Complex Number as a Point in The Plane

A complex number a + ib, from the origin (0, 0) ending at the point (a, b) on the complex plane. This vector with both direction and magnitude, similar to an arrow in geometry or physics. The length of the vector a + bi is: 

Magnitude = a2+b2 . For example, 3 + 4i it will be: 

32+42 = 9+16 = 25 = 5
Here, the magnitude is 5

Modulus of a Complex Number as a geometric property.

A complex number is written as z = a + bi, where a is the real part, b is the imaginary part. 
The modulus of a number is its distance of that number from zero on the number line.  
For any real number x: 
            |x| = x (if x0)
            |x| = x (if x0)  
For example: 
|3| = 3 
|-3| = 3 

Both positive and negative numbers result in positive results.
In both cases, the number is 3 units away from 0, 
Because distance is always positive or zero and never negative
Formula: 
           |z| =  a2+b2 

Example: if you plot the point z = 3 + 4i, it corresponds to the point (3, 4) on the plane.
So, |3 + 4i| =  32+42 = 9+16 = 25 = 5
5 is the length of the line from the origin to the point (3,4).

Example:  if you plot the point z = -2 + 0i, it corresponds to the point (2, 0) on the plane.
So,  |-2 + 0i| =  (-2)2+02 = 4+0 = 4 = 2
2 is the length of the line from the origin to the point (2,0).
 

The argument of a complex number is the angle between the positive real axis and the line joining the origin to the point (a, b) in the complex plane.

For a complex number, z = a + bi, the argument of z: 
 = tan-1ba

Let us try to understand below how the argument  of a complex number z = a + bi is in different quadrants

1. First quadrant (a > 0, b > 0):  = tan-1ba

2. Second quadrant (a < 0, b > 0):  - tan-1ba

3. Third quadrant (a < 0, b < 0): - + tan-1ba

4. Fourth quadrant (a > 0, b < 0):    = -tan-1ba

The principal argument of a complex number lies in -π < θ <π.

The main value of a complex number is restricted to a specific range here. The principal argument is also known as amplitude.
The  is such that
Arg(z)  (-, ] (in radians) 
Arg(z)  (-180, 180)

It is the unique angle formed between the positive real axis and the line representing the complex number in the complex plane. 

Example: For z = 1+i
 = tan-111 = 4
Thus, the principal argument of z = 1+ i is 4

Let us now understand the principal argument of Z = a =+ ib in different quadrants

 
1. First quadrant (a > 0, b > 0): Arg Z = tan-1ba
 

2. Second quadrant (a < 0, b > 0) : Arg Z = π + tan-1ba

3. Third quadrant (a < 0, b < 0): Arg Z = - + tan-1ba
 

In some cases,

Case 1: 

z = 3 + 0i, written as z = (3, 0) 
This lies on the positive real axis, which is the x-axis
This has no angle, and it lies flat on the axis.  
The principal argument for this is  = 0

Case 2: 

z = 0 + 2i, written as z = (0,2)
It lies on the positive imaginary axis, that is y-axis
There is no angle for this, and it lies along the axis 
The principal argument of this complex number is π/2

A complex number like z = 1 + i is plotted as the point (1, 1). A diagram of the complex plane can show the real axis, imaginary axis, and the vector from (0, 0) to (1, 1), with |z| = 2,  = 4

Complex numbers corresponding to vector addition

Addition or subtraction of complex numbers is similar to adding and subtracting polynomials; we combine real and imaginary parts and then operate. 

The formula for adding and subtracting complex numbers is
Adding complex numbers z1= a + bi and z2= c + di is, 
z1+ z2 = (a + c) + (b + d)i
For example: Add z1= 3 + 4i and z2= 2 + 5i 
Solution: 
Given,  
           z1= 3 + 4i (a=3, b=4) 
           z2= 2 + 5i (c=2, d=5)
z1+ z2= (3 + 2) + (4 + 5)i
z1+ z2= 5 + 9i
Representing this in a graph below

 
Geometrically, adding two complex numbers z1 and z2 corresponds to vector addition using the parallelogram law. The sum is represented by the diagonal of the parallelogram formed by the vectors representing z1 and z2. 

