Prove that : The sum of complex number and its conjugate is real
Answers
Answer:
Let z
1
=x
1
+iy
1
and z
2
=x
2
+iy
2
be two complex numbers.
First suppose z
1
,z
2
are conjugate of each other. Then,
z
2
=
z
ˉ
1
=x
1
+iy
1
Hence z
1
+z
2
=z
1
+
z
ˉ
1
=2x
1
, which is real and z
1
z
2
=z
1
z
ˉ
1
=(x
1
+iy
1
)(x
1
−iy
1
)=x
1
2
+y
1
2
, which is also real.
Thus sum z
1
+z
2
and product z
1
z
2
are real when z
1
and z
2
are conjugate complex.
Now let the sum z
1
+z
2
and product z
1
z
2
be real. We have z
1
+z
2
=(x
1
+x
2
)+i(y
1
+y
2
)
and z
1
z
2
=x
1
x
2
−y
1
y
2
+i(x
1
y
2
+x
2
y
1
)
Since z
1
+z
2
and z
1
z
2
are both real, we must have y
1
+y
2
=0 and x
1
y
2
+x
2
+y
1
=0
These give y
2
=−y
1
and x
2
=x
1
.
Hence z
2
=x
2
+iy
2
=x
1
−iy
1
=
z
ˉ
1
, that is, z
1
and z
2
are conjugate complex.