Geometrical Representation of Complex Numbers

Introduction

In this section you are going to learn about geometrical representation of a complex number. You will learn about modulus and argument of a complex number and how to evaluate them in detail for any complex number. There is a small section on conjugate of a complex number in the end. We will discuss the properties of modulus, argument and conjugate later. But knowing them now only will give us a clear picture of what a complex number is.

Argand Plane

The Argand plane, also known as the complex plane, is a way to visualize complex numbers just as we visualize real numbers on the number line. The real number line is a one-dimensional line where every point corresponds to a real number, with positive numbers lying to the right of the origin and negative numbers to the left. The real number line is sufficient for representing all real numbers, but it cannot represent complex numbers that have both a real and an imaginary component. A real line has one dimension. For a complex number because two components we need two dimensions. Here is how we do it:

The geometrical representation of complex numbers is elegantly depicted on the Argand plane, also known as the complex plane. This plane is a Cartesian coordinate system where a complex number \( z = a + ib \) is represented as an ordered pair \( (a, b) \). Here, each point on the complex plane corresponds uniquely to a complex number in the set \( \mathbb{C} \), establishing a one-to-one correspondence between the set of complex numbers and points on the plane.

The distance of the complex number \( z = a + ib \) from the origin \( 0 + i0 \) is termed the modulus or magnitude of the complex number, denoted as \( |z| \), and is computed as \( \sqrt{a^2 + b^2} \).

The angle \( \theta \), formed by the line connecting the complex number to the origin with the positive x-axis, is known as the argument or amplitude of the complex number. It is symbolized by \( \text{arg}(z) \) or sometimes \( \text{amp}(z) \).

Through this representation, complex numbers gain a spatial interpretation, allowing for a deeper understanding of their properties and interactions.

Non-Uniqueness of Argument

Argument of a complex number is not unique. If \(\theta\) is the argument of the complex number so can be \(\theta + 2\pi\) or \(\theta - 2\pi\) or \(\theta + 4\pi\) and so on. In general the argument of a complex number is \(\theta + 2n\pi\) where \(\theta\) is any one of the argument.

Principal Argument

To make the argument unique we define principal argument.

Principal Argument of a complex number is that argument of the complex number which lies in the interval \((- \pi, \pi]\). Exactly one such argument out of infinite arguments lie in this interval. \((- \pi, \pi]\) denotes one complete revolution around the circle with centre at origin. An argument lying in \((- \pi, \pi]\) also has the smallest magnitude.

Suppose a complex number has the general argument \(2n\pi + \frac{4\pi}{3}\), \(n \in \mathbb{Z}\).

For \(n = 0\), we get \(\frac{4\pi}{3}\)

For \(n = 1\), we get \(\frac{10\pi}{3}\)

For \(n = -1\) we get \(-\frac{2\pi}{3}\)

For \(n = -2\) we get \(-\frac{8\pi}{3}\)

Thus we can see clearly that only one argument which is \(-\frac{2\pi}{3}\) lies in the interval \((- \pi, \pi]\) and we call it the principal argument.

Caution

Since the idea is that only one argument lies in one revolution around a circle, some authors also use the principal argument as \([0,2π)\). So be careful!

How to calculate the argument of a complex number?

Consider the following examples:

Example

Find the arguments of the following complex numbers:

a. \(z=5\)

b. \(z=-5\)

c. \(z=5i\)

d. \(z=-5i\)

e. \(z=1+\sqrt3i\)

f. \(z=-1+\sqrt3i\)

g. \(z=-1-\sqrt3i\)

h. \(z=1-\sqrt3i\)

Solution:

a. \(z = 5\) (This is a real number with no imaginary part, meaning it lies on the positive x-axis.)

• \(a = 5, b = 0\)
• principal \(\arg(z) = 0\) while the general \(\arg(z)=2n\pi\), \(n\in\mathbb Z\), because it lies on the positive x-axis.

b. \(z = -5\) (This is a real number with no imaginary part, meaning it lies on the negative x-axis.)

