Argand Plane and Polar Representation of Complex Number

In two-dimensional geometry, we represent pairs of numbers (x, y) on the XY plane, where x is known as the abscissa and y is the ordinate. Similarly, complex numbers can also be plotted on a plane called the Argand plane (or complex plane). Much like the XY plane, the Argand plane has two axes:

• The horizontal axis is referred to as the real axis.
• The vertical axis is known as the imaginary axis.

A complex number in the form x + iy corresponds to the ordered pair (x, y), and is represented geometrically as the unique point (x, y) on the Argand plane, analogous to points on the XY-plane in coordinate geometry.

1.0What is an Argand Plane?

The Argand Plane, named after the Swiss mathematician Jean-Robert Argand, is a two-dimensional coordinate plane used to graphically represent complex numbers.

Definition: 

The plane in which each point corresponds to a complex number is known as the complex plane or Argand plane.

 Here, the real part of a complex number is plotted on the horizontal axis (called the real axis), and the imaginary part is plotted on the vertical axis (called the imaginary axis).

2.0How Do You Graph a Complex Number?

Graphing a complex number on the Argand plane is straightforward:

1. The real part ‘a’ determines the horizontal position of the point.
2. The imaginary part ‘b’ determines the vertical position.
3. The complex number z = a + ib is plotted as the point (a, b).

3.0Modulus of the Complex Plane

A complex number z = a + ib is represented as a point (a, b) in the Argand plane.

For example, if we take z = 3 + 4i, it corresponds to the point (3, 4) on the Argand plane. The distance of this point from the origin (0, 0) represents the modulus or absolute value of the complex number, which can be found using the formula: $|z|=\sqrt{a^2+b^2}$

4.0Conjugate of Complex Number

If a complex number z = a + ib is represented as a point (a, b) in the Argand plane then its conjugate is z = a – ib. 

Geometrically Pt (a, –b) is a mirror image of (a, b) on the real axis.

5.0What Are the Different Forms of Representation of Complex Numbers?

Complex numbers can be represented in multiple forms, including:

1. Cartesian Form: This is the standard form z = 3 + 4i, where a is the real part, and b is the imaginary part.
2. Polar Form: In this form, the complex number is represented in terms of its modulus r and argument θ (the angle made with the positive real axis). The polar form is given by:

z = r(cos θ + i sin θ)

where $r=|z|=\sqrt{a^2+b^2}$ and $\theta =\arg (z)={\tan }^{-1}\left(\frac{b}{a}\right)$.

3. Exponential Form: Using Euler’s formula, e^iθ = cos θ + i sin θ, the polar form can be further simplified to:

z = re^iθ

e^iθ = cos θ + i sin θ

6.0Polar Representation of Complex Numbers

The polar representation of a complex number expresses it in terms of its distance from the origin (modulus) and the angle it forms with the positive real axis (argument). As mentioned earlier, this form is especially useful in analyzing the properties of complex numbers geometrically.

A complex number z = a + ib can be represented in polar form using its modulus r (which represents the distance of the point from the origin) and its argument θ (the angle formed with the positive real axis). 

Components of Polar Representation:

1. Modulus (r): 

The modulus r of the complex number z = x + iy is the distance of the point from the origin, calculated as:

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

2. Argument (θ): 

The argument θ (or phase) is the angle that the complex number makes with the positive real axis, measured in radians. It can be determined using the formula:

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

The argument is typically taken between – π and π radians, π, called Principal Argument of Z and is denoted by arg z.

For principal argument –π < θ ≤ π special care must be taken with the quadrant in which the complex number lies.

• First quadrant, 

(x > 0, y > 0) $\theta ={\tan }^{-1}\frac{y}{x}$

• Second quadrant, 

(x < 0, y > 0) $\theta =\pi +{\tan }^{-1}\frac{y}{x}$

• Third quadrant, 

(x < 0, y < 0) $\theta =-\pi +{\tan }^{-1}\frac{y}{x}$

• Fourth quadrant, 

(x > 0, y < 0) $\theta ={\tan }^{-1}\frac{y}{x}$

Thus, the polar form of a complex number z is written as:

$z=r(cos\theta +\iota sin\theta )$

Exponential Form:

Using Euler's formula, $e^{i\theta }=\cos \theta +i\sin \theta$, the polar representation can be further simplified to the exponential form: $z=r{e}^{i\theta }$

7.0De Moivre’s Theorem

De Moivre’s Theorem provides a powerful tool for raising complex numbers in polar form to powers or extracting roots. The theorem states:

(r(cos θ + i sin θ))ⁿ = = rⁿ(cos (nθ) + i sin (nθ))

This is especially useful when calculating powers of complex numbers or solving equations involving complex roots. Using this theorem, we can express higher powers of a complex number by simply multiplying the modulus by the desired power and multiplying the argument by the exponent.

