Complex Numbers

For centuries, mathematicians were stumped by a simple problem: What is the square root of a negative number? Since any number multiplied by itself (positive or negative) results in a positive value, the equation x² = -1 seemed impossible.

Enter the Complex Number. Far from being “imaginary” in the sense of being fake, complex numbers are essential tools in modern physics, engineering, and computer graphics.

What is a Complex Number?

A complex number is written in the form:

z = a + bi

Where:

• a = Real part denoted by Re z
• b = Imaginary part denoted by Im z
• i = Imaginary unit

The imaginary unit is defined as:

i² = −1

Example:

3 + 4i
Real part = 3
Imaginary part = 4i

The Imaginary unit (i)

The imaginary unit allows us to represent the square root of −1.

i = √−1

Powers of i follow a repeating cycle:

i¹ = i
i² = −1
i³ = −i
i⁴ = 1

Then the pattern repeats:

i⁵ = i
i⁶ = −1
i⁷ = −i
i⁸ = 1

This cycle repeats every 4 powers.

In general, for any integer k:

i⁴ᵏ = 1

i⁴ᵏ⁺¹ = i

i⁴ᵏ⁺² = −1

i⁴ᵏ⁺³ = −i

The Modulus and the Conjugate of a Complex Number

Let z = a + ib be a complex number. Then the modulus of z, denoted by |z|, is defined as:

|z| = √(a² + b²)

The conjugate of z, denoted by z, is defined as:

z = a − ib

Examples

|3 + i| = √(3² + 1²) = √10

|2 − 5i| = √(2² + (−5)²) = √29

3 + i = 3 − i

2 − 5i = 2 + 5i

−3i − 5 = 3i − 5

Multiplicative Inverse

The multiplicative inverse of a non-zero complex number z is:

z⁻¹ = 1 / (a + ib)

= a/(a² + b²) − i·b/(a² + b²)

= (a − ib)/(a² + b²)

= z / |z|²

Important Result

z × z = |z|²

Properties

For any two complex numbers z₁ and z₂:

1. |z₁ z₂| = |z₁| |z₂|
2. |z₁ / z₂| = |z₁| / |z₂|   (provided z₂ ≠ 0)
3. z₁ z₂ = z₁ z₂
4. z₁ ± z₂ = z₁ ± z₂
5. z₁ / z₂ = z₁ / z₂   (provided z₂ ≠ 0)

Visualising the Complex Plane or Argand Plane

Unlike real numbers, which sit on a single line, complex numbers require two dimensions. We use the Complex Plane (Argand Diagram):

1. Horizontal Axis (x): The Real part (a).
2. Vertical Axis (y): The Imaginary part (bi).

Example: 3 + 4i is plotted 3 units right and 4 units up.

The plane having a complex number assigned to each of its point is called the complex plane or the Argand plane

The Argument (𝛳) in Complex Numbers

In the context of complex numbers, 𝛳(theta) represents the argument, often written mathematically as arg(z)

The argument is the angle formed between the positive Real axis (the right side of the horizontal line) and the line connecting the origin to your complex number.

When defining a point in the complex plane:

• The Modulus (r) tells you how far the point is from the center (origin).
• The Argument (𝛳) tells you which direction to point.

It is a crucial part of writing complex numbers in polar form. Instead of standard rectangular coordinates (a + bi), polar form looks like:

z = r(cos 𝛳 + i sin 𝛳)

z = r eⁱᶿ (Euler’s Formula)

To calculate 𝛳, basic trigonometry can be used based on the real part (a) and the imaginary part (b):

tan 𝛳 = b/a

For example, if z = 3+4i :

Real part (a) = 3

Imaginary part (b) = 4

tan 𝛳 = b/a = 4/3

𝛳 = 53.1°

Conjugate of Complex Number in Argand Plane

The representation of a complex number z = x + iy and its conjugate z = x – iy in the Argand plane are, respectively, the points P (x, y) and Q (x, – y). Geometrically, the point (x, – y) is the mirror image of the point (x, y) on the real
axis.

Algebra of Complex Numbers

In this section, we develop the algebra of complex numbers

1. Addition of two complex numbers

Let$$z_1 = a + ib$$

and$$z_2 = c + id$$

be any two complex numbers.

Then the sum of $z_1$​ and $z_2$ is defined as$$z_1 + z_2 = (a + c) + i(b + d)$$

Thus, the sum of two complex numbers is again a complex number.

