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

Example of Complex Subtraction

$\paren {6 - 2 i} - \paren {2 - 5 i} = 4 + 3 i$

Proof 1

\(\ds \paren {6 - 2 i} - \paren {2 - 5 i}\)

\(=\)

\(\ds \paren {6 - 2} + \paren {-2 - \paren {-5} } i\)

Definition of Complex Subtraction

\(\ds \)

\(=\)

\(\ds 4 + 3 i\)

$\blacksquare$

Proof 2

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.

Using Geometrical Interpretation of Complex Addition, 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 $2 - 5 i$.

$\blacksquare$