For example, to compute (1 + i)⁵, we first convert 1 + i into polar form, then apply De Moivre’s Theorem.

8.0Application of De Moivre’s Theorem

De Moivre’s Theorem finds applications in several areas, including:

1. Simplifying Powers of Complex Numbers: Instead of expanding complex numbers through binomial expansion, De Moivre’s Theorem simplifies the computation.
2. Finding Roots of Complex Numbers: It is useful for calculating complex roots, such as cube roots, which are crucial in solving certain types of algebraic equations.
3. Fourier Analysis: De Moivre's Theorem is used in Fourier transforms and signal processing to simplify sinusoidal expressions involving complex exponentials.

9.0Solved Example of Argand Plane and Polar Representation of Complex Numbers 

Example 1: Consider the complex number z = 1 + i. To find its polar form:

Solution: 

1. Modulus: $r=\sqrt{1^2+1^2}=\sqrt{2}$
2. Argument: $\theta ={\tan }^{-1}\left(\frac{1}{1}\right)=\frac{\pi }{4}$

Thus, the polar form of 1 + i is: $z=\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)$

or in exponential form: $z=\sqrt{2}{e}^{i\frac{\pi }{4}}$

Example 2: Find the modulus and argument of each the complex number $z=-\sqrt{3}+i$

Solution: 

Let $z=-\sqrt{3}+i$, then

1. Modulus: $|z|=\sqrt{(\sqrt{3}{)}^2+1^2}=\sqrt{4}=2$
2. Argument: $\theta ={\tan }^{-1}\left(\frac{1}{-\sqrt{3}}\right)=\frac{5\pi }{6}$

Example 3: Convert the complex number ($2\sqrt{3}-2i$) into polar form.

Solution: 

The given complex number $z=2\sqrt{3}-2i$

1. Modulus: $|z|=\sqrt{(2\sqrt{3}{)}^2+(-2{)}^2}$

$=\sqrt{12+4}=\sqrt{16}=4$

2. Argument:

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

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

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

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

Thus, the polar form of $2\sqrt{3}-2i$ is

$\Rightarrow z=4\left[\cos \left(-\frac{\pi }{6}\right)+i\sin \left(-\frac{\pi }{6}\right)\right]$

$\Rightarrow \mathrm{z}=4\left[\cos \frac{\pi }{6}-i\sin \frac{\pi }{6}\right]$

Example 4: Convert $\frac{(1+7i)}{(2-i{)}^2}$ into polar form.

Solution: 

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

$\Rightarrow z=\frac{(1+7i)}{\left(4+i^2-4i\right)}$

$\Rightarrow \mathrm{z}=\frac{(1+7\mathrm{i})}{(3-4\mathrm{i})}\times \frac{(3+4\mathrm{i})}{(3+4\mathrm{i})}$

$\Rightarrow z=\frac{(1+7i)(3+4i)}{(9+16)}$

$\Rightarrow \mathrm{z}=\frac{-25+25i}{25}$

$\Rightarrow \mathrm{z}=-1+\mathrm{i}$

1. Modulus:$|z|=\sqrt{(-1{)}^2+1^2}=\sqrt{2}$
2. Argument:

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

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

$\Rightarrow \theta ={\tan }^{-1}(-1)$

$\Rightarrow \theta =\frac{3\pi }{4}$

Thus, the polar form is

$z=\sqrt{2}\left(\cos \frac{3\pi }{4}+i\sin \frac{3\pi }{4}\right)$

Example 5: Express the complex number $\sin \frac{\pi }{5}+i\left(1-\cos \frac{\pi }{5}\right)$ in polar form.

