Let
ξ
\xi
be an irrational number with simple continued fraction expansion
ξ
=
[
a
0
;
a
1
,
…
,
a
i
,
…
]
\xi = [{a_0};{a_1}, \ldots ,{a_i}, \ldots ]
, and
p
i
/
q
i
{p_i}/{q_i}
be its
i
i
th convergent. In this paper we first prove the duality of some inequalities, and then prove the following conjugate properties for symmetric and asymmetric Diophantine approximations. (i) Among any three consecutive convergents
p
i
/
q
i
(
i
=
n
−
1
,
n
,
n
+
1
)
{p_i}/{q_i}(i = n - 1,n,n + 1)
, at least one satisfies
\[
ξ
−
p
i
/
q
i
|
>
1
/
(
a
n
+
1
2
+
4
q
i
2
)
,
\xi - {p_i}/{q_i}| > 1/\left ( {\sqrt {a_{^{n + 1}}^2 + 4q_i^2} } \right ),
\]
and at least one does not satisfy this inequality. (ii) Let
τ
\tau
be a positive real number. Among any four consecutive convergents
p
i
/
q
i
(
i
=
n
−
1
,
n
,
n
+
1
,
n
+
2
)
{p_i}/{q_i}(i = n - 1,n,n + 1,n + 2)
, at least one satisfies
\[
−
1
/
(
c
n
2
+
4
τ
q
i
2
)
>
ξ
−
p
i
/
q
i
>
τ
/
(
c
n
2
+
4
τ
q
i
2
)
,
- 1/\left ( {\sqrt {c_{^n}^2 + 4\tau q_i^2} } \right ) > \xi - {p_i}/{q_i} > \tau /\left ( {\sqrt {c_n^2 + 4\tau q_i^2} } \right ),
\]
and at least one does not satisfy this inequality, where
c
n
=
a
n
+
1
{c_n} = {a_{n + 1}}
if
n
n
is odd,
c
n
=
a
n
+
2
{c_n} = {a_{n + 2}}
if
n
n
is even.