Complex Numbers

The concept of complex numbers was first introduced by the Greek mathematician called the Hero of Alexandria in the 1st century while attempting to find the square root of a negative number. However, he converted the negative into a positive before taking the square root. Then in the 16th century, Italian mathematician Gerolamo Cardano formally defined complex numbers while solving negative roots in cubic and quadratic polynomials.

A complex number is represented in the form a + bi, where a is the real part and b is the imaginary part. Complex numbers help in determining the square root of negative numbers. 

The i in the complex number is called ‘iota’. It is used to represent the imaginary part of the complex number. Thereby, it helps to find the square root of negative numbers as the value of i² = –1. 
 

In order to understand more about complex numbers, first let’s learn about its properties.

 

• If x, y are real numbers and x + iy = 0 then x = 0, y = 0. 
  
  That is,   x + iy = 0 = 0 + i0   
  
  According to the definition of equality for complex numbers, the real parts and imaginary parts must be equal. Therefore, we conclude that:
  
  x = 0, y = 0
  
   
• If x + iy = u + iv, then x = u, and y = v
  
  Given that x, y, u and v are real numbers, we know:
  
   x + iy = u + iv
  
  By definition of complex number equality, their real and imaginary components must be equal, leading to: 
  
  x = u, y = v
  
   
• For any three complex numbers u, v, and z, the following fundamental laws hold:
   

1.  Commutative Law:  u + v = v + u
    u . v = v. u
    
2. Associative Law: (u + v) + z = u + (v + z)
    (u . v) . z = u . (v . z)
    
3. Distributive Law: x . (v + z) = x . v + x . z
   
    

• The product of two conjugate complex numbers is a real number. Let z = x + iy, where x and y are real. The conjugate is:
  
  z = x – iy
  
  Multiplying z by its conjugate:
  
  z . z = (x + iy) (x – iy)
  
  Expanding using the difference of squares: 
  
  x² – i²y²
  
  Since i^² = – 1, this simplifies to: 
  
  x² + y²
  
  Which is a real number.

The standard form of a complex number is given by z = a + ib, where a, b ∈ ℝ and i (iota) represents the imaginary unit. Based on the values of a (the real part) and b (the imaginary part), complex numbers can be categorized in

• Zero Complex Numbers: If a = 0 and b = 0, the complex number is called a zero complex number. The only example of this is 0. 
   
• Purely Real Numbers: If a ≠ 0 and b = 0, the number is considered a purely real number, meaning it has no imaginary component. Examples include 2, 3, 5, 7, and all the other real numbers. 
   
• Purely Imaginary Numbers: If a = 0 and b ≠ 0, the number is considered a purely imaginary number, as it has no real part. Examples include – 7i, – 5i, – i, 5i, 7i, and other numbers with only an imaginary component. 
   
• General Complex Numbers: If a ≠ 0 and b ≠ 0, the number is classified as a complex number that includes both real and imaginary parts. Examples include (–1 – i), (1 + i), (1 – i), (2 + 3i), and similar expressions. 

Complex numbers can be represented in different forms, each with its own advantages depending on the application. The three most common representations are Rectangular (standard) Form, Polar Form, and Exponential Form. These forms allow for easier calculations in various fields.

• Rectangular Form: Also known as the Standard Form, this representation of a complex number is written as a + ib, where a and b are real numbers.
  
  Examples: (5 + 5i), – 7i, (– 3 – 4i), etc.

• Polar Form: In this representation, a complex number is expressed in terms of its magnitude (r) and angle () relative to the positive x-axis. It is written as: 
  
  r [cos θ + i sin θ]
  
  Here, r represents the distance from the origin, and is the angle between the radius vector and the positive x-axis. 
  
  Examples: [cos 2 + i sin 2], 5 [cos 6 + i sin 6], etc.

• Exponential Form: This form uses Euler’s Formula to express a complex number as: 
  
  \(re^{iθ} \space \text{or}\space r·e^{iθ}\) 
  
  Where r is the magnitude and θ is the angle. 
  
  Examples: eⁱ⁽⁰⁾, e^i/2, 5e^i/6, etc.

