.3
Exercise 1.3