1.1.4 Modulus-Argument Form

What is Modulus-Argument Form?

infoNote

The modulus-argument form (also known as the polar form) of a complex number allows us to express the number using its modulus and argument, rather than its real and imaginary parts.

A complex number $z = a + bi$ can be written in a modulus-argument form as:

z = r(\cos \theta + i \sin \theta)

where:

$r = |z|$ is the modulus of the complex number (the distance from the origin to the point),
$\theta = \arg(z)$ is the argument of the complex number (the angle with the positive real axis),
$\cos \theta \ \text{and} \sin \theta$ represent the directions of the real and imaginary parts. This form is sometimes written as:

z = r \text{cis} \theta

where cis stands for $\cos \theta + i \sin \theta$

Converting to Modulus-Argument Form

To convert a complex number from Cartesian form $z = a + bi$ to modulus-argument form, follow these steps:

Method

Step 1: Find the modulus

Step 2: Find the argument

Step 3: Write the number in the form

Step 1: Find the modulus using the formula:

r = |z| = \sqrt{a^2 + b^2}

Step 2: Find the argument using:

\theta = \tan^{-1}\left(\frac{b}{a}\right)

Make sure you adjust the angle based on which quadrant the complex number lies in.

Step 3: Write the number in the form:

z = r(\cos \theta + i \sin \theta)

lightbulbExample

Example 1: Convert $z = 1 + i$ to Modulus-Argument Form

Step 1: Find the modulus:

r = |z| = \sqrt{1^2 + 1^2}

= \sqrt{1 + 1} = \sqrt{2}

Step 2: Find the argument:

\theta = \tan^{-1}\left(\frac{1}{1}\right)

= \frac{\pi}{4} \text{ radians} = 45^\circ

Step 3: Write in modulus-argument form:

z = \sqrt{2} (\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})

lightbulbExample

Example 2: Convert $z = -3 + 4i$ to Modulus-Argument Form Step 1: Find the modulus:

r = |z| = \sqrt{(-3)^2 + 4^2}

= \sqrt{9 + 16} = \sqrt{25} = 5

Step 2: Find the argument:

\theta = \tan^{-1}\left(\frac{4}{-3}\right)

This gives an angle in the second quadrant.

Using the correct angle for this quadrant:

\theta = \pi - \tan^{-1}\left(\frac{4}{3}\right)

Approximate angle:

= \theta \approx 2.21 \text{ radians}

Step 3: Write in modulus-argument form:

z = 5 (\cos 2.21 + i \sin 2.21)

Note Summary

infoNote

Why is the Modulus-Argument Form Useful?

The modulus-argument form is extremely useful when multiplying or dividing complex numbers.

It simplifies these operations as the modulus and argument can be handled separately.

For multiplication:

For division:

. ,,

z_1 \times z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2))

\frac{z_1}{z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2))

It is also helpful for raising complex numbers to powers and finding roots (as we will see in de Moivre's Theorem).

infoNote

Recap of what you MUST know:

The modulus-argument form of a complex number is written as $z = r(\cos \theta + i \sin \theta)$ , where $r$ is the modulus and $\theta$ is the argument.
You can convert a complex number from the Cartesian form to the modulus-argument form by finding the modulus $r$ and argument $\theta$
This form simplifies operations like multiplication, division, and finding powers of complex numbers.

Modulus-Argument Form Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

1.1.4 Modulus-Argument Form

What is Modulus-Argument Form?

Converting to Modulus-Argument Form

Method

Example 1: Convert $z = 1 + i$ to Modulus-Argument Form

Note Summary

Why is the Modulus-Argument Form Useful?

Recap of what you MUST know:

500K+ Students Use These Powerful Tools to Master Modulus-Argument Form For their A-Level Exams.

Other Revision Notes related to Modulus-Argument Form you should explore

Modulus-Argument Form Simplified Revision Notes for A-Level Edexcel Further Maths Core Pure

1.1.4 Modulus-Argument Form

What is Modulus-Argument Form?

Converting to Modulus-Argument Form

Method

Example 1: Convert z=1+iz = 1 + iz=1+i to Modulus-Argument Form

Note Summary

Why is the Modulus-Argument Form Useful?

Recap of what you MUST know:

500K+ Students Use These Powerful Tools to Master Modulus-Argument Form For their A-Level Exams.

Other Revision Notes related to Modulus-Argument Form you should explore

Example 1: Convert $z = 1 + i$ to Modulus-Argument Form