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

The argument of $z = a + bi$z=a+bi (where $z \neq 0$z=0) is the angle $\theta$θ that the line from the origin to $z$z makes with the positive real axis, measured anticlockwise.

$\arg(z) = \theta$arg(z)=θ

The principal argument is defined to lie in the range:

$-\pi < \theta \leq \pi$−π<θ≤π

Finding the Argument

Start by finding the acute angle $\alpha = \arctan\left(\frac{|b|}{|a|}\right)$α=arctan(∣a∣∣b∣​), then adjust according to the quadrant:

Quadrant	Sign of $a$a and $b$b	$\arg(z)$arg(z)
First ($a > 0, b > 0$a>0,b>0)	$+, +$+,+	$\alpha$α
Second ($a < 0, b > 0$a<0,b>0)	$-, +$−,+	$\pi - \alpha$π−α
Third ($a < 0, b < 0$a<0,b<0)	$-, -$−,−	$-(\pi - \alpha) = \alpha - \pi$−(π−α)=α−π
Fourth ($a > 0, b < 0$a>0,b<0)	$+, -$+,−	$-\alpha$−α

Special cases:

• $\arg(r) = 0$arg(r)=0 for $r > 0$r>0 (positive real)
• $\arg(-r) = \pi$arg(−r)=π for $r > 0$r>0 (negative real)
• $\arg(ki) = \frac{\pi}{2}$arg(ki)=2π​ for $k > 0$k>0 (positive imaginary)
• $\arg(-ki) = -\frac{\pi}{2}$arg(−ki)=−2π​ for $k > 0$k>0 (negative imaginary)

Worked Example 1: Find the modulus and argument of $z = 1 + \sqrt{3}\, i$z=1+3​i.

$|z| = \sqrt{1 + 3} = 2$∣z∣=1+3​=2

$a = 1 > 0$a=1>0, $b = \sqrt{3} > 0$b=3​>0, so $z$z is in the first quadrant.

$\alpha = \arctan\left(\frac{\sqrt{3}}{1}\right) = \frac{\pi}{3}$α=arctan(13​​)=3π​

$\arg(z) = \frac{\pi}{3}$arg(z)=3π​

Worked Example 2: Find the modulus and argument of $z = -3 - 3i$z=−3−3i.

$|z| = \sqrt{9 + 9} = 3\sqrt{2}$∣z∣=9+9​=32​