复数乘法的交换律、结合律及乘法
对加法的分配律证明过程
复数的乘法满足交换律、结合律以及乘法对加法的分配律，证明如下。 设i b a z 111+=，i b a z 222+=，i b a z 333+=
（1）∵))((221121i b a i b a z z ++=
i b a a b b b a a )()(21212121-+-=
))((112212i b a i b a z z ++=
i b a a b b b a a )()(12121212++-=
12122121b b a a b b a a -=-，12122121b a a b b a a b +=+
∴ 1221z z z z =
（2）∵))]()([()(332211321i b a i b a i b a z z z +++=
)(])()[(3321212121i b a i b a a b b b a a +⨯++-=
])()[(3212132121b b a a b a b b a a +--=])()[(3212132121b b b a a a b a a b -+++i )(321321321321b b a b a b a b b a a a ---=i b b b b a a a b a a a b )(321321321321-+++ )])()[(()(332211321i b a i b a i b a z z z +++=
])()[()(3232323211i b a a b b b a a i b a ++-⨯+=
)]()([3232132321b a a b b b b a a a +--=i b a a b a b b a a b )]()([3232132321++-+
)(321321321321b a b a b b b b a a a a ---=i b a a a b a b b b a a b )(321321321321++-+ )(321321321321b b a b a b a b b a a a ---=i b b b b a a a b a a a b )(321321321321-+++ ∴)()(321321z z z z z z =
（3）∵)]())[(()(332211321i b a i b a i b a z z z ++++=+ ])())[((323211i b b a a i b a ++++=
)]()([321321b b b a a a +-+=i b b a a a b )]()([321321++++
)(31213121b b b b a a a a --+=i b a b a a b a b )(31212121++++ ))(())((331122113121i b a i b a i b a i b a z z z z +++++=+ i b a a b b b a a )()(21212121++-=i b a a b b b a a )()(31313131++-+ )(31312121b b a a b b a a -+-=i b a a b b a a b )(31312121++++ )(31213121b b b b a a a a --+=i b a b a a b a b )(31213121++++ ∴3121321)(z z z z z z z +=+ 根据乘法法则得出的22z z z z ==，通常也写成.z z z z ==这个公式很重要。
