Geometric Interpretation of Complex Number

The set of the complex numbers, which is typically indicated by the notation C  is, by definition, the vector space R², also known as the set of pairs of real numbers, with the operations of addition and multiplication defined as follows for any z₁, z₂ that are within the set of complex numbers:

The “naive” operations with numbers including the square root of –1 to get the roots of a cubic polynomial were the impetus for defining multiplication exactly as it is expressed above.

 If you recall the prior lecture, this was done in order to locate the roots of the cubic polynomial.
Recall that the standard basis of R2 which essentially means that any vector from R² (and, by extension, from C), may be represented in a unique form as a linear combination.

z =[xy] = xe₁ + ye₂.

Now, keep in mind that using the multiplication established above for every z, 

We have e₁ z = ze₁  = z.

 Because of this, I will denote e₁ as 1 to highlight the fact that it is my unit, which is a small abuse of the notation system. 

Furthermore, e₂•e₂ = e₂ 

 so if I introduce the notation i = e₂

 i²= –1, and because of this, the common phrase z = x + iy may (and should be regarded as) a representation of z. 

There are no longer any “imaginary” quantities! 

In any case, using the symbol I rather than writing vectors saves so much space and time (instead of writing vectors), and makes the computations so much easier, that we will always use this notation.

However, a student should always remember that behind it there are always a couple of our very familiar vectors that form the standard basis of R².

Complex Plane

In mathematics, the complex plane is the plane that is formed by complex numbers.

 It uses a Cartesian coordinate system, in which the x-axis, also referred to as the real axis, is formed by the real numbers and the y-axis, also referred to as the imaginary axis, is formed by the imaginary numbers. 

The real numbers and the imaginary numbers both form the plane.

Complex numbers can be interpreted geometrically if they are plotted on a complex plane. 

They add up as vectors when subjected to addition.

In polar coordinates, the multiplication of two complex numbers can be expressed in a way that is more straightforward:

the magnitude or modulus of the product is the product of the two absolute values, or moduli, and the angle or argument of the product is the sum of the two angles, or arguments.

This makes the multiplication of two complex numbers easier to understand.

Specifically, the rotational effect of modulus 1 multiplication by a complex number is seen below.

Occasionally referred to as the Argand plane or the Gauss plane, the complex plane has a variety of names.

Imaginary Axis

Imaginary axis (plural imaginary axes) (mathematics) The line that runs vertically through the complex plane, along which every point corresponds to a complex number that contains no real component at all.

We are aware that the border between the zones of stability and instability in the s-plane is determined by the imaginary axis that runs through the plane and that this boundary is denoted by the equation s = j.

 Because z equals esT, this boundary has to be mapped onto the z-plane in the following way:

z=e^jωT

In other words, this equation establishes the limit between the stable and unstable regions in the z-plane, which serves as the boundary between the two.

In point of fact, the circle centred on the origin and having a radius of one is defined by the equation z = e^jt

This is not at all self-evident, and in order to demonstrate it, we will make use of Euler’s identity.

 This demonstrates to us that:

ejθ=cosθ+jsinθ

As a consequence of this, we can rewrite z = e^jωT as follows:

z=cos ωT+jsin ωT

Argands Plane

A diagram in which complex numbers are represented by the points in the plane, the coordinates of which are respectively the real and imaginary parts of the number. 

For example, the number x + I y can be represented by the point (x, y), or by the corresponding vector <x, y>.

 r is the modulus and is the argument of x+iy.

which is a Cartesian coordinate system consisting of two perpendicular axes for graphing complex numbers, with the real part of a number being plotted along the horizontal axis and the imaginary part being plotted along the vertical axis. 

If the polar coordinates of (x, y) are (r,θ), then r is the modulus and is the argument of x.

Conclusion

The crucial part is that operations made the set C into a field, which is introduced in the way that the set of complex numbers is written above. 

 For a complex z = x + iy, 

I introduce its conjugate z = x+ iy and make the observation that zz¯ = (x + iy)(x iy) = x₂ + y₂ is a real number (which, once more, 

should be understood as a vector in R₂ with the second coordinate zero, 

or even better as (x₂ + y₂)e₁ in order to be considered.