Complex Subtraction/Examples/(7 + i) - (4 - 2i)

Example of Complex Subtraction

$\paren {7 + i} - \paren {4 - 2 i} = 3 + 3 i$

Proof 1

\(\ds \paren {7 + i} - \paren {4 - 2 i}\)

\(=\)

\(\ds \paren {7 - 4} + \paren {1 - \paren {-2} } i\)

Definition of Complex Subtraction

\(\ds \)

\(=\)

\(\ds 3 + 3 i\)

$\blacksquare$

Proof 2

By definition of complex subtraction:

$\paren {7 + i} - \paren {4 - 2 i} = \paren {7 + i} + \paren {-4 + 2 i}$

Let the complex numbers $7 + i$ and $-4 + 2 i$ 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 $3 + 3 i$, which is the sum of $7 + i$ and $-4 + 2 i$.

Hence, $3 + 3 i$ is the difference of $7 + i$ and $2 - 5 i$.

$\blacksquare$