Complex Subtraction/Examples/(6 - 2i) - (2 - 5i)/Proof 2

Proof

 * Complex-Subtraction-(6-2i)-(2-5i).png

By definition of complex subtraction:


 * $\paren {6 - 2 i} - \paren {2 - 5 i} = \paren {6 - 2 i} + \paren {-2 + 5 i}$

Let the complex numbers $6 - 2 i$ and $-2 + 5 i$ be represented by the points $P_1$ and $P-2$ respectively in the complex plane.

Complete the parallelogram with $OP_1$ and $OP_2$ as the adjacent sides.

By the Parallelogram Law, the point $P$ represents the complex number $4 + 3 i$, which is the sum of $6 - 2 i$ and $-2 + 5 i$.

Hence, $4 + 3 i$ is the difference of $6 - 2 i$ and $2 - 5 i$.