Question

irrational value of n where 2<n<3​

Answer 1

Explanation:
Adding on to the other answer, we can easily generate as many such numbers as we'd like by noting that the sum of an irrational with a rational is irrational. For example, we have the well known irrationals
e
=
2.7182
...
and
π
=
3.1415
...
.
So, without worrying about the exact bounds, we can definitely add any positive number less than
0.2
to
e
or subtract a positive number less than
0.7
and get another irrational in the desired range. Similarly, we can subtract any positive number between
0.2
and
1.1
and get an irrational between
2
and
3
.
2
<
e
<
e
+
0.1
<
e
+
0.11
<
e
+
0.111
<
...
<
e
+
1
9
<
3
2
<
π
−
1.1
<
π
−
1.01
<
π
−
1.001
<
...
<
π
−
1
<
3
This can be done with any irrational for which we have an approximation for at least the integer portion. For example, we know that
1
<
√
2
<
√
3
<
2
. As
√
2
and
√
3
are both irrational, we can add
1
to either of them to get further irrationals in the desired range:
2
<
√
2
+
1
<
√
3
+
1
<
3