Example

$$(2 + 3i) + (-6 + 5i)$$ $$= (2 – 6) + i(3 + 5)$$$$= -4 + 8i$$

Properties of Addition of Complex Numbers

The addition of complex numbers satisfies the following properties:

(i) Closure Law

The sum of two complex numbers is always a complex number.$$z_1 + z_2$$

is a complex number for all complex numbers $z_1$​ and $z_2$

(ii) Commutative Law

For any two complex numbers $z_1$ and $z_2$,$$z_1 + z_2 = z_2 + z_1$$

(iii) Associative Law

For any three complex numbers $z_1, z_2, z_3$$$(z_1 + z_2) + z_3 = z_1 + (z_2 + z_3)$$

(iv) Existence of Additive Identity

There exists a complex number$$0 + i0$$

denoted simply as 0, called the additive identity or zero complex number, such that$$z + 0 = z$$

for every complex number $z$.

(v) Existence of Additive Inverse

For every complex number$$z = a + ib$$

there exists a complex number$$-a + i(-b)$$

denoted by$$-z$$

This number is called the additive inverse (or negative) of $z$

Thus,$$z + (-z) = 0$$z+(−z)=0

which gives the additive identity.

2. Subtraction of Two Complex Numbers:

Given any two complex numbers z₁ and z₂ , the difference z₁ − z₂ is defined as:

z₁ − z₂ = z₁ + (−z₂)

Examples

Example 1

(6 + 3i) − (2 − i)

= (6 + 3i) + (−2 + i)
= 4 + 4i

Example 2

(2 − i) − (6 + 3i)

= (2 − i) + (−6 − 3i)
= −4 − 4i

3. Multiplication of Two Complex Numbers

Let

z₁ = a + ib
z₂ = c + id

be any two complex numbers.

The product z₁z₂ is defined as:

z₁z₂ = (ac − bd) + i(ad + bc)

Example:

(3 + i5)(2 + i6)

= (3 × 2 − 5 × 6) + i(3 × 6 + 5 × 2)

= −24 + i28

Properties of Multiplication of Complex Numbers

1. Closure Law

The product of two complex numbers is also a complex number.

If z₁ and z₂ are complex numbers, then
z₁z₂ is also a complex number.

2. Commutative Law

For any two complex numbers z₁ and z₂:

z₁z₂ = z₂z₁

3. Associative Law

For any three complex numbers z₁, z₂, z₃:

(z₁z₂)z₃ = z₁(z₂z₃)

4. Multiplicative Identity

There exists a complex number

1 + 0i (denoted as 1)

such that for every complex number z:

z × 1 = z

5. Multiplicative Inverse

For every non-zero complex number

z = a + ib (where a ≠ 0 or b ≠ 0)

there exists a complex number:

1/z = a/(a² + b²) − i b/(a² + b²)

This number is called the multiplicative inverse of z.

It satisfies:

z × (1/z) = 1

6. Distributive Law

For any three complex numbers z₁, z₂, z₃:

(a) z₁(z₂ + z₃) = z₁z₂ + z₁z₃

(b) (z₁ + z₂)z₃ = z₁z₃ + z₂z₃

4. Division of Two Complex Numbers:

Given any two complex numbers $z_1$​ and $z_2$​, where $z_2 \neq 0$, the quotient $\frac{z_1}{z_2}$ is defined by$$\frac{z_1}{z_2} = z_1 \times \frac{1}{z_2}$$

Example

Let$$z_1 = 6 + 3i \quad \text{and} \quad z_2 = 2 – i$$

Step 1: Write the Division

$$\frac{z_1}{z_2} = \frac{6 + 3i}{2 – i}$$

Step 2: Find the Multiplicative Inverse of the Denominator

$$\frac{1}{2-i} = \frac{2}{2^2 + (-1)^2} + i\frac{-(-1)}{2^2 + (-1)^2}$$$$= \frac{2}{5} + \frac{i}{5}$$$$= \frac{2+i}{5}$$

Step 3: Multiply

$$\frac{z_1}{z_2} = (6+3i)\left(\frac{2+i}{5}\right)$$

Step 4: Expand

$$(6+3i)(2+i)$$$$= 12 + 6i + 6i + 3i^2$$$$= 12 + 12i – 3$$$$= 9 + 12i$$

Step 5: Final Answer

$$\frac{z_1}{z_2} = \frac{9 + 12i}{5}$$

or$$\frac{9}{5} + \frac{12}{5}i$$

Equality of Complex Numbers

Two complex numbers are equal if their real parts and imaginary parts are equal.

If$$a + ib = c + id$$

then

• $a = c$
• $b = d$

Why Do They Matter?

It’s easy to think these are theoretical, but they drive our modern world:

• Electrical Engineering: Essential for calculating AC circuits and impedance.
• Fluid Dynamics: Used to model airflow over airplane wings.
• Quantum Physics: The Schrödinger equation relies on complex numbers.
• Fractals: The Mandelbrot Set is generated through iterations of complex numbers.