Subtracting complex numbers z1= a + bi and z2= c + di is, 
z1- z2 = (a - c) + (b - d)i
For example: Add z1= 3 + 4i and z2= 2 + 5i 
Solution: 
Given,  
           z1= 3 + 5i (a=3, b=5) 
           z2= 2 + 4i (c=2, d=4)
z1+ z2= (3 - 2) + (5 - 4)i
z1+ z2= 1 + 1i

Multiplication of complex numbers Formula 

Multiplying complex numbers follows the same process as multiplying binomials or polynomials. To multiply complex numbers, we use the distributive property.
(a + ib) (c + id) = ac + iad + ibc + i2bd
 (a + ib) (c + id) = (ac - bd) + i(ad + bc) [because i2 = -1]
Multiplying complex numbers formula 
         (a + ib) (c + id) = (ac - bd) + (ad + bc)i

Multiplication of complex numbers in polar form

 The polar form of a complex number is z = r (cos + i sin), where r is the complex number's modulus and  its argument. The formula for multiplying complex numbers would be:

z1= r1(cos 1+ i sin  1)   
z2= r2(cos 2+ i sin  2) 
z1z2 = [r1(cos 1+ i sin  1)] [r2(cos 2+ i sin  2)] 
= r1r2 (cos1cos2 + i cos1 sin2 + i sin1cos2 + i2 sin1 sin2)
= r1r2  (cos1cos2 + i cos1 sin2 + i sin1cos2 -  sin1 sin2) {because i2= -1}
=   r1r2  [cos1cos2 -  sin1 sin2+ i (cos1 sin2 +  sin1cos2)]
= r1r2 [cos (1+ 2) + i sin(1+ 2)] {because cos a cos b - sin a sin b = cos (a + b) and sin a cos b + sin b cos a = sin (a + b)}

Hence,  [r1(cos 1+ i sin  1)] [r2(cos 2+ i sin  2)] = r1r2 [cos (1+ 2) + i sin(1+ 2)] 

The geometric effect of the conjugation of complex numbers

Z = a + ib represents (a, b) in the complex plane
z = a - ib represents (a, -b), which is the conjugate on the complex plane

Conjugation geometrically reflects a point across the real axis by changing the sign of the imaginary part, thus creating a mirror image
Below is a graphical representation of the above condition

Polar form connection of a complex number

Any complex number z = a + bi is written in polar form as
z = r (cosθ + i sinθ)
And using Euler's formula, it is written as 
     z = reiθ
And how are they connected?

Algebra and geometry: In the complex plane, z = a + bi represents a point, which can also be viewed geometrically as a vector from the origin (0, 0) to the point (a, b). 
This will be in rectangular form, which  is expressed as z = a + bi, and this represents coordinates (a, b)
Modulus and Argument: The modulus r = |z| =  a2+b2 represents the length of the vector, while the argument θ=arg(z) =tan-1(b/a) represents the angle the vector makes with the real axis, thus describing the same point using polar coordinates in terms of distance and direction.

Trigonometry: Applying trigonometric principles to a right triangle

cos θ = ar
sin θ = br 
then , a = r cos θ and b = r sin θ
z = a + bi = r cos θ + ir sin θ = r(cos θ + i sin θ), this is the polar form.
This is in polar form, which is expressed as z = r(cos θ + i sin θ), and this represents length and angle

Euler’s formula connecting to complex numbers

The trigonometric expression, which is the complex number in polar form,
         z = r (cosθ + i sinθ) 
This is replaced by Euler's formula, 
         ei = cosθ + i sinθ
We get,
            z = rei
This is in Euler's form, which is expressed as z = r(cos θ + i sin θ), and this is the exponential representation 

Distance Between Complex Numbers

The distance between two complex numbers, z1 = a + bi and z2 = c + di, is the same as the distance between their corresponding points (a, b) and (c, d) in the complex plane.

You can calculate this distance using either the standard distance formula from geometry:
Distance = (a-c)2+(b-d)2

Or by finding the modulus (absolute value) of their difference:
Distance = | z1- z2| = |(a - c) + (b - d)i| = (a-c)2+(b-d)2 
Both methods yield the same result, rooted in the Pythagorean theorem.

 Subtracting two complex numbers, z1 - z2 , and then taking the modulus, |z1 - z2|, directly calculates the distance between the points representing those complex numbers in the complex plane.
 

Complex numbers are helpful in various real-world applications, in fields like oscillations, waves, rotations, and many more. Here are some examples of it

1.  In Signal Processing (Image and Audio), beyond basic signal analysis, complex numbers are used in advanced image and audio processing techniques, including compression and restoration.

2. When medical imaging techniques like MRI (Magnetic Resonance Imaging) and CT scans utilize Fourier analysis with complex numbers to reconstruct detailed images of the human body.

3. In Quantum Mechanics, the wave function in quantum mechanics, which describes the state of a particle, is inherently complex-valued. Complex numbers are not just a mathematical convenience here; they are fundamental to the theory.

4. Complex numbers are used in Financial Mathematics, where it is applied in modeling stochastic processes, such as stock price movements, and the analysis of financial derivatives.

5. In Navigation Systems (GPS), Signal processing algorithms in GPS receivers use techniques involving complex numbers to determine precise location and timing information.