Complex Arithmetic/Examples/3(1+2i) - 2(2-3i)

Example of Complex Arithmetic

$3 \paren {-1 + 4 i} - 2 \paren {7 - i} = -17 + 14 i$

Proof 1

\(\ds 3 \paren {1 + 2 i} - 2 \paren {2 - 3 i}\)

\(=\)

\(\ds \paren {3 \times 1 + 3 \times 2 i} - \paren {2 \times 2 - 2 \times 3 i}\)

\(\ds \)

\(=\)

\(\ds \paren {3 + 6 i} - \paren {4 - 6 i}\)

\(\ds \)

\(=\)

\(\ds \paren {3 - 4} + \paren {6 - \paren {-6} } i\)

Definition of Complex Subtraction

\(\ds \)

\(=\)

\(\ds -1 + 12 i\)

$\blacksquare$

Proof 2

By definition of complex subtraction:

$3 \paren {1 + 2 i} - 2 \paren {2 - 3 i} = \paren {7 + i} + 2 \paren {-2 + 3 i}$

Let the complex numbers $3 \paren {1 + 2 i}$ and $2 \paren {-2 + 3 i}$ 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 $-1 + 12 i$, which is the sum of $3 \paren {1 + 2 i}$ and $2 \paren {-2 + 3 i}$.

Hence, $-1 + 12 i$ is the difference of $3 \paren {1 + 2 i}$ and $2 \paren {2 - 3 i}$.

$\blacksquare$