• \(a = -5, b = 0\)
• principal \(\arg(z) = \pi\) and the general \(\arg(z)=2n\pi+\pi\), \(n\in\mathbb Z\), because it lies on the negative x-axis.

c. \(z = 5i\) (This is a purely imaginary number with no real part, meaning it lies on the positive y-axis.)

• \(a = 0, b = 5\)
• principal \(\arg(z) = \frac{\pi}{2}\), while the general \(\arg(z)=2n\pi+\frac{\pi}{2}\), \(n\in\mathbb Z\) because it lies on the positive y-axis.

d. \(z = -5i\) (This is a purely imaginary number with no real part, meaning it lies on the negative y-axis.)

• \(a = 0, b = -5\)
• \(\arg(z) = -\frac{\pi}{2}\), while the general \(\arg(z)=2n\pi+\frac{\pi}{2}\), because it lies on the negative y-axis.

e.

For \(z = 1 + \sqrt{3}i\), the argument (\(\theta\)) is found by visualizing \(z\) on the complex plane and using trigonometry.

1. Visualization: Plot \(z\) on the complex plane. The real part (\(1\)) is on the x-axis, and the imaginary part (\(\sqrt{3}\)) is on the y-axis. This creates a right triangle with the line from the origin to \(1 + \sqrt{3}i\) as the hypotenuse.

2. Use Trigonometry: The argument \(\theta\) is the angle the line makes with the positive x-axis. The tangent of this angle is the ratio of the opposite side (\(\sqrt{3}\)) to the adjacent side (1):

   \[ \tan(\theta) = \frac{\sqrt{3}}{1} = \sqrt{3} \]

3. Finding \(\theta\): The angle whose tangent is \(\sqrt{3}\) is \(60^\circ\) or \(\frac{\pi}{3}\) radians. This is a standard value from trigonometry.

Therefore, the principal argument of \(z = 1 + \sqrt{3}i\) is \(\frac{\pi}{3}\) radians, indicating it's in the first quadrant where both components (real and imaginary) are positive. The general argument is \(2n\pi+\frac{\pi}{3}\)

f.

For \( z = -1 + \sqrt{3}i \), which lies in the second quadrant where the real part is negative and the imaginary part is positive:

1. Visualization: On the complex plane, \( z \) is 1 unit to the left of the origin and \(\sqrt{3}\) units up.

2. Trigonometry: The magnitude of the tangent of angle \( \alpha \) is the same as before as we have the triangle with same dimensions.

   \[ \tan(\alpha) = \frac{\sqrt{3}}{1} = \sqrt{3} \]

   This means \( \alpha \) is \( 60^\circ \) or \( \frac{\pi}{3} \) radians.

3. Finding \( \theta \): The argument \( \theta \) is measured from the positive x-axis to the line connecting the origin to our complex number, moving counterclockwise. Since \( \alpha \) is \( 60^\circ \) and \( z \) lies in the second quadrant, \( \theta \) is:

\[ \theta = \pi - \alpha = \pi - \frac{\pi}{3} = \frac{2\pi}{3} \]

The principal argument of \( z = -1 + \sqrt{3}i \) is therefore \( \frac{2\pi}{3} \) radians. The general argument is \(2n\pi+\frac{2\pi}{3}\).

g.

For the complex number \( z = -1 - \sqrt{3}i \), which lies in the third quadrant:

1. Visualization: On the complex plane, \( z \) is 1 unit to the left (negative on the real axis) and \(\sqrt{3}\) units down (negative on the imaginary axis).

2. Right-Angled Trigonometry: The acute angle \( \alpha \) in the triangle formed can be found by considering the opposite side over the adjacent side in the triangle. This gives us:

   \[ \tan(\alpha) = \frac{\sqrt{3}}{1} \]

   The acute angle \( \alpha \) is \( 60^\circ \) or \( \frac{\pi}{3} \) radians since the tangent of \( 60^\circ \) is \( \sqrt{3} \).

