Answer:
Any irrational number, e.g.
√
2
Step-by-step explanation:
x
+
1
4
is irrational if and only if
x
is irrational.
Equivalently,
x
+
1
4
is rational if and only if
x
is rational.
To prove this we can proceed as follows:
First suppose that
x
+
1
4
is rational.
Then there are some integers
p
,
q
, with
q
>
0
such that:
x
+
1
4
=
p
q
Subtracting
1
4
from both sides, this becomes:
x
=
p
q
−
1
4
=
4
p
−
q
4
q
which is rational.
Conversely, if
x
is rational, then there are integers
m
,
n
with
n
>
0
such that
x
=
m
n
and we find:
x
+
1
4
=
m
n
+
1
4
=
4
m
+
n
4
n
which is also rational.
