020 基本代数性质 Basic Algebraic Properties

复数的加法和乘法与实数的加法和乘法类似，有很多属性。

交换律： $$z_1+z_2=z_2=z_1,z_1z_2=z_2z_1\tag{1}$$ 证明：令 $z_1=(x_1,y_1),z_2=(x_2,y_2)$，那么 $$\begin{aligned} z_1+z_2&=(x_1+x_2,y_1+y_2)\\ &=(x_2+x_1,y_2+y_1)\\ &=z_2+z_1 \end{aligned}$$ $$\begin{aligned} z_1z_2&=(x_1x_2-y_1y_2,x_1y_2+y_1x_2)\\ &=(x_2x_1-y_2y_1,x_2y_1+y_2x_1)\\ &=z_2z_1 \end{aligned}$$

结合律： $$(z_1+z_2)+z_3=z_1+(z_2+z_3),(z_1z_2)z_3=z_1(z_2z_3)\tag{2}$$ 证明：令 $z_1=(x_1,y_1),z_2=(x_2,y_2),z_3=(x_3,y_3)$，那么 $$\begin{aligned} (z_1+z_2)+z_3&=(x_1+x_2,y_1+y_2)+(x_3+y_3)\\ &=(x_1+x_2+x_3,y_1+y_2+y_3)\\ &=(x_1,y_1)+(x_2+x_3,y_2+y_3)\\ &=z_1+(z_2+z_3) \end{aligned}$$ $$\begin{aligned} (z_1z_2)z_3&=(x_1x_2-y_1y_2,x_1y_2+y_1x_2)(x_3,y_3)\\ &=(x_1x_2x_3-y_1y_2x_3-x_1y_2y_3-y_1x_2y_3,x_1x_2y_3-y_1y_2y_3+x_1y_2x_3+y_1x_2x_3)\\ &=(x_1(x_2x_3-y_2y_3)-y_1(y_2x_3+x_2y_3),x_1(x_2y_3+y_2x_3)+y_1(x_2x_3-y_2y_3))\\ &=(x_1,y_1)(x_2x_3-y_2y_3,y_2x_3+x_2y_3)\\ &=z_1(x_2,y_2)(x_3,y_3)\\ &=z_1(z_2z_3) \end{aligned}$$

分配律： $$z(z_1+z_2)=zz_1+zz_2\tag{3}$$ 证明：令 $z=(x,y),z_1=(x_1,y_1),z_2=(x_2,y_2)$，那么 $$\begin{aligned} z(z_1+z_2)&=(x,y)(x_1+x_2,y_1+y_2)\\ &=(xx_1+xx_2-yy_1-yy_2,xy_1+xy_2+yx_1+yx_2)\\ &=(xx_1-yy_1,xy_1+yx_1)+(xx_2-yy_2,xy_2+yx_2)\\ &=(x,y)(x_1,y_1)+(x,y)(x_2,y_2)\\ &=zz_1+zz_2 \end{aligned}$$

由于交换率，那么 $yi=iy$，因此 $z=x+yi=x+iy$。由于结合律，$z_1+z_2+z_3$ 或 $z_1z_2z_3$ 可以不写括号，结果也是明确的。