3. Finding \( \theta \): The overall argument \( \theta \) for \( z \) is measured from the positive x-axis to the line connecting the origin to the point \( -1 - \sqrt{3}i \), moving clockwise. In the third quadrant, \( \theta \) is \( -\pi \) plus the acute angle \( \alpha \):

   \[ \theta = -\pi + \alpha = -\pi + \frac{\pi}{3} = -\frac{2\pi}{3} \]

Thus, the princiipal argument \( \theta \) of \( z = -1 - \sqrt{3}i \) is \( -\frac{2\pi}{3} \) radians. The general argument is \(2n\pi-\frac{2\pi}{3}\)

h.

For the complex number \( z = 1 - \sqrt{3}i \), which lies in the fourth quadrant:

1. Visualization: On the complex plane, \( z \) is 1 unit to the right (positive on the real axis) and \(\sqrt{3}\) units down (negative on the imaginary axis).

2. Right-Angled Trigonometry: The acute angle \( \alpha \) in the triangle formed can be found by considering the opposite side over the adjacent side in the triangle:

\[ \tan(\alpha) = \frac{\sqrt{3}}{1} = \sqrt{3} \]

The acute angle \( \alpha \) is \( 60^\circ \) or \( \frac{\pi}{3} \) radians since the tangent of \( 60^\circ \) is \( \sqrt{3} \).

1. Finding \( \theta \): The argument \( \theta \) for \( z \) is measured from the positive x-axis to the line connecting the origin to the point \( 1 - \sqrt{3}i \), moving clockwise. In the fourth quadrant, \( \theta \) is the negative of the acute angle \( \alpha \):

\[ \theta = -\alpha = -\frac{\pi}{3} \]

Thus, the principal argument \( \theta \) of \( z = 1 - \sqrt{3}i \) is \( -\frac{\pi}{3} \) and the general argument is \(2n\pi-\frac{\pi}{3}, n\in\mathbb Z\)

Given a complex number \( z = x + iy \)

Case I) z lies on axes

1. \( x = 0 \) and \( y = 0 \).

   • The complex number lies at origin.
   • In this case the argument is not defined.

2. \( x > 0 \) and \( y = 0 \)

   • The complex number lies on the positive real axis and thus the argument is \( 2n\pi, n \in \mathbb{Z} \)

   • The principal argument is 0

3. \( x < 0 \) and \( y = 0 \)

   • The complex number lies on the negative real axis and thus the argument is \( (2n+1)\pi, n \in \mathbb{Z} \)

   • The principal argument is \( \pi \).

4. \( x = 0 \) and \( y > 0 \)

   • The complex number lies on the positive imaginary axis and thus the argument is

     \[ {2n\pi + \frac{\pi}{2}}, n \in \mathbb{Z} \]

   • The principal argument is \( \frac{\pi}{2} \)

5. \( x = 0 \) and \( y < 0 \)

   • The complex number lies on the negative imaginary axis and thus the argument is

     \[ {2n\pi - \frac{\pi}{2}}, n \in \mathbb{Z} \]

   • The principal argument is \( -\frac{\pi}{2} \)

Case II) For complex numbers not lying on axes:

1. Calculate \( \alpha = \tan^{-1}\left(\left|\frac{y}{x}\right|\right) \)

2. Decide the quadrant in which the complex number lies.

3. Use the following figure to evaluate principal argument \( \theta \).

Alternative way

If you are familiar with inverse trigonometry, then for complex numbers not lying on axes, we can use the following rules:

Case I) \( x > 0 \) and \( y \neq 0 \) [Complex number lies in the 1st and 4th quadrant]

• The principal argument is \( \tan^{-1}\left(\frac{y}{x}\right) \) and the general argument is \( \tan^{-1}\left(\frac{y}{x}\right) + 2n\pi \)

Case II) \( x < 0 \) and \( y > 0 \) [Complex number lies in the 2nd quadrant]

• The principal argument is \( \tan^{-1}\left(\frac{y}{x}\right) + \pi \) and the general argument is \( \tan^{-1}\left(\frac{y}{x}\right) + (2n+1) \pi \)

