Modulus & Argument (Edexcel A-Level Further Mathematics): Revision Notes

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.

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.

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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