PX153 - B1 - basic operations and the argand diagram

complex numbers

a complex number can be written in the form $z = x + i y$
$R e (z) = x$
$I m (z) = y$
$x, y \in R$
$i^{2} = - 1$
- the set of real numbers $R e$ is a subset of the set of complex numbers $C$
- the modulus $| z | = \sqrt{x^{2} + y^{2}}$
- the argument $a r g (z) = θ = \arctan \frac{y}{x}$
- we use the convention $0 \leq θ < 2 π$ ; $a r g (z) \in [0, 2 π)$

argand diagram

complex conjugate

$z^{*} = x - i y$
$| z | = \sqrt{z . z^{*}}$

standard operations

$z_{1} = x_{1} + i y_{1}$
$z_{2} = x_{2} + i y_{2}$

addition and subtraction

z_{1} \pm z_{2} = (x_{1} \pm i y_{1}) + (x_{2} \pm i y_{2})

multiplication

z_{1} z_{2} = (x_{1} + i y_{1}) (x_{2} + i y_{2}) = (x_{1} x_{2} - y_{1} y_{2}) + i (x_{2} y_{1} + x_{1} y_{2})

division

\frac{1}{z} = \frac{1}{x + i y} = \frac{x - i y}{(x + i y) (x - i y)} = \frac{x - i y}{x^{2} + y^{2}}

\frac{z_{1}}{z_{2}} = z_{1} z_{2}^{- 1} = (x_{1} + i y_{1}) \frac{(x_{2} - i y_{2})}{x_{2}^{2} + y_{2}^{2}} = \frac{(x_{1} x_{2} - y_{1} y_{2}) + i (x_{2} y_{1} + x_{1} y_{2})}{x_{2}^{2} + y_{2}^{2}}

\frac{z_{1}}{z_{2}} = \frac{z_{1} z_{2}^{*}}{z_{2} z_{2}^{*}}

powers of i

i^{n} = i^{r}

where, $r =$ remainder from division by 4

note: added during tutorial