Complex Number

#fundamental/math

Complex Number (z)

Extension of real numbers using the imaginary unit $j = \sqrt{- 1}$ .

z = a + j b = | z | e^{j ϕ}

Representations

Form	Notation	Components
Cartesian	$z = a + j b$	Real: $a = Re (z)$ Imaginary: $b = Im (z)$
Polar/Phasor	$z = \| z \| e^{j ϕ} = \| z \| ∠ ϕ$	Magnitude: $\| z \|$ Phase: $ϕ$

Cartesian → Polar

$| z | = \sqrt{a^{2} + b^{2}}$
$ϕ = \tan^{- 1} (b / a) quadrant aware$

Polar → Cartesian (using Euler’s Formula)

$a = | z | \cos ϕ$
$b = | z | \sin ϕ$

Operations

Operation	Formula
Addition and Subtraction	$z_{1} \pm z_{2} = (a 1 \pm a_{2}) + j (b_{1} \pm b_{2})$
Multiplication	$z_{1} z_{2} = \| z_{1} \| \| z_{2} \| ∠ (ϕ_{1} + ϕ_{2})$
Division	$z_{1} / z_{2} = \| z_{1} \| / \| z_{2} \| ∠ (ϕ_{1} - ϕ_{2})$
Inversion	$1 / z = z^{*} / \| z \|^{2} = 1 / \| z \| ∠ - ϕ$
Conjugate	$z^{*} = a - j b$

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