Algebraic Operations on Complex Numbers

Three operations are defined on complex numbers $z=x+iy,\ z_1=x_1+iy_1,\ z_2=x_2+iy_2$.

1. Scalar multiplication. For real $k$, $kz=kx+iky$ — every part is scaled by $k$. In particular $0\cdot z=0$, $1\cdot z=z$, and $(-1)z=-z$.
2. Addition. For $x_1,x_2,y_1,y_2\in\mathbb R$,

   $z_1+z_2=(x_1+iy_1)+(x_2+iy_2)=(x_1+x_2)+i(y_1+y_2).$

   Since vectors are characterised by length and direction, and are unchanged under translation, when $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$ the parallelogram law of addition applies: the sum $z_1+z_2$ corresponds to the point $(x_1+x_2,\,y_1+y_2)$, obtained geometrically as the fourth vertex of the parallelogram with sides $Oz_1$ and $Oz_2$.
3. Subtraction. Defined via addition of the negative: $z_1-z_2=z_1+(-z_2)=(x_1-x_2)+i(y_1-y_2)$. The vector representing $z_1-z_2$ can be drawn either as a position vector from the origin to $(x_1-x_2,\,y_1-y_2)$, or — an equally valid, equivalent picture — as the vector joining the tip of $z_2$ to the tip of $z_1$.
4. Multiplication. Expanding $(x_1+iy_1)(x_2+iy_2)$ and using $i^2=-1$:

   $z_1z_2=(x_1x_2-y_1y_2)+i(x_1y_2+x_2y_1).$

   Although the product of two complex numbers is again a complex number represented by a vector in the same plane, this product is neither the scalar product nor the vector (cross) product from ordinary vector algebra — it is its own, distinct operation. …