A complex number z = a + ib is uniquely represented by the point P (a, b) on the complex plane. Similarly, every point on the complex plane corresponds to a unique complex number.

Complex Plane: The plane where complex numbers are uniquely represented is known as the Complex Plane, also called the Argand Plane or Gaussian Plane. 

The Complex Plane has two axes, they are:
 

• X-axis or Real Axis
   
• Y-axis or Imaginary Axis

X-Axis (Real Axis)
 

• Every purely real complex number is represented by a unique point on this axis.
   
• The real part, Re(z), of any complex number is plotted along this axis.
   
• For this reason, the axis is known as the Real Axis.

Y-Axis (Imaginary Axis)
 

• Every purely imaginary complex number is represented by a unique point on this axis. 
   
• The imaginary part, Img(z), of any complex number is plotted along this axis.
   
• Therefore, the Y-axis is referred to as the imaginary Axis. 

To plot a complex number z = a + ib on the complex plane, follow these rules:

• The real part of z, denoted as Re(z) = a, is assigned as the X-coordinate of the point P. 
   
• The imaginary part of z, denoted as Im(z) = b, is assigned as the Y-coordinate of the point P.
   
• Thus, the complex number z = a + ib corresponds to the point P (a, b) on the complex plane. 

The modulus of a complex number represents its absolute value and is defined as the distance from the origin to the corresponding point on the complex plane. It is also known as the magnitude of the complex number. 

For a complex number z = a + ib, the modulus is calculated as:

z = √(a^² + b^²)

The argument of a complex number is the angle θ between its vector and the positive real (x) axis. 

Mathematically, for z = a + ib, it is given by:

 
  \(\theta = \arctan\left(\frac{b}{a}\right) \)

Complex numbers follow specific mathematical operations similar to real numbers, but with additional rules for handling the imaginary unit i. These operations include addition, subtraction, multiplication, and division, are carried out using fundamental algebraic properties. Mastering these operations is crucial for handling complex numbers in both mathematical and real-world applications. 

• Addition: To add complex numbers, sum their real and imaginary parts separately.

            
Example: (3 + 2i) + (1 + 4i) = 4 + 6i.

• Subtraction: Subtract the real parts and the imaginary parts separately.

Example: (3 + 2i) – (1 + 4i) = 2 – 2i

• Multiplication: Multiply two complex numbers using the distributive property and the rule i2 = –1.

Example: (3 + 2i) (1 + 4i) = 3 + 12i + 2i + 8i² = 3 + 14i – 8 = – 5 + 14i.

• Division: To divide a complex number by another, multiply both numerator and denominator by the conjugate of the denominator, then simplify.
   

Example: (3 + 2i) / (1 + 4i) = (3 + 2i) (1 - 4i) / (1 + 4i)(1 – 4i) = (11 - 10i)/17.

Solving complex numbers can be difficult, but with the right approach, it can be easy and simple. Here are some tips and tricks to master the complex numbers:

1. Memorize the powers of i:
   i = √–1
   i² = –1
   i³ = –i 
   i⁴ = 1
    
2. When adding or subtracting two complex numbers, add or subtract the real parts and the imaginary parts separately and then combine both.
    
3. When solving for large power of i, simplify the power in factor of 4, for easy calculation.
    
4. Remember the values of some common expressions:
   (1 + i)² = 2i
   (1 – i)² = –2i
   w³ = 1
   1 + w + w² = 0
    
5. Always rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.

Complex numbers are more than just numbers or theoretical math, they are essential in technology, science, and engineering. Here are some of the real applications of complex numbers. 

• Electrical Engineering: Complex numbers are used in AC circuit analysis, where voltage and current are represented as complex numbers to simplify calculations involving phase differences.
   
• Signal processing: They help in processing audio, radio, and TV signals by representing waves mathematically. They are also used in fluid dynamics.
   
• Computer graphics and animation: They are used to performing transformations, rotations, and scaling of objects in 2D and 3D graphics.
   
• Physics: Complex numbers are used in quantum mechanics to describe wave functions, in electromagnetism to solve field equations, and in fluid dynamics.
   
• Control systems: Engineers use complex numbers to evaluate stability and performance in systems such as airplane autopilot and robotic motion control.