# Definition:Complex Arithmetic

## Definition

Complex arithmetic is the branch of arithmetic which concerns the manipulation of complex numbers

## Examples

### Example: $\dfrac {\paren {1 + 2 i}^2} {1 - i}$

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

### Example: $\dfrac 1 {1 + i} + \dfrac 1 {1 - 2 i}$

$\dfrac 1 {1 + i} + \dfrac 1 {1 - 2 i} = \dfrac 7 {10} - \dfrac 1 {10} i$

### Example: Sum of Powers of $i$ from $0$ to $7$

$1 + i + i^2 + i^3 + i^4 + i^5 + i^6 + i^7 = 0$

### Example: $\dfrac 1 {\paren {4 + 2 i} \paren {2 - 3 i} }$

$\dfrac 1 {\paren {4 + 2 i} \paren {2 - 3 i} } = \dfrac 7 {130} + \dfrac {2 i} {65}$

### Example: $\dfrac {5 + 5 i} {3 - 4 i} + \dfrac {20} {4 + 3 i}$

$\dfrac {5 + 5 i} {3 - 4 i} + \dfrac {20} {4 + 3 i} = 3 - i$

### Example: $\dfrac {3 i^{30} - i^{19} } {2 i - 1}$

$\dfrac {3 i^{30} - i^{19} } {2 i - 1} = 1 + i$

## Also see

• Results about complex arithmetic can be found here.