Case III) \( x < 0 \) and \( y < 0 \) [Complex number lies in the 3rd quadrant]

• The principal argument is \( \tan^{-1}\left(\frac{y}{x}\right) - \pi \) and the general argument is \( \tan^{-1}\left(\frac{y}{x}\right) + (2n-1) \pi \)

Conjugate of a complex number

The conjugate of a complex number is obtained by changing the sign of the imaginary part of the complex number. If a complex number is represented as \( z = x + iy \), then its conjugate, denoted as \( \bar{z} \), is given by \( \bar{z} = x - iy \). The process of taking a conjugate involves putting a bar over the complex number and then switching the sign of the imaginary component.

For example, if the complex number is \( 2 - 3i \), its conjugate would be \( 2 + 3i \). The sum and product of a complex number and its conjugate are always real numbers. The equation provided for this is \( (x + iy)(x - iy) = x^2 + y^2 \) and \( x + iy + x - iy = 2x \).

Geometrically, the conjugate of a complex number \( z \) is the reflection of \( z \) in the real axis. Therefore, the argument of the conjugate of \( z \), denoted as \( \arg(\bar{z}) \), is the negative of the argument of \( z \), which can be written as \( \arg(\bar{z}) = -\arg(z) \).

Example

Problem:

Given the quadratic equation \( x^2 - 3x + 9 = 0 \), find the roots.

Solution:

The solutions to the equation \(x^2 - 3x + 9 = 0\) are complex numbers because the discriminant \( b^2 - 4ac \) is negative. The solutions are:

\[ x = \frac{3}{2} + \frac{\sqrt{3}}{2}i \quad \text{and} \quad x = \frac{3}{2} - \frac{\sqrt{3}}{2}i \]

These solutions are conjugate complex numbers. In general, when you have a quadratic equation with real coefficients and a negative discriminant, the solutions will always be a pair of complex conjugates \(a + bi\) and \(a - bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Conjugate Roots of a quadratic equation

For a quadratic equation \( ax^2 + bx + c = 0 \) with real coefficients \( a \), \( b \), and \( c \) and a discriminant \( \Delta = b^2 - 4ac < 0 \), the roots of the equation are complex and are conjugates of each other.

Proof:

Let the quadratic equation be given by \( ax^2 + bx + c = 0 \), where \( a, b, c \in \mathbb{R} \) and \( a \neq 0 \).

Consider the discriminant \( \Delta = b^2 - 4ac \). For complex roots, we have \( \Delta < 0 \).

The quadratic formula for finding the roots of \( ax^2 + bx + c = 0 \) is:

\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]

Since \( \Delta < 0 \), we cannot take the square root of a negative number in the real number system. Therefore, we introduce the imaginary unit \( i \) where \( i^2 = -1 \). This allows us to write \( \sqrt{\Delta} \) as \( i\sqrt{-\Delta} \) since \( \Delta \) is negative.

Substituting \( i\sqrt{-\Delta} \) for \( \sqrt{\Delta} \) in the quadratic formula, we get:

\[ x = \frac{-b \pm i\sqrt{-\Delta}}{2a} \]

Hence, the roots of the quadratic equation are:

\[ x_1 = \frac{-b}{2a} + \frac{i\sqrt{-\Delta}}{2a} \]

\[ x_2 = \frac{-b}{2a} - \frac{i\sqrt{-\Delta}}{2a} \]

Notice that \( x_1 \) and \( x_2 \) are of the form \( p + qi \) and \( p - qi \) respectively, where \( p = -\frac{b}{2a} \) and \( q = \frac{\sqrt{-\Delta}}{2a} \).

Therefore, \( x_1 \) and \( x_2 \) are complex conjugates of each other, as they have the same real part \( p \) and imaginary parts that are negatives of each other \( \pm q \).

Thus, for any quadratic equation with real coefficients and a negative discriminant, the roots will always be complex conjugates of the form \( \frac{-b}{2a} \pm \frac{i\sqrt{-\Delta}}{2a} \).

QED