Complex Number

                                                                      Complex Number 

Introduction:

In Real Number System there is no solution to the equation  x² = - 1 . But in a new number system there is a solution to this equation .The backbone of this new number system is' i ' , known as imaginary unit .

       i² = - 1 

∴ i = √ - 1 

By taking multiple of this imaginary unit we can make many imaginary numbers such as 2i , -12i ,√5i etc .

By addition of real number with  imaginary number we get complex number.For example 1 + i ,3 - 5i  etc 

This type of combination of real and imaginary number is called complex  number .

History of Complex Number :

Sometimes solution to cubic equation contains square root of negative numbers .This situation led Italian Mathematician Gerolamo Cardano to conceive of complex numbers in 1545 , though his concept was rudimentary. 

Work on the general polynomials ultimately led to the fundamental theorem of algebra which shows that every polynomial of degree one  or higher has solution with complex numbers. Complex numbers thus form a closed field where every polynomial has a root. 

Many mathematicians contributed to the development of complex numbers. The rules of addition ,substraction and multiplication were developed by Italian mathematician Rafael Bombelli A more abstract formalism for complex numbers was further developed by Irish mathematician William Rowan Hamilton who extended the abstraction to theory of quaternions 

The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek Mathematician Hero of Alexandria in the first century AD .

The impetus to study complex number as a separate topic arose in the 16 century when the algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (Niccolò Fontana Tartaglia, Gerolamo Cardano ) 

In the 18th  century complex numbers gained wider use as it was noticed that formal manipulation of complex expression could be used to simplify calculation of trigonometric functions.In 1730 Abraham de Moivere  used a theorem  to simplify  trigonometric expressions. This theorem is known as de Moivere theorem.

(cos θ +i sin θ)ⁿ = cosnθ + i sin nθ

In 1748 Eular went further and discovered Eular Formula of complex analysis.

cos θ +i sin θ = e^iθ 

This formula could be used to reduce any trigonometric identity to a much simpler exponential identities.

The idea of complex number as points in the complex plane was first described by Casper Wessel in 1799.In 1806 Jean-Robert Argand issued a pamphlet on complex number and gave a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss published an essentially topological proof of the theorem in 1797 .In the beginning of 19th century other mathematicians developed independently the geometrical representations of complex numbers The English mathematician G.H.Hardy that Gauss was the first mathematician to use complex numbers in a really confident and scientific way.

Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.

Argand called cos φ + i sin φ the direction factor, and ${\displaystyle r={\sqrt {a^{2}+b^{2}}}}$ the modulus .Cauchy (1821) called cos φ + i sin φ the reduced form ,called argument.Gauss used i for , introduced the term complex number for a + bi,and called a² + b² the norm. 

Development of Complex numbers :

A complex number is a number of the form a + i b where a and b are real numbers and i² = 1 .

For example 3 + i 4 is a complex number. This way complex number is defined as polynomial 

 with real coefficients in the single indeterminate i for which the relation i² = -1 or i²  +1 = 0 holds.

 and i^4k = 1, i^4k+1 = i, i^4k+2 = −1, and i^4k+3 = −i, k = integer 

a is called the real part and b is called  imaginary part.

    

The cartesian complex plane :

the definition of complex number involving two real numbers suggest the use of cartesian

 co-ordinates in the complex plane.Horizontal axis is used to display real part 

and vertical axis is used to specify imaginary part.

A complex number may be viewed as a co-ordinatized  point or as a position vector

from the origin to the point.Coordinate values of complex number is expressed in cartesian ,

rectangular form.When complex number viewed as position vector addition correspond to vector

 addition.Multiplication corresponds to multiplying their amplitudes and adding the angles they

 make with real axis.The multiplication of a complex number by  i corresponds to the rotation of 

position vector by 90 degree counterclockwise.about the origin . This can written algebraically as 

(a + i b)  = a i + b i²  = - b + a i 

Polar complex plane :

We can express complex number in polar coordinates also .The absolute value com complex 

number z = x + i y is r  =| z | = √ ( x² + y²) 

The argument  of z is defined as the angle the radius vector makes with the real axis.

φ = arg z = arg ( x + i y ) = tan⁻¹ ( y/x ) 

Together with r and φ any complex number can be represented as

z = r ( cosφ + i sinφ )

using Euler's formula it can be expressed as  

z  = r e ^iφ 

Equality - Two complex numbers  a₁ +i b₁ and a₂ + i b₂ only if both their real and imaginary parts

 are equal

i.e  a₁ = a₂ and b₁ = b₂ 

In polar form two complex numbers are equal if their magnitudes are equal and their arguments 

differ by an integral multiple of 2π 

Ordering - 

unlike real numbers there is no natural ordering in complex numbers .Complex numbers cannot have 

the structure of ordered field.This is because in an ordered field every nontrivial sum of squares ≠  0 

and i² + 1 = 0 is a non trivial sum of squares.Complex numbers are naturally thought of existing on two 

dimensional plane.

Conjugate - The complex conjugate of a complex number z = x + i y is defined as z✡  = x  - i y 

Geometrically conjugate is a  reflection of the  complex number about the real axis.

Conjugating twice gives the original complex number. The product of a complex number z = x +iy  with 

its conjugate is known as absolute square.It is always as a positive real number.

z.z✡ = x² + y² = |z|²  = |z✡|²

The real and imaginary part can be extracted from from the conjugation'

Re( z) = (z + z✡)/2 and Im(z) = (z - z✡)/2i 

Conjugation distributes over basic arithmetic operations i.e 

(z +w)✡ =  z✡ + w✡ , (z -w)✡ =  z✡ - w✡ 

(z .w) = z✡ .w✡ 

Addition and Subtraction - Two complex numbers a and b are most easily added by separately adding 

real and imaginary parts i.e 

a + b = (x + i y ) +( u + i v) =  (x + u )+ i ( y + v) 

Similarly subtraction can be performed as 

a - b = (x + i y ) - ( u + i v) =  (x - u )+ i ( y -  v) 

Multiplication- Two complex numbers are multiplied under the rules of distributive property 

, the commutative property and the defining property i² = - 1 in the following way -

z.w =( x +iy).(u +iv) 

= x(u + iv) + iy ( u +iv ) ( by right distributive law )

= xu + x iv + ivu + iy iv ( left distributive law   )

xu + iy iv+ i  xv  + i vu

= xu +yv i² + i(xv + yu )

= (xu - yv) + i ( xy + yu) 

Reciprocal and Division - using the conjugation reciprocal of a nonzero complex number

z = x + i y is defined as 1/z = z✡/(z z✡) =  z✡/|z|² =  z✡/( x² + y²)

= x/( x² + y²) - i y/( x² + y²)  where ( ( x² + y²) >0 )

It can be used for division of an arbitrary complex number by a non zero complex number i.e

w/z = w.1/z = (u + iv)[ x/( x² + y²) - i y/( x² + y²)] 

= 1/( x² + y²)[ (ux + vy) + i(vx - uy)]

Multiplication and division in polar form -

Formulas for multiplication and division in polar form are simpler than in corresponding cartesian

 forms. Given two complex numbers z₁ = r₁(cos φ₁ + i sin φ₁)  and z₂ = r₂(cos φ₂ + i sin φ₂) we may derive 

Where we use the trigonometric identities 

 

We see here absolute values are multiplied and arguments are added to give the polar form of the product.

Multiplying by i corresponds to a quarter-turn counter clockwise which gives back i² = -1 .

    

The picture illustrates the multiplication of two complex numbers i.e

(2 + i) (3 + i) = 5 + 5i

Application of Complex numbers :

Complex numbers have application in many scientific areas such as signal processing ,

control theory,electromagnetism ,fluid dynamics and quantum mechanics etc.

Improper Integral - In applied fields certain improper integrals are evaluated with use of complex

valued functions such as method of contour integrations.

        Dynamics equations - To obtain a solution for linear differential equations complex numbers

 are used.

Control Theory- In control theory systems are often transformed from time domain to frequency 

domain using Laplace Transforms.The systems zeros and poles are then analyzed in complex plane.

Signal analysis - Complex numbers are used in signal analysis.For given real functions representing 

physical quantities in terms of sine and cosine corresponding complex function are considered.

For a sine wave of a given frequency absolute value |z|  is the amplitude and the argument of z is

 phase.In Fourier analysis a real valued signal is written as sum of some periodic functions

 expressed in complex valued

functions in the forms 

x(t) = Re {X(t)} 

where X(t) =  A e^iωt  , ω represent angular frequency and the complex 

number A encodes the phase and amplitude of the signal.

In physics voltage in AC circuits can be represented as

V(t) = V₀e ^ iωt , a complex function 

Inferences :

Complex number system ,C can be seen an extension of real number system ,R.The process of 

extending the field R to C is known as Cayley - Dickson construction.It can be further extended to 

higher dimensions,yielding H quaternions and octonions with dimension 4 and 8 respectively.

In this context  complex number can re treated as binarions 

Set of Complex number is  a  field.This means any two complex numbers can be added to

 yield another complex number.For any complex number  z there is an additive inverse - z ,

which is also a complex number.Every non zero complex number has a reciprocal complex number.

Law of commutativity holds for addition and multiplication for two complex numbers.Unlike real 

numbers complex field is not an ordered field.