設(shè) 復(fù)數(shù)為 a + bi,a,b為實(shí)數(shù).
1,與自身共軛的復(fù)數(shù)
a + bi = a - bi,
b = -b,
2b = 0,
b = 0.
因此,與自身共軛的復(fù)數(shù)是實(shí)數(shù).
2,與自身的平方共軛的復(fù)數(shù)
(a + bi)^2 = a^2 - b^2 + 2abi
a + bi = a^2 - b^2 - 2abi,
也即,
a = a^2 - b^2,
b = -2ab.
若
b = 0.
則,a = a^2,a(a-1)=0,
a = 0 或者 a = 1.
若
b 不等于 0.
則,1 = -2a,a = -1/2.
b^2 = a^2 - a = 1/4 + 1/2 = 3/4.
b = 3^(1/2)/2,或者,b = -3^(1/2)/2.
綜合,得,
a = 0,b = 0,與自身的平方共軛的復(fù)數(shù)為 0,
或者,a = 1,b = 0,與自身的平方共軛的復(fù)數(shù)為 1,
或者,a = -1/2,b = 3^(1/2)/2,與自身的平方共軛的復(fù)數(shù)為 [-1+i3^(1/2)]/2,
或者,a = -1/2,b = -3^(1/2)/2,與自身的平方共軛的復(fù)數(shù)為 [-1-i3^(1/2)]/2.