Modulus and Argument (Polar Form)

This lesson covers the modulus and argument of a complex number, and the important modulus-argument form (also called polar form). This representation is often more natural than Cartesian form, particularly for multiplication, division, and powers.

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The Modulus

The modulus of $z = a + bi$z=a+bi is the distance from the origin to the point $(a, b)$(a,b) on the Argand diagram:

$|z| = \sqrt{a^2 + b^2}$∣z∣=a2+b2​

Complex number	Modulus
$3 + 4i$3+4i	$\sqrt{9 + 16} = 5$9+16​=5
$-5 + 12i$−5+12i	$\sqrt{25 + 144} = 13$25+144​=13
$1 - i$1−i	$\sqrt{1 + 1} = \sqrt{2}$1+1​=2​
$7$7	7
$-3i$−3i	3

Properties of the Modulus

Property	Formula
Non-negative	$
Conjugate	$
Product	$
Quotient	$\left
Square	$
Triangle inequality	$

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The Argument