Archimedes, about 2300 years ago, showed that the rational number
22
7
is greater than
π
, so it is a potential candidate. Let’s see if it’s less than
√
10
. We can do that by seeing if it’s square is less than
10
.
(
22
7
)
2
=
22
2
7
2
=
484
49
<
490
49
=
10
.
So it looks like the very common rational approximation of
π
is in fact a rational number between
π
and
√
10
.
By the way, it appears that the second most common rational approximation of
π
, namely
355
113
, is also greater than
π
, so forms a second answer to this question, with
π
<
355
113
<
22
7
<
√
10
.
