Application of complex number in engineering

INTRODUCTION

A complex number is a number comprising area land imaginary part. It can be written in the form a+ib, where a and b are real numbers, and i is the standard imaginary unit with the property i2=-1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the understanding of addition and multiplication.

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.

COMPLEX NUMBER INTERPRETATION:

A number in the form of x+iy where x and y are real numbers and i = -1 is called a complex number.

Let z = x+iy

X is called real part of z and is denoted by R (z)

Y is called imaginary part of z and is denoted by I (z)

CONJUGATE OF A COMPLEX NUMBER:

A pair of complex numbers x+iy and x-iy are said to be conjugate of each other.

PROPERTIES OF COMPLEX NUMBERS ARE:

1. If x1+ iy1 = x2 + iy2 then x1- iy1 = x2 - iy2
2. Two complex numbers x1+ iy1 and x2 + iy2 are said to be equal
                  If R (x1 + iy1) = R (x2 + iy2)
                  I (x1 + iy1) = I (x2 + iy2)
3. Sum of the two complex numbers is
                  (x1 + iy1) + (x2 + iy2) = (x1+ x2) + i(y1+ y2)
4. Difference of two complex numbers is
                  (x1 + iy1) - (x2 + iy2) = (x1-x2) + i(y1 - y2)
5. Product of two complex numbers is
                  (x1+ iy1) x ( x2 + iy2) = x1x2 - y1y2 + i(y1x2 + y2 x1)
6. Division of two complex numbers is
                  (x1 + iy1) / (x2 + iy2) = (x1x2+y1y2)/x2x2+y2y2 +( iy1x2 -iy2x1)/    x2x2+y2y2
7. Every complex number can be expressed in terms of r (cosθ + i sinθ)
                  R (x+ iy) = r cosθ
                  I (x+ iy) = r sinθ
                  r = x2+y2 and θ = tan-1(y/x)

REPRESENTATION OF COMPLEX NUMBERS IN PLANE

The set of complex numbers is two-dimensional, and a coordinate plane is required to illustrate them graphically. This is in contrast to the real numbers, which are one-dimensional, and can be illustrated by a simple number line. The rectangular complex number plane is constructed by arranging the real numbers along the horizontal axis, and the imaginary numbers along the vertical axis. Each point in this plane can be assigned to a unique complex number, and each complex number can be assigned to a unique point in the plane.

Modulus and Argument of a complex number:

The number r = x2+y2 is called modulus of x+ iy and is written by mod (x+ iy) or x+iy

θ = tan-1yx is called amplitude or argument of x + iy and is written by amp (x + iy) or arg (x + iy)

Application of imaginary numbers:

For most human tasks, real numbers (or even rational numbers) offer an adequate description of data. Fractions such as 2/3 and 1/8 are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Negative numbers such as -3 and -5 are meaningless when measuring the mass of an object, but essential when keeping track of monetary debits and credits. Similarly, imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, cartography, vibration analysis, and many others.

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