Complex numbers and It’s application

HISTORY OF COMPLEX NUMBERS:

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.

The rules for addition, subtraction and multiplication of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton.

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation x2 = −1, that is, i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number.

Discrete Fourier TransformThe DFT is a ubiquitous algorithm in computer science, used in image processing, digital communication, compression and countless other uses in and around signal processing. It is likely the most useful and common transformation (linear or otherwise) in computer science, often being implemented at the hardware level itself.

Given a sequence of numbers  , ,the DFT is defined as :

Complex numbers are beautiful, because they encode geometric information through algebra.

QuaternionIn mathematics, the quaternions are a number system that extends the complex numbersQuaternions provide a very convenient way of representing rotations of three-dimensional space. Even more importantly, when rotations are represented with quaternions (as opposed to Euler angles), it becomes much easier to smoothly interpolate one rotation to another, which is something computers need to do repeatedly when animators are working on the next 3D feature film.

Many game programmers have already discovered the wonderful world of quaternions and have started to use them extensively. Several third-person games, including both TOMB RAIDER titles, use quaternion rotations to animate all of their camera movements. Every third-person game has a virtual camera placed at some distance behind or to the side of the player's character. Because this camera goes through different motions (that is, through arcs of a different lengths) than the character, camera motion can appear unnatural and too "jerky" for the player to follow the action. This is one area where quaternions come to rescue.

struct Quat{ float x; float y; float z; float w;};

Video Games and Quaternion

"The shortest route between two truths in the real domain passes through the complex domain."

A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale.

FractalWith computers, we can generate beautiful art from complex numbers.

These designs are called fractals.

Fractals are produced using an iteration process.Common fractals are based on the Julia Set and the Mandelbrot Set.

The Julia Set equation is:Zn+1 = (Zn)2 + c

For the Julia Set, the value of c remains constant and the value of Zn changes

The Mandelbrot Set

The Mandelbrot is the same as the Julia Set, but the value of c is allowed to change.

Analytic combinatoricsIn mathematics, analytic combinatorics is one of the many techniques of counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions and then complex analysis techniques to get asymptotics.

It is used for analysis of algorithms.

The fast multipole method has been called one of the ten most significant algorithms in scientific computation discovered in the 20th century.

Fast multipole method

The fast multipole method (FMM) is a numerical technique that was developed to speed up the calculation of long-ranged forces in the n-body problem. It does this by expanding the system Green's function using a multipole expansion, which allows one to group sources that lie close together and treat them as if they are a single source.

N-Body problems inevitably come up when doing any most any kind of physical simulation work, particularly when particles are involved.

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