Geometric Interpretation

The usage of Cartesian coordinates in the complex plane is directly suggested by the formulation of complex numbers, which requires the involvement of two arbitrary real values. The real part is typically represented by the horizontal (real) axis, with increasing values to the right, and the imaginary part is marked by the vertical (imaginary) axis, with increasing values upwards. Both axes increase in value from left to right.

It is possible to look at a charted number either as the point that is being coordinated or as a position vector that is moving from the origin to this point. The Cartesian, rectangular, or algebraic form of a complex number, z, can therefore be used to indicate the coordinate values of the complex number.

When complex numbers are viewed as position vectors, the operations of addition and multiplication take on a very natural geometric character. Specifically, addition corresponds to vector addition, and multiplication corresponds to multiplying their magnitudes and adding the angles they make with the real axis. This gives the operations of addition and multiplication a very natural geometric flavour. When observed in this manner, the operation of multiplying a complex number by i is equivalent to rotating the position vector anticlockwise by a quarter turn (90°) about the origin. This fact can be expressed algebraically as follows:

            (a+bi) . i = ai + b(i)² = -b + ai

Complex graph:

When trying to visualise complex functions, you need to have both a complex input and complex output. Graphing a complex function visually would require the perception of a four dimensional space, which is something that can only be done through the use of projections. This is because each complex number is only represented in two dimensions. As a result of this, various different methods of visually representing complicated functions have been developed.

The output dimensions of a domain colouring are represented, respectively, by colour and brightness. The argument of a complex number is often represented by the colour, and the magnitude of the complex number is typically represented by the brightness of the point’s ornation in the complex plane as domain. The gradation may be discontinuous, but it is assumed to be monotonous. Dark spots mark moduli that are close to zero, and brighter spots are spots that are further away from the origin. The colours frequently shift between red, yellow, green, cyan, and blue before arriving at magenta. These shifts take place in steps of π / 3 from 0 to 2π. Color wheel graphs are the name given to these types of visualisations. This offers a straightforward approach of visualising the functions while preserving the relevant information. 

Another method for visualising complex functions is to use Riemann surfaces.

One way to think of Riemann surfaces is as deformations of the complex plane. The real and imaginary inputs are represented along the horizontal axes, while the real or imaginary output is only represented along the single vertical axis.

The complex plane( gaussian plane):

The following definition should be used to talk about the geometric representation of complex numbers:

• On the complex plane, the coordinates (a, b) are the location of the point that is assigned to the complex number z = a + b i. Although the complex plane and the Cartesian coordinate system are quite comparable to one another, the complex plane’s axes are referred to differently.

• The real component of the complex number is represented along the x-axis. This axis is known as the real axis, and it is denoted by the letters R or R_e.

• The imaginary component of the complex number is depicted along the y-axis. This axis is referred to as the imaginary axis, and it is denoted by the letters iR or I_m.

The starting point of the coordinate system is referred to as the zero point.

The number 1 is located one unit to the right of the zero point on the real axis of the complex plane, whereas the number i is located one unit above the zero point on the imaginary axis of the complex plane.

Conclusion:

A complex number, denoted by the letter z, can be associated with an ordered pair of real numbers. These real numbers, in turn, can be regarded as the coordinates of a point in a space with just two dimensions. The usage of Cartesian coordinates in the complex plane is directly suggested by the formulation of complex numbers, which requires the involvement of two arbitrary real values.

The Cartesian, rectangular, or algebraic form of a complex number, z, can therefore be used to indicate the coordinate values of the complex number.

Graphing a complex function visually would require the perception of a four dimensional space, which is something that can only be done through the use of projections. 

Another method for visualising complex functions is to use Riemann surfaces. The real and imaginary inputs are represented along the horizontal axes, while the real or imaginary output is only represented along the single vertical axis.

The real component of the complex number is represented along the x-axis. This axis is known as the real axis. And, The imaginary component of the complex number is depicted along the y-axis. This axis is referred to as the imaginary axis.