Solution: 

1. Modulus:

$\Rightarrow r^2=|z{|}^2={\sin }^2\frac{\pi }{5}+{\left(1-\cos \frac{\pi }{5}\right)}^2$

$\Rightarrow r^2={\sin }^2\frac{\pi }{5}+1-2\cos \frac{\pi }{5}+{\cos }^2\frac{\pi }{5}$

$\Rightarrow r^2=\left({\sin }^2\frac{\pi }{5}+{\cos }^2\frac{\pi }{5}\right)+1-2\cos \frac{\pi }{5}$

$\Rightarrow r^2=1+1-2\cos \frac{\pi }{5}=2-2\cos \frac{\pi }{5}$

$\Rightarrow r^2=2\left(1-\cos \frac{\pi }{5}\right)$

$\Rightarrow r^2=4{\sin }^2\frac{\pi }{10}$

$\Rightarrow r=2\sin \frac{\pi }{10}$

2. Argument:

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

$\Rightarrow \theta ={\tan }^{-1}\left(\frac{1-\cos \frac{\pi }{5}}{\sin \frac{\pi }{5}}\right)$

$\Rightarrow \theta ={\tan }^{-1}\left(\frac{2{\sin }^2\frac{\pi }{5}}{2\sin \frac{\pi }{10}\cdot \cos \frac{\pi }{10}}\right)$

$\Rightarrow \theta ={\tan }^{-1}\left(\tan \frac{\pi }{10}\right)$

$\Rightarrow \theta =\frac{\pi }{10}$

Thus, the polar form is

$z=2\sin \frac{\pi }{10}\left(\cos \frac{\pi }{10}+i\sin \frac{\pi }{10}\right)$

10.0Practice Questions on Argand Plane and Polar Representation of Complex Numbers

1. Find the modulus and argument of complex number $-1-\mathrm{i}\sqrt{3}$.
2. Convert the complex number ($-2+2\mathrm{i}\sqrt{3}$) into polar form.
3. Convert the complex number $\frac{1}{(1+i)}$ into polar form.
4. Convert the complex number $\frac{(\mathrm{i}-1)}{\left(\cos \frac{\pi }{3}+\mathrm{i}\sin \frac{\pi }{3}\right)}$ into polar form.
5. Express $\frac{1}{(4+3i)}$ of the following in the form (a + ib) and find its conjugate.

11.0Sample Questions on Argand Plane and Polar Representation of Complex Numbers

1. How do you plot a complex number on the Argand Plane?

Ans: To plot a complex number z = x + iy on the Argand plane:

• The real part x is plotted on the horizontal axis.
• The imaginary part y is plotted on the vertical axis. The complex number is then represented as the point (x, y) on this plane.

2. What is the polar form of a complex number?

Ans: The polar form of a complex number z = x + iy expresses it in terms of its modulus r and argument θ as:

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

Where:

• $r=|z|=\sqrt{x^2+y^2}$ is the modulus (distance from the origin).
• $\theta =\arg (z)={\tan }^{-1}\left(\frac{y}{x}\right)$ is the argument (angle with the positive real axis).

3. What is the difference between Cartesian and Polar representation? 

Ans: 

• Cartesian form of a complex number is z = x + iy, where x and y are the real and imaginary components.
• Polar form represents a complex number in terms of its modulus and argument: $z=r(cos\theta +\iota sin\theta )$. This is particularly useful for multiplication, division, and finding powers/roots of complex numbers.

4. What is the modulus of a complex number? 

Ans: The modulus of a complex number z = x + iy is the distance from the origin to the point (x, y) on the Argand plane. It is given by:

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

The modulus represents the magnitude of the complex number.

5. What is the argument of a complex number?

Ans: The argument θ of a complex number z = x + iy is the angle that the line from the origin to the point (x, y) makes with the positive real axis. It is calculated as:

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

The argument is usually expressed in radians and lies between –π and π.

6. What is De Moivre’s Theorem?

Ans: De Moivre’s Theorem is a formula that allows us to raise complex numbers in polar form to powers or extract roots. It states:

$(r(\cos \theta +i\sin \theta ){)}^n=r^n(\cos (n\theta )+i\sin (n\theta ))$

This theorem is especially useful for finding powers and roots of complex numbers.