Representation of a Complex Number

A complex number is one that can be written as a + ib, where a represents the real component and b represents the imaginary part; I represent the imaginary unit, which is defined as the square root of -1 or the solution of x2 = -1. This article will teach you how to express a complicated number.

A complex number with no real part is considered to be totally imaginary, as well as the points of these number fall on the complex plane’s vertical axis. Similarly, a complex number with a zero imaginary element can be seen as a real number, and it’s point can be found on the complex plane’s horizontal axis.

Geometrical Representation of Complex Numbers

Complex numbers use the horizontal axis for the real part and vertical axis for the imaginary part to expand the concept of a one-dimensional number line to the two-dimensional complex plane. As a result, a complex number a+bi can be identified as a complex plane point P(a,b). Complex numbers can be represented in Polar form, which correlates each complex number with its magnitude (distance from the origin) and a certain angle (the argument of the complex number).

Graphing a Complicated Number

We use the s-plane, which is comparable to the x-y plane in the regular cartesian system, to draw a complex number a + ib, which can be thought of as a point P(a,b).

The S-plane, also known as the Argand plane, Complex-plane, or Argand diagram, is a graph that plots the real and imaginary values a and b of the point P on the real and imaginary axes, respectively, on the horizontal and vertical axes.

Consider the complex plane point (2,3). Because both the real and imaginary terms are positive, it can be written as 2+3i in terms of a complex number and plotted in the first quadrant. Similarly, in the second quadrant, -2+3i can be written as (-2,3); in the third quadrant, (-2, -3) can be written as -2-3i; and in the fourth quadrant, (2, -3) can be written as 2-3i.

POLAR COMPLEX PLANE

The graphic above depicts a line in the complex plane that connects certain locations and forms an angle with the real axis. Draw a line parallel to the Real axis and intersecting the Imaginary axis from the line, and label it b. A line drawn from the plot contacting the Real axis and parallel to the imaginary axis is called a. The Pythagoras theorem then gives us r = √(a2 +b2)

It is frequently referred to as magnitude It’s also known as the complex number’s absolute value or modulus, or r = |z|. Similarly, the parameter tan ϴ= b/a or simply = tanϴ-1(y/x) will be used. It’s also known as the complex number’s phase or angle.

If the complex number seems to have no imaginary parts or b=0, then r=|x|, i.e., the real number’s absolute value equals the complex number’s absolute value.

Take the polar form z = r cosϴ + r i sinϴ => z = r (cosϴ + i sinϴ) as an example.

We get z = r eiϴ

Important Complex Numbers Concepts

Analytical Complexity

The study of functions of a complex variable is known as complex analysis. They have 4-dimensional graphs, which are represented by color-coding a triple graph in four dimensions.

Functions that are holomorphic

If a function f: C -> C satisfies the Cauchy-Riemann equations, it is called holomorphic.

Consider the equation f(z) = u z + v z*, where u and v are complex coefficients. If and only if b = 0 is this map holomorphic.

CONCLUSION

Complex numbers make finding square root of negative integers easier. In the first century, a Greek mathematician named Hero of Alexandria discovered the concept of complex numbers while attempting to find the square root of a negative number.

However, all he did was convert the negative to a positive & calculate the numerical root value. Furthermore, Italian mathematician Gerolamo Cardano defined the real identity of a complex number in the 16th century while looking for the negative roots of cubic & quadratic polynomial formulas.