Complex Numbers

COMPLEX NUMBERS Presented to you by• • • • • •

MANDAR SHEMILA SUPRIYA SHILPA ANKITA RALSTON

**COMPLEX NUMBERS** • A complex number can be visually represented as a pair of numbers forming a vector on a diagram called an Argand diagram • In mathematics, the complex numbers are an extension of the real numbers obtained by adjoining an imaginary unit, denoted i, which satisfies: • Every complex number can be written in the form a + bi, where a and b are real numbers called the real part and the imaginary part of the complex number, respectively. • Complex numbers are a field, and thus have addition, subtraction, multiplication, and division operations. These operations extend the corresponding operations on real numbers, although with a number of additional elegant and useful properties, e.g., negative real numbers can be obtained by squaring complex (imaginary) numbers.

***DISCOVERED*** • Complex numbers were first conceived and defined by the Italian mathematician Girolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. • The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. • This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, it is always possible to find solutions to polynomial equations of degree one or higher. • The rules for addition, subtraction, multiplication, and division 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, who extended this abstraction to the theory of quaternions.

**DEFINATION** • complex numbers are very often written in the form where a and b are real numbers, and i is the imaginary unit, which has the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. • The real numbers, R, may be regarded as a subset of C by considering every real number a complex number with an imaginary part of zero; that is, the real number a is identified with the complex number a + 0i. Complex numbers with a real part of zero are called imaginary numbers; instead of writing 0 + bi, that imaginary number is usually denoted as just bi. If b equals 1, instead of using 0 + 1i or 1i, the number is denoted as i. • In some disciplines (in particular, electrical engineering, where i is a symbol for current), the imaginary unit i is instead written as j, so complex numbers are sometimes written as a + bj.

***HISTORY*** • Complex numbers became more prominent in the 16th century, when closed formulas for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolo Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. For example, Tartaglia's cubic formula gives the following solution to the equation x³ − x = 0: • At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions –i, and . Substituting these in turn for in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3 – x = 0.

***HISTORY*** • The term "imaginary" for these quantities was coined by René Descartes in 1637 and was meant to be derogatory. • A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity , which is valid for positive real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. • The incorrect use of this identity (and the related identity ) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of to guard against this mistake.

***HISTORY*** • The 18th century saw the labors of Abraham de Moivre and Leonhard Euler. To de Moivre is due (1730) the well-known formula which bears his name, de Moivre's formula: and to Euler (1748) Euler's formula of complex analysis: • The existence of complex numbers was not completely accepted until the geometrical interpretation (see below) had been described by Caspar Wessel in 1799; it was rediscovered several years later and popularized by Carl Friedrich Gauss, and as a result the theory of complex numbers received a notable expansion.

The field of complex numbers  A field is an algebraic structure with addition, subtraction, multiplication, and division operations that satisfy certain algebraic laws. The complex numbers form a field, known as the complex number field, denoted by C. In particular, this means that the complex numbers possess:  An additive identity ("zero"), 0 + 0i.  A multiplicative identity ("one"), 1 + 0i.  An additive inverse of every complex number. The additive inverse of a + bi is −a − bi.  A multiplicative inverse (reciprocal) of every nonzero complex number. The multiplicative inverse of a + bi is  Examples of other fields are the real numbers and the rational numbers. When each real number a is identified with the complex number a + 0i, the field of real numbers R becomes a subfield of C.  The complex numbers C can also be characterized as the topological closure of the algebraic numbers or as the algebraic closure of R

 The complex plane Geometric representation of z and its conjugate in the complex plane.  A complex number z can be viewed as a point or a position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001) named after Jean-Robert Argand. The point and hence the complex number z can be specified by Cartesian (rectangular) coordinates.  The Cartesian coordinates of the complex number are the real part x = Re(z) and the imaginary part y = Im(z). The representation of a complex number by its Cartesian coordinates is called the Cartesian form or rectangular form or algebraic form of that complex number.

 Absolute value, conjugation and distance  The absolute value (or modulus or magnitude) of a complex number z = reiφ is defined as | z | = r. Algebraically, if z = x + yi, then  The absolute value has three important properties:  where if and only if (triangle inequality) for all complex numbers z and w. These imply that | 1 | = 1 and | z / w | =|z|/|w|.  By defining the distance function d(z,w) = | z − w | , we turn the set of complex numbers into a metric space and we can therefore talk about limits and continuity.

POLAR FORM  CARTESIAN REPRESENTATION OF POLAR FORM Z=X+iY.  DEFINATION OF “POLAR FORM”.  “PRINCIPAL VAULES” OF POLAR FORM.  CONVERSION FORM CARTESIAN TO POLAR FORM.  NOTATION OF POLAR FORM  “MULTIPLICATION FORMULA”.  “DIVISION FORMULA”.  “ROOT EXTRACTION & EXPONENTIATION”.

Euler's identity  The exponential function ez can be defined as the limit of (1 + z/N)N, as N approaches infinity, and thus eiπ is the limit of (1 + iπ/N)N. In this animation N takes various increasing values from 1 to 100. The computation of (1 + iπ/N)N is displayed as the combined effect of N repeated multiplications in the complex plane, with the final point being the actual value of (1 + iπ/N)N. It can be seen that as N gets larger (1 + iπ/N)N approaches a limit of −1.

 In mathematical analysis, Euler's identity, named after Leonhard Euler, is the equation where is Euler's number, the base of the natural logarithm, is the imaginary unit, one of the two complex numbers whose square is negative one (the other is ), and is pi, the ratio of the circumference of a circle to its diameter.  Euler's identity is also sometimes called Euler's equation.

DERIVATION • Euler's formula for a general angle. • The identity is a special case of Euler's formula from complex analysis, which states that • for any real number x. (Note that the arguments to the trigonometric functions sin and cos are taken to be in radians.) In particular, • Since and it follows that which gives the identity

**APPLICATIONS** • Complex numbers are used in many different fields including applications in engineering, electromagnetism , quantum physics, applied mathematics, and chaos theory. When the underlying field of numbers for a mathematical construct is the field of complex numbers, the name usually reflects that fact. Examples are complex analysis, complex matrix, complex polynomial and complex Lie algebra. – 1 Control theory – 2 Signal analysis – 3 Improper integrals – 4 Quantum mechanics – 5 Relativity – 6 Applied mathematics – 7 Fluid dynamics – 8 Fractals

 1.CONTROL THEORY:  IN CONTROL THEORY ,DYNAMICAