# Complex Arithmetic/Examples

## Examples of Complex Arithmetic

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

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

### Example: $\paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10}$

$\paren {\dfrac {1 + \sqrt 3 i} {1 - \sqrt 3 i} }^{10} = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$

### Example: $\paren {\dfrac {\sqrt 3 - i} {\sqrt 3 + i} }^4 \paren {\dfrac {1 + i} {1 - i} }^5$

$\paren {\dfrac {\sqrt 3 - i} {\sqrt 3 + i} }^4 \paren {\dfrac {1 + i} {1 - i} }^5 = -\dfrac {\sqrt 3} 2 - \dfrac 1 2 i$

### Example: $3 \paren {-1 + 4 i} - 2 \paren {7 - i}$

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

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

$\paren {i - 2} \paren {2 \paren {1 + i} - 3 \paren {1 - i} } = -9 + 7 i$

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

$\dfrac {\paren {2 + i} \paren {3 - 2 i} \paren {1 + 2 i} } {\paren {1 - i}^2} = -\dfrac {15} 2 + 5 i$

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

$\paren {2 i - 1}^2 \paren {\dfrac 4 {1 - i} + \dfrac {2 - i} {1 + i} } = -\dfrac {11} 2 - \dfrac {23} 2 i$

### Example: $\dfrac {i^4 + i^9 + i^{16} } {2 - i^5 + i^{10} - i^{15} }$

$\dfrac {i^4 + i^9 + i^{16} } {2 - i^5 + i^{10} - i^{15} } = 2 + i$

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

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

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

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

### Example: $3 \paren {1 + i} + 2 \paren {4 - 3 i} - \paren {2 + 5 i}$

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

### Example: $\dfrac 1 2 \paren {4 - 3 i} + \dfrac 3 2 \paren {5 + 2 i}$

$\dfrac 1 2 \paren {4 - 3 i} + \dfrac 3 2 \paren {5 + 2 i} = \dfrac {19} 2 + \dfrac 3 2 i$

Let $z_1 = 2 + i$, $z_2 = 3 - 2 i$ and $z^3 = -\dfrac 1 2 + \dfrac {\sqrt 3} 2 i$.

### Example: $\cmod {3 z_1 - 4 z_2}$

$\cmod {3 z_1 - 4 z_2} = \sqrt {157}$

### Example: $z_1^3 - 3 z_1^2 + 4 z_1 - 8$

$z_1^3 - 3 z_1^2 + 4 z_1 - 8 = -7 + 3 i$

### Example: $\paren {\overline {z_3} }^4$

$\paren {\overline {z_3} }^4 = -\dfrac 1 2 - \dfrac {\sqrt 3} 2 i$

### Example: $\cmod {\dfrac {2 z_2 + z_1 - 5 - 1} {2 z_1 - z_2 + 3 - 1} }^2$

$\paren {\overline {z_3} }^4 = -\dfrac 1 2 - \dfrac {\sqrt 3} 2 i$

Let $z_1 = 1 - i$, $z_2 = -2 + 4 i$ and $z_3 = \sqrt 3 - 2 i$.

### Example: ${z_1}^2 + 2 z_1 - 3$

${z_1}^2 + 2 z_1 - 3 = -1 - 4 i$

### Example: $\cmod {2 z_2 - 3 z_1}^2$

$\cmod {2 z_2 - 3 z_1}^2 = 170$

### Example: $\paren {z_3 - \overline {z_3} }^5$

$\paren {z_3 - \overline {z_3} }^5 = -1024 i$

### Example: $\cmod {z_1 \overline {z_2} + z_2 \overline {z_1} }$

$\cmod {z_1 \overline {z_2} + z_2 \overline {z_1} } = 12$

### Example: $\cmod {\dfrac {z_1 + z_2 + 1} {z_1 - z_2 + i} }$

$\cmod {\dfrac {z_1 + z_2 + 1} {z_1 - z_2 + i} } = \dfrac 3 5$

### Example: $\dfrac 1 2 \paren {\dfrac {z_3} {\overline z_3} + \dfrac {\overline z_3} {z_3} }$

$\dfrac 1 2 \paren {\dfrac {z_3} {\overline z_3} + \dfrac {\overline z_3} {z_3} } = -\dfrac 1 7$

### Example: $\overline {\paren {z_2 + z_3} \paren {z_1 - z_3} }$

$\overline {\paren {z_2 + z_3} \paren {z_1 - z_3} } = -7 + 3 \sqrt 3 + \sqrt 3 i$

### Example: $\cmod { {z_1}^2 + \overline {z_2}^2}^2 + \cmod {\overline {z_3}^2 - {z_2}^2}^2$

$\cmod { {z_1}^2 + \overline {z_2}^2}^2 + \cmod {\overline {z_3}^2 - {z_2}^2}^2 = 765 + 128 \sqrt 3$

### Example: $\map \Re {2 {z_1}^3 + 3 {z_2}^2 - 5 {z_3}^2}$

$\map \Re {2 {z_1}^3 + 3 {z_2}^2 - 5 {z_3}^2} = -35$

### Example: $\map \Im {\dfrac {z_1 z_2} {z_3} }$

$\map \Im {\dfrac {z_1 z_2} {z_3} } = \dfrac {6 \sqrt 3 + 4} 7$