Argument of Complex Number

A complex number has two parts: a real part and an imaginary part. The argument is the angle between the positive x-axis and the line representing the direction or angle of the complex number relative to the positive real axis. In the Argand plane, we can plot any complex number with its real part on the x-axis and its imaginary part on the y-axis. The complex number Z = a + ib can be plotted as a point A (a, b), and the angle can be found using the inverse tangent of the imaginary part divided by the real part.

Any complex number can be written as:

    Z = a + ib, where a is the real part and b is the imaginary part. 

When the complex number lies in the first quadrant, we can use the formula for finding the argument:

     θ = tan ^{- 1} (b/a)
 

The inverse tangent function is represented as tan -1. 

For other quadrants, different formulas and some adjustments are needed to find the correct argument. 

In the Argand plane, the argument of complex numbers is determined by the quadrant where the point (a, b) is located. Therefore, the formula for the argument differs from others depending on the quadrant. 

• First quadrant (a > 0, b > 0)

The complex number is located in the first quadrant, and the argument is calculated using: 

θ = tan ^{- 1} (b/a)

• Second quadrant (a < 0, b >0)

When the complex number is located in the second quadrant, the direction can be adjusted by adding π or 180°. 
The formula for argument is:

θ = π - tan ^{- 1} (b/a)

• Third quadrant (a < 0, b < 0)

π or 180° is added to adjust the direction when the complex number is in the third quadrant. The formula is: 

θ = π + tan ^{- 1} (b/a)

• Fourth quadrant (a > 0, b < 0)

The argument of a complex number is negative when the number is located in the fourth quadrant. The principal value of the argument is:

θ = -tan - 1 (b/a)

There are some special cases where the formula varies, mainly when the real part (a) or the imaginary part (b) is zero. 

• If a = 0, b > 0 
  
  If the real part of a complex number is 0, and the imaginary part is greater than zero, the complex number is called a purely imaginary positive number. 
  The argument is: 
  
  θ = π/2 or (90°)

• If a = 0, b < 0
  
  The complex number is called a purely imaginary negative number if the real part is 0 and the imaginary part is less than 0. 
  
  θ = -π/2

• If b = 0, a > 0
  
  If the imaginary part is 0 and the real part of the complex number is greater than zero, the complex number is called a purely real positive number. 
  The argument is: 
  
  θ = 0

• If b = 0, a < 0
  
  The complex number is called a purely real negative number if the imaginary part is 0 and the real part is less than 0. 
  
  θ = π (180°)

An angle has both a principal value and a general value, giving us the principal and general arguments. The argument of the complex number is determined using the inverse tangent function, which follows the general solution of the trigonometric tangent function.

The values of the principal argument of a complex number is denoted as Ard(z) and range from -π  <  θ   π for the first and fourth quadrants. Starting from the positive x-axis, the angle is measured in the counterclockwise direction in the first and second quadrants, so 0 < θ < π. In the third and fourth quadrants, the angle is measured in the clockwise direction from the positive x-axis, so -π < θ < 0. 

2nπ + θ is the general argument of complex numbers, where θ is the principal argument, and n is any integer. The argument of the complex number has both a principal and a general argument, determined using the tangent function. 

Two fundamental characteristics, the argument of a complex number and the modulus of a complex number, completely describe the complex number in the Argand plane. The modulus of a complex number explains how far it is from the origin, while the argument is the angle it makes with the line representing the number and the positive x-axis. Now, let us look at each characteristic in detail.

Modulus of complex number: In the Argand plane, modulus is the distance from the origin (0,0) to the point (a, b) representing the complex number. The standard form of a complex number is Z = a + ib, and the modulus is represented as |z|. The modulus is the square root of the sum of the squares of the real and imaginary parts of a complex number, which is expressed as |Z| = √a²+ b². 

Argument of a complex number: The argument of a complex number is the angle measured from the positive x-axis to the line representing the complex number on the Argand plane. It is represented as θ = tan ^{- 1} (b/a).
 
The point A (a,b) with the origin point O (0, 0) represents the complex number. 

Understanding the real-life significance of the argument of a complex number helps students apply it more easily in various situations. The real-world applications of the argument of a complex number are listed below:

• In electrical engineering, engineers use the argument of complex numbers to determine the phase difference between voltage and current. The voltage and current are complex numbers in alternating current circuits.

• In communication technology, engineers use complex numbers and the argument to transmit and modulate data efficiently. The signals of radio, mobile networks, and Wi-Fi are optimized and minimized by calculating their argument. 

• In aerospace, pilots use the argument of complex numbers to model velocity vectors, which contain both motion direction and speed. It helps them to find an aircraft or plane’s accurate angle of the velocity vector, and minimize the turbulence.