Argument of Complex Numbers

The argument of a complex number is a crucial concept in understanding the geometric interpretation of complex numbers. It refers to the angle that a complex number (when represented as a point on the Argand plane (a 2D coordinate system)), makes with the positive real axis. This angle helps in visualizing the direction of the complex number relative to the origin.

A complex number is expressed in the form z = a + bi, where a represents the real part and b represents the imaginary part. When plotted on the Argand plane, the real part is taken along the x-axis (real axis), and the imaginary part is taken along the y-axis (imaginary axis).

1.0What is a Complex Number?

A complex number is of the form z = a + bi, where ‘a‘ represents the real part, ‘b’ represents the imaginary part, ‘i’ represents the imaginary unit defined as i² = –1.

Complex numbers can be represented on the Argand plane (a 2D plane), where the x-axis corresponds to the real part, and the y-axis corresponds to the imaginary part. 

Example : Z = 3 + 4i, Z = 7 – 2i 

2.0What is the Argument of a Complex Number?

The argument of a complex number is the angle θ formed by the line connecting the complex number to the origin and the positive real axis. The argument is measured in radians and can take any value based on the position of the complex number in the complex plane.

Consider a right-angle triangle whose hypotenuse is r. if θ is the acute angle opposite to side y and adjacent to side x, then

$\cos \theta =\frac{x}{r};\sin \theta =\frac{y}{r},$

Which gives

x = r cos θ; y = r s in θ.

3.0Formula to Find the Argument of a Complex Number

To find the argument θ of a complex number z = a + bi, use:

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

However, special care must be taken with the quadrant in which the complex number lies, as it affects the value of the argument:

• First quadrant (a > 0, b > 0): $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$

• Second quadrant (a < 0, b > 0) : $\theta =\pi +{\tan }^{-1}\left(\frac{b}{a}\right)$

• Third quadrant (a < 0, b < 0): $\theta =\pi +{\tan }^{-1}\left(\frac{b}{a}\right)$

• Fourth quadrant (a > 0, b < 0): $\theta =2\pi +{\tan }^{-1}\left(\frac{b}{a}\right)$

4.0Principal Argument of a Complex Number

The principal argument, denoted as Arg(z), is the unique value of the argument of a complex number restricted to the interval (−π, π].

For θ ∈ (−π, π], the principal argument ensures that only one value is used to represent the angle.

Let us understand the principal argument of Z = a + ib in different Quadrants.

• First quadrant (a > 0, b > 0) : $\mathrm{Arg}Z={\tan }^{-1}\left(\frac{b}{a}\right)$

• Second quadrant (a < 0, b > 0) : $\mathrm{Arg}Z=\pi +{\tan }^{-1}\left(\frac{b}{a}\right)$

• Third quadrant (a < 0, b < 0): $\mathrm{Arg}Z=-\pi +{\tan }^{-1}\left(\frac{b}{a}\right)$

• Fourth quadrant (a > 0, b < 0): $\mathrm{Arg}Z={\tan }^{-1}\left(\frac{b}{a}\right)$

The principal argument is most commonly used in applications because it gives a standard, unique value for the angle. It is especially important when converting a complex number to polar form.

5.0General Argument of a Complex Number

The general argument of a complex number can take infinite values. If θ is one argument of a complex number z, then the general argument is expressed as:

θ + 2nπ where n ∈ Z

Here, n can be any integer (positive or negative), which means any multiple of 2π can be added to θ and still represent the same angle geometrically.

This reflects the periodic nature of complex numbers, where rotating the complex number by 2π radians results in the same direction on the Argand plane.

6.0Steps to Find Modulus and Argument of a Complex Number

To summarize, here’s how you can find the modulus and argument of a complex number:

1. Find the Modulus: Use the formula $|z|=\sqrt{a^2+b^2}$ .
2. Find the Argument: Use $\theta ={\tan }^{-1}\left(\frac{b}{a}\right)$ , adjusting for the correct quadrant.
3. Determine Principal Argument: If needed, adjust the argument to lie in (–π, π].

7.0Solved Example of Argument of Complex Numbers 

Example 1: Find modulus, argument and principal argument of $z=-1-\sqrt{3}i$ :

Solution: 

• Modulus: $|z|=\sqrt{(-1{)}^2+(-\sqrt{3}{)}^2}=2$
• Argument: Since a = –1 and $b=-\sqrt{3}$ , it lies in the third quadrant.

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

$\theta =-\left(\pi -\frac{\pi }{3}\right)$

$\theta =-\frac{2\pi }{3}\text{ radians }$

Thus, the principal argument is $-\frac{2\pi }{3}$.

General Argument = $2n\pi -\frac{2\pi }{3}$ where n ∈ z

Example 2: Find the modulus and argument of complex number (2 \sqrt{3}-2 i) .

Solution: 

Let $z=(2\sqrt{3}-2i)$.

Here real part $a=2\sqrt{3}$,

Imaginary part b = –2.

1. Modulus: 

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

$=\sqrt{12+4}$

$=\sqrt{16}=4$

2. Argument:

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

Since the complex number is in the fourth quadrant, argument is 

$\Rightarrow \theta ={\tan }^{-1}\left(\frac{-2}{2\sqrt{3}}\right)$

$\Rightarrow \theta ={\tan }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$

$\Rightarrow \theta =\frac{-\pi }{6}$

General Argument = $2n\pi -\frac{\pi }{6}$  where n ∈ Z

Example 3: Find the principal argument of the complex number z = –3 – 3i.

Solution: 

Let z = –3 – 3i.

Here real part a = –3,

Imaginary part b = –3.

Since the complex number is in the third quadrant, argument is 

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

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

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

Example 4: Find the argument of the complex number z = –1 + i.

Solution: 

Let z = –1 + i.

Here real part a = –1,

Imaginary part b = 1.

Since the complex number is in the second quadrant, argument is 

• m$\theta =\pi +{\tan }^{-1}\left(\frac{b}{a}\right)$
• $\theta =\pi +{\tan }^{-1}\left(\frac{1}{-1}\right)$
• $\theta =\pi -{\tan }^{-1}1$
• $\theta =\left(\pi -\frac{\pi }{4}\right)$
• $\theta =\frac{3\pi }{4}$

Example 5: Find the argument of the complex number $\frac{(1+7i)}{(2-i{)}^2}$ .

Solution: 

Let $z=\frac{(1+7i)}{(2-i{)}^2}$

• $z=\frac{(1+7i)}{\left(4+i^2-4i\right)}$
• $z=\frac{(1+7i)}{(3-4i)}\times \frac{(3+4i)}{(3+4i)}$
• $z=\frac{(1+7i)(3+4i)}{(9+16)}$
• $z=\frac{-25+25i}{25}$
• z=(-1+i)

Here real part a = –1,

Imaginary part b = 1.

Since the complex number is in the second quadrant, argument is 

• $\theta =\pi +{\tan }^{-1}\left(\frac{b}{a}\right)$
• $\theta =\pi +{\tan }^{-1}\left(\frac{1}{-1}\right)$
• $\theta =\left(\pi -\frac{\pi }{4}\right)$
• $\theta =\frac{3\pi }{4}$

8.0Sample Questions on Argument of Complex Numbers

1. What is the difference between the general argument and the principal argument? 

Ans: The general argument of a complex number refers to all possible values of the angle θ, which can be written as:

$\theta +2n\pi ,where\mathrm{n}\in \mathrm{Z}$

The principal argument is the unique value of θ restricted to the range (−π, π], meaning the angle is adjusted to fall within this range for standard representation.

2. What is the modulus of a complex number, and how does it relate to the argument?

The modulus |z| of a complex number z = a + bi represents the distance from the origin to the point representing the complex number from the origin in the complex plane. It is calculated as:

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

The modulus and argument are used together to express the complex number in polar form as:

$z=|z|(\cos \theta +i\sin \theta )$ where θ is the argument.

3. What is the argument of the conjugate of a complex number?

If z = a + bi, then the conjugate of z is $\bar{z}=a-bi$. The argument of the conjugate $\bar{Z}$ is the negative of the argument of z. That is: $\arg (\bar{z})=-\arg (z)$