Mastering Complex Numbers: Modulus and Argument for Competitive Exams

Welcome to this module on Modulus and Argument of Complex Numbers, a crucial topic for competitive exams like JEE Mathematics. Understanding these concepts is fundamental to solving a wide range of algebraic problems.

What is a Complex Number?

A complex number is a number that can be expressed in the form $a + bi$ , where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying $i^2 = -1$ . The term ' $a$ ' is called the real part, and ' $b$ ' is called the imaginary part.

In the complex number

3 - 4i

, what is the real part and what is the imaginary part?

The real part is 3, and the imaginary part is -4.

The Modulus of a Complex Number

The modulus of a complex number $z = a + bi$ represents its distance from the origin (0,0) in the complex plane. It is denoted by $|z|$ and is calculated using the Pythagorean theorem: $|z| = \sqrt{a^2 + b^2}$ .

Modulus is the distance from the origin.

The modulus of a complex number $z = a + bi$ is its magnitude, calculated as $\sqrt{a^2 + b^2}$ . It's always a non-negative real number.

Geometrically, a complex number $z = a + bi$ can be represented as a point $(a, b)$ in the complex plane. The modulus $|z|$ is the length of the line segment connecting the origin $(0, 0)$ to this point $(a, b)$ . Applying the distance formula (which is derived from the Pythagorean theorem), we get $|z| = \sqrt{(a-0)^2 + (b-0)^2} = \sqrt{a^2 + b^2}$ . For example, if $z = 3 + 4i$ , then $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$ .

Calculate the modulus of the complex number

z = -5 + 12i

$|z| = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$ .

The Argument of a Complex Number

The argument of a complex number $z = a + bi$ is the angle $\theta$ that the line segment from the origin to the point $(a, b)$ makes with the positive real axis. It is denoted by $\arg(z)$ or $\theta$ . The principal argument is usually taken in the interval $(-\pi, \pi]$ .

The argument $\theta$ of a complex number $z = a + bi$ can be found using trigonometric relations in the complex plane. If $z$ is represented by point $(a, b)$ , then $a = |z| \cos(\theta)$ and $b = |z| \sin(\theta)$ . From these, we can derive $\tan(\theta) = \frac{b}{a}$ . However, simply using $\arctan(\frac{b}{a})$ is not enough, as the arctan function typically returns values in $(-\frac{\pi}{2}, \frac{\pi}{2})$ . We must consider the quadrant in which the complex number lies to determine the correct argument. For example, if $z = -1 - i$ , then $a = -1$ and $b = -1$ . $\tan(\theta) = \frac{-1}{-1} = 1$ . While $\arctan(1) = \frac{\pi}{4}$ , the point $(-1, -1)$ is in the third quadrant, so the correct argument is $\theta = \frac{5\pi}{4}$ or $-\frac{3\pi}{4}$ (principal argument).

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Remember to always check the quadrant of the complex number when calculating its argument to ensure you get the correct angle.

For the complex number

z = 1 + i

, what is the value of

\tan(\theta)

and what is its principal argument?

$\tan(\theta) = \frac{1}{1} = 1$ . Since $z$ is in the first quadrant, the principal argument is $\frac{\pi}{4}$ .

Polar Form of a Complex Number

A complex number $z = a + bi$ can also be expressed in polar form as $z = r(\cos(\theta) + i \sin(\theta))$ , where $r = |z|$ is the modulus and $\theta = \arg(z)$ is the argument. This form is often written more compactly as $z = r e^{i\theta}$ (Euler's formula).

Form	Components	Key Use
Rectangular Form ( $a+bi$ )	Real part ( $a$ ), Imaginary part ( $b$ )	Basic arithmetic operations, identifying real/imaginary components
Polar Form ( $r(\cos\theta + i\sin\theta)$ )	Modulus ( $r$ ), Argument ( $\theta$ )	Multiplication, division, powers, roots of complex numbers; geometric interpretation

Practice Problems and Strategies

To excel in competitive exams, practice converting between rectangular and polar forms. Pay close attention to the quadrant when finding the argument. Understanding properties like $|z_1 z_2| = |z_1| |z_2|$ and $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$ will be immensely helpful.

|z_1| = 2

and

|z_2| = 3

, what is

|z_1 z_2|

$|z_1 z_2| = |z_1| |z_2| = 2 \times 3 = 6$ .

Learning Resources

Complex Numbers - Modulus and Argument | JEE Mathematics(video)

A comprehensive video tutorial explaining the modulus and argument of complex numbers with JEE-specific examples.

Complex Numbers: Modulus and Argument - Brilliant.org(documentation)

An interactive explanation of modulus and argument, including visual aids and practice problems.

Introduction to Complex Numbers - Khan Academy(documentation)

A foundational overview of complex numbers, including their representation and basic properties.

Complex Numbers - Modulus and Argument | Byju's(blog)

Detailed explanation of complex numbers, covering modulus, argument, and their applications in competitive exams.

Understanding Complex Numbers: Modulus and Argument(wikipedia)

A community forum discussing various aspects of complex numbers, including modulus and argument, with expert answers.

JEE Advanced Mathematics - Complex Numbers(blog)

A practice problem with a detailed solution focusing on the modulus of a complex number.

The Polar Form of Complex Numbers(documentation)

Explains the polar form of complex numbers, linking modulus and argument to this representation.

Complex Numbers: Argument and Principal Argument(wikipedia)

A discussion on the nuances of calculating the argument and principal argument of complex numbers.

Geometric Interpretation of Complex Numbers(documentation)

Provides a visual understanding of complex numbers in the complex plane, highlighting modulus and argument.

Complex Numbers for JEE Main & Advanced(blog)

A resource covering complex numbers for JEE, with specific sections on modulus and argument.

Modulus and Argument of a Complex Number