# Complex Arithmetic/Examples/3(1+i) + 2(4-3i) - (2+5i)

## Example of Complex Arithmetic

$3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i} = 9 - 8 i$

## Proof 1

 $\ds$  $\ds 3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i}$ $\ds$ $=$ $\ds \paren {3 + 3} + \paren {8 - 6 i} - \paren {2 + 5 i}$ $\ds$ $=$ $\ds \paren {\paren 3 + 8 - 2} + \paren {3 - 6 - 5} i$ $\ds$ $=$ $\ds 9 - 8 i$

$\blacksquare$

## Proof 2

We have:

 $\ds 3 \paren {1 + i}$ $=$ $\ds 3 + 3 i$ $\ds 2 \paren {4 - 3 i}$ $=$ $\ds 8 - 6 i$ $\ds -\paren {2 + 5 i}$ $=$ $\ds -2 - 5 i$

These can be depicted in the complex plane as follows:

To find the required sum, proceed as in the following diagram:

Construct $8 - 6 i$ with its initial point placed at the terminal point of $3 + 3 i$.

Construct $-2 - 5 i$ with its initial point placed at the terminal point of this instance of $8 - 6 i$.

The required resultant $3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i}$ is therefore represented by the terminal point of $-2 - 5 i$.

$\blacksquare$