1.1.3 Modulus & Argument

Overview

These two concepts are important for understanding how to represent complex numbers in a geometric form using the Argand diagram.

Modulus of a Complex Number

The modulus of a complex number $z = a + bi$ is the distance of the point representing the complex number from the origin on an Argand diagram (a 2D plane where the real part is on the $x-axis$ and the imaginary part is on the $y-axis$ ).

The modulus is denoted by $|z|$ . The formula for the modulus is:

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

where $a$ is the real part and $b$ is the imaginary part of the complex number.

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Example 1: Finding the Modulus For $z = 3 + 4i$

|z| = \sqrt{3^2 + 4^2}

= \sqrt{9 + 16}

= \sqrt{25} = 5

So, the modulus of $z = 3 + 4i$ is $5$

Argument of a Complex Number

The argument of a complex number $z = a + bi$ is the angle $\theta$ that the line representing the complex number makes with the positive real axis ( $x-axis$ ) on the Argand diagram.

The argument is denoted by $\arg(z)$ and is measured in radians.

To find the argument, use the following formula:

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

However, you must consider which quadrant the complex number is in, as the angle might need adjustment depending on whether $a$ and $b$ are positive or negative.

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Example 2: Finding the Argument For $z = 1 + i$

Step 1: Find the modulus

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

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

Step 2: Find the argument:

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

= \tan^{-1}(1)

= \frac{\pi}{4} \text{ radians}

= 45^\circ

So, the argument of $z = 1 + i$ is $\frac{\pi}{4}$ radians (or $45$ $degrees$ ).

Argand Diagram

The Argand diagram is a 2D plane where:

The $x-axis$ represents the real part of a complex number.
The $y-axis$ represents the imaginary part.

A complex number $z = x + yi$ is plotted as the point $(a, b)$ .

The modulus gives the distance from the origin to this point, and the argument gives the angle made with the positive real axis.

Note Summary

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Common Mistakes:

Incorrect Calculation of Modulus: Forgetting to square both the real and imaginary parts or making arithmetic errors in $|z| = \sqrt{a^2 + b^2}$
Wrong Quadrant for Argument: Misidentifying the quadrant of the complex number and not adjusting the angle accordingly.
Ignoring Radian Measures: Using degrees instead of radians when solving for the argument in contexts requiring radians.
Mistaking Real and Imaginary Parts: Confusing $a$ (real part) and $b$ (imaginary part) when calculating the argument.
Skipping Argand Diagram Checks: Failing to visualize the complex number on the Argand diagram to verify modulus and argument.

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Key Formulas:

Modulus of $z = a + bi$

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

Argument of $z = a + bi$ (when $a \neq 0$ ):

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

(Adjust $\theta$ based on the quadrant.)

Polar Form of a Complex Number:

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

General Angle Adjustments:

Quadrant I: $\theta$ as calculated.
Quadrant II: $\pi - |\theta|$
Quadrant III: $\pi + |\theta|$
Quadrant IV: $-|\theta|$ or $2\pi - |\theta|$ Converting Between Forms:
From Cartesian ( $a + bi$ ) to Polar ( $|z|, \theta$ ): Use $|z|$ and $\arg(z)$
From Polar to Cartesian:

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

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