1.2 Algebraic Operations on Complex Numbers

1.2.1 Scalar Multiplication of Complex Numbers

If $z = x + iy$ and $k\in R$, then we define $kz = (kx) + (ky)i$ or $(kx) + i(ky)$

The following diagram shows $kz$ for $k = 2$, $\frac{1}{2},-1$$\frac{1}{2}, - 1$

1.2.2 Addition of Two Complex Numbers

If $z_{1} = x_{1} + i y_{1}$ and $z_{2} = x_{2} + i y_{2}$ where $x_{1},x_{2},y_{1}$ and $y_{2}\in R$, then

$z_{1} + z_{2} = (x_{1} + i y_{1}) + (x_{2} + i y_{2}) = (x_{1} + x_{2}) + i(y_{1} + y_{2})$

From figure 1.6, also by the parallelogram law of addition,

$z_{1} + z_{2} = (x_{1} + x_{2}) + i(y_{1} + y_{2})$