Argument of Complex Numbers - Definition, Formula, Example

Formula for the Argument of a Complex Number

The argument of a complex number is calculated using its real and imaginary parts. Although the inverse tangent function provides the reference angle, the final argument depends on the quadrant in which the complex number lies. Therefore, both the formula and the signs of the coordinates should always be considered together.

Standard Formula

If $z=x+iy$

then the argument of the complex number is given by

$\theta=\arg(z)=\tan^{-1}\left(\frac{y}{x}\right)$

This formula gives the correct angle only after making the necessary quadrant adjustment.

Meaning of the Variables

In the formula

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

• $x$ represents the real part of the complex number.
• $y$ represents the imaginary part.
• $\theta$ represents the argument.
• $\tan^{-1}$ denotes the inverse tangent function.

The values of $x$ and $y$ determine both the reference angle and the quadrant.

How to Find the Argument of Complex Numbers?

To calculate the argument of a complex number:

1. Identify real part ($x$) and imaginary part ($y$).

2. Substitute values in $\theta = \tan^{-1}\left(\frac{y}{x}\right)$.

3. Adjust $\theta$ based on the quadrant where $(x,y)$ lies.

Argument Using the Inverse Tangent Function

The inverse tangent function is the most common method for finding the argument.

First, calculate

$\tan^{-1}\left(\frac{y}{x}\right)$

This gives the reference angle.

Next, determine the correct quadrant using the signs of the real and imaginary parts, and adjust the angle if necessary.

For example, if

$z=-3+3i$

then

$\tan^{-1}\left(\frac{3}{-3}\right)=\tan^{-1}(-1)=-45^\circ$

Since the point lies in the second quadrant, the correct argument is

$135^\circ$

and not $-45^\circ$.

Argument of a complex number in different quadrants

If $\theta$ lies between $-\pi<\theta \leq \pi$, then $\theta$ is called a principal argument. The value of the argument differs depending on which quadrant point $(x, y)$ lies.

Note:

If $\arg (\mathrm{z})=\frac{\pi}{2}$ or $-\frac{\pi}{2}, \mathrm{z}$ is purely imaginary.
If $\arg (\mathrm{z})=0$ or $\pi, \mathrm{z}$ is purely real.

Properties of Argument of Complex Numbers

i) Argument of product
$\arg (z_1 z_2) = \arg (z_1) + \arg (z_2) + 2k\pi, k \in \mathbb{Z}$

Explanation: The angle of the product of two complex numbers equals the sum of their individual angles, adjusted by multiples of $2\pi$.

ii) Argument of quotient
$\arg \left(\frac{z_1}{z_2}\right) = \arg (z_1) - \arg (z_2) + 2k\pi, k \in \mathbb{Z}$

Explanation: The angle of the quotient of two complex numbers equals the difference of their arguments.

iii) Argument of conjugate
$\arg (\bar{z}) = -\arg (z)$

Explanation: The conjugate of a complex number reflects it across the real axis in the Argand plane, so its angle changes sign.

iv) Argument of powers
$\arg (z^n) = n \cdot \arg (z) + 2k\pi, k \in \mathbb{Z}$

Explanation: Raising a complex number to a power multiplies its angle by $n$

v) Equal arguments condition (collinear vectors in same direction)
If $|z_1 + z_2| = |z_1| + |z_2|$, then
$\arg (z_1) = \arg (z_2)$

Explanation: This condition means $z_1$ and $z_2$ lie along the same line in the Argand plane, pointing in the same direction.

vi) Opposite arguments condition (collinear vectors in opposite direction)
If $|z_1 + z_2| = \big||z_1| - |z_2|\big|$, then
$\arg (z_1) - \arg (z_2) = \pi$

Explanation: This means $z_1$ and $z_2$ are collinear but point in opposite directions on the Argand plane.

Applications Of Argument Of Complex Number

The argument of a complex number has numerous applications in transforming the complex number to polar form, and also in finding the relationship between the real part and the imaginary part of the complex number.

Polar Form of Complex Number: The polar form of the complex number is P = r(Cosθ + iSinθ). Here θ is the argument of the complex number, and r is the argument of the complex number. Polar form is another important form of representing the complex number in the argand plane. The polar form of the complex number represented in cartesian form is (rCosθ, rSinθ).

Polar Form and Argument

The argument and modulus together provide a convenient way to represent a complex number in polar form. This representation simplifies multiplication, division, powers, and roots of complex numbers, making it an important concept in higher mathematics and engineering.

Converting to Polar Form

A complex number $z=x+iy$ can be expressed in polar form as $z=r(\cos\theta+i\sin\theta)$

where

• $r$ is the modulus of the complex number.

• $\theta$ is its argument.

To convert a complex number into polar form:

1. Find the modulus using

$|z|=\sqrt{x^2+y^2}$

2. Find the argument using

$\theta=\arg(z)$

3. Substitute both values into the polar form.

This representation is especially useful when multiplying or dividing complex numbers.

Relationship Between Modulus and Argument

Every non-zero complex number is uniquely determined by its modulus and principal argument.

If $z=x+iy$ then

• Modulus gives the distance from the origin.

• Argument gives the direction from the positive real axis.

Together,

$(r,\theta)$

represent the polar coordinates of the complex number.

The relationship can be written as

$x=r\cos\theta$

$y=r\sin\theta$

These equations connect the Cartesian and polar forms of a complex number.

Euler's Form of a Complex Number

Using Euler's identity,

$e^{i\theta}=\cos\theta+i\sin\theta$

the polar form can be written more compactly as

$z=re^{i\theta}$

This form is known as the exponential form or Euler's form of a complex number.

It is widely used in:

• Electrical engineering

• Signal processing

• Fourier analysis

• Differential equations

• Quantum mechanics

Connection with De Moivre's Theorem

The argument plays a central role in De Moivre's theorem, which is used to calculate powers and roots of complex numbers.

If $z=r(\cos\theta+i\sin\theta)$ then

$z^n=r^n\left(\cos n\theta+i\sin n\theta\right)$

The theorem shows that:

• The modulus is raised to the power $n$.

• The argument is multiplied by $n$.

This greatly simplifies calculations involving higher powers of complex numbers.

Applications of the Argument of Complex Numbers

The argument of a complex number is not just a theoretical concept. It has numerous applications in mathematics, engineering, physics, and computer science wherever direction, phase, or rotation needs to be represented.

Applications in Mathematics

In mathematics, the argument is used to:

• Convert complex numbers into polar form.

• Apply De Moivre's theorem.

• Find powers and roots of complex numbers.

• Solve trigonometric equations.

• Study analytic functions in complex analysis.

Applications in Electrical Engineering

Electrical engineers represent alternating current (AC) voltages and currents using complex numbers.

The argument represents the phase angle between electrical quantities, making it easier to analyze electrical circuits involving capacitors and inductors.

Applications in Signal Processing

Complex numbers are widely used to represent digital and analog signals.

The argument represents the phase of a signal, which is essential in:

• Audio processing.

• Wireless communication.

• Image processing.

• Radar systems.

Applications in Physics

Physicists use the argument of complex numbers in:

• Wave mechanics.

• Quantum mechanics.

• Electromagnetic waves.

• Oscillations.

• Vibrations.

Many physical quantities are represented as rotating vectors whose direction is described using the argument.

Argument vs Modulus of a Complex Number

Although the argument and modulus describe the same complex number, they represent different geometric properties.

Key Differences

The modulus measures the distance of a complex number from the origin.

The argument measures the angle made by the complex number with the positive real axis.

The modulus answers "How far?", while the argument answers "In which direction?"

Similarities

Both the modulus and the argument:

• Describe the position of a complex number.

• Are required to write the polar form.

• Are used in De Moivre's theorem.

• Play important roles in complex number operations.

• Are widely used in mathematics and engineering.

Formula Comparison

For $z=x+iy$

Modulus: $|z|=\sqrt{x^2+y^2}$

Argument: $\arg(z)=\tan^{-1}\left(\frac{y}{x}\right)$

(with proper quadrant adjustment)

These two formulas are the foundation of the polar representation of complex numbers.

Comparison Table

Best Books for the Argument of Complex Numbers

A strong understanding of complex numbers and their geometric interpretation is essential for mastering the argument of a complex number. The following books explain the concept with clear theory, illustrations, and exam-oriented practice.

Shortcut Tips and Tricks

Finding the argument becomes much easier if you identify the quadrant correctly before applying the formula. These shortcuts help reduce calculation errors.

Important Formula Table

The following formulas are frequently used while solving problems related to the argument of complex numbers.

Solved Examples Based on Argument of a Complex Number

Example 1: Arg $\left(\frac{2 i}{\sqrt{3}-i}\right)$ equals:

Solution:

As we have learned in Definition of Argument/Amplitude of z in Complex Numbers:

$\theta=\tan ^{-1}\left|\frac{y}{x}\right|, z \neq 0$

$\theta, \pi-\theta,-\pi+\theta,-\theta$ are Principal Arguments if z lies in the first, second, third, or fourth quadrant respectively.

Now,

$\begin{aligned} & \frac{2 i}{\sqrt{3}-i}=\frac{2 i}{\sqrt{3}-i} \times \frac{\sqrt{3}+i}{\sqrt{3}+i}=\frac{2 \sqrt{3} i-2}{4} \\ & \Rightarrow \frac{2 i}{\sqrt{3}-i}=\frac{-1}{2}+\frac{i \sqrt{3}}{2}\end{aligned}$

$\because$ it lies in 2nd quadrant so

argument= $\pi-\tan ^{-1}\left|\frac{\frac{\sqrt{3}}{2}}{\frac{-1}{2}}\right|=\pi-\frac{\pi}{3}=\frac{2 \pi}{3}$

Hence, the required answer is $\frac{2 \pi}{3}$.

Example 2: Let $z_0$ be a root of the quadratic equation, $x^2+x+1=0$. If $z=3+6 i z_0^{81}-3 i z_0^{93}$ then arg z is equal to :

Solution:

Definition of Argument/Amplitude of z in Complex Numbers -

$\theta=\tan ^{-1}\left|\frac{y}{x}\right|, z \neq 0$

$\theta, \pi-\theta,-\pi+\theta,-\theta$ are Principal Arguments if z lies in the first, second, third, or fourth quadrant respectively.

Cube roots of unity -

$z=(1)^{\frac{1}{3}} \Rightarrow z=\cos \frac{2 k \pi}{3}+i \sin \frac{2 k \pi}{3}$

k=0,1,2 so z gives three roots

$\Rightarrow 1, \frac{-1}{2}+i \frac{\sqrt{3}}{2}(\omega), \frac{-1}{2}-i \frac{\sqrt{3}}{2}\left(\omega^2\right)$

- wherein

$\omega=\frac{-1}{2}+\frac{i \sqrt{3}}{2}, \omega^2=\frac{-1}{2}-\frac{i \sqrt{3}}{2}, \omega^3=1,1+\omega+\omega^2=0$

$1, \omega, \omega^2$ are cube roots of unity.

Quadratic Equation

$x^2+x+1=0$, roots are, $\omega$ and $\omega^2$ where $\omega$ is the cube root of unity.

$
z=3+6 i\left(z_0\right)^{81}-3 i\left(z_0\right)^{90}
$

$z_0=\omega$ and $\omega^2$

$
\begin{aligned}
& z=3+6 i(\omega)^{81}-3 i(\omega)^{93} \\
& z=3+3 i \quad \because \omega^3=1 \\
& \arg (z)=\frac{1}{4}
\end{aligned}
$
Hence, the required answer is $\frac{\pi}{4}$.

Example 3: Which of the following is one of the arguments of $z=-\sqrt{3}-3 i$:

1) $\frac{\pi}{3}$
2) $\frac{-5 \pi}{3}$
3) $\frac{-4 \pi}{3}$
4) $\frac{4 \pi}{3}$

Solution:

As we have learned in, General Argument of a Complex Number: $2 n \pi+0$

- wherein

$\theta$ is the principal argument of complex numbers.

$\because$ z lies in 3rd quadrant

$\therefore$ its principal argument $=-\pi+\tan ^{-1}\left|\frac{-3}{-\sqrt{3}}\right|=-\pi+\tan ^{-1} \sqrt{3}=-\pi+\frac{\pi}{3}=\frac{-2 \pi}{3}$

$\therefore 2 n \pi-\frac{2 \pi}{3}$ will be its general argument.

for n=1, we get argument = $\frac{4 \pi}{3}$

Hence, the required answer is the option (4).

Example 4: The point of intersection the curves arg $(z-i+2)=\frac{\pi}{6}$ & $\arg (z+4-3 i)=-\frac{\pi}{4}$ is given by:

Solution:

Definition of Argument/Amplitude of $z$ in Complex Numbers -

$\theta=\tan ^{-1}\left|\frac{y}{x}\right|, z \neq 0$

$\theta, \pi-\theta,\pi+\theta,-\theta$ are Principal Argument if $z$ lies in first, second, third or fourth quadrant respectively.

$\arg (z-i+2)=\frac{\pi}{6} \Rightarrow \tan \frac{\pi}{6}=\frac{y-1}{x+2}$

$\Rightarrow x-\sqrt{3 y}=-(\sqrt{3}+2), x>-2, y>1$

$\arg (z+4-3 i)=-\frac{\pi}{4} \Rightarrow \tan \left(-\frac{\pi}{4}\right)=\frac{y-3}{x+4}$

$\Rightarrow y+x=-1, x>-4, y<3$

So, there is no point of intersection.

Example 5: Arg $\left(i^{18}+\frac{1}{i^{25}}\right)$ equals:

Solution:

$i^{18}+\frac{1}{i^{25}}=\left(i^4\right)^4 \cdot i^2+\left(\frac{1}{i^4}\right)^6 \cdot \frac{1}{i}=i^2+\frac{1}{i}=-1-i$

As this lies in the third quadrant, so Arg(z) = $-\pi+\tan ^{-1} \left\lvert\, \frac{y}{x}\right.$

$\therefore$ Arg $(-1-i)=-\pi+\tan ^{-1}\left|\frac{-1}{-1}\right|=-\pi+\frac{\pi}{4}=\frac{-3 \pi}{4}$

Hence, the required answer is $\frac{-3 \pi}{4}$.

List of topics related to the Argument of a Complex Number

Below is the list of important concepts connected with the argument of a complex number, including its definition, formulas, properties, quadrant-wise cases, and applications. These topics provide a complete understanding of how arguments work in complex number geometry.

NCERT Resources

Below are the key NCERT resources for Class 11 Maths Chapter 5 – Complex Numbers and Quadratic Equations, including step-by-step NCERT solutions, exemplar problems with detailed explanations, and concise notes to strengthen understanding and practice.

NCERT Solutions for Class 11 Maths Chapter 5 -Complex Numbers and Quadratic Equations

NCERT Exemplar Class 11 Maths Solutions for Chapter 5 - Complex Numbers and Quadratic Equations

NCERT Class 11 Maths Notes for Chapter 5 - Complex Numbers and Quadratic Equations

Practice Questions based on the Argument of a Complex Number

Below are carefully chosen practice questions on the argument of a complex number. These questions will help you apply formulas, understand quadrant-wise cases, and strengthen problem-solving skills for exams.

Argument Of Complex Number - Practice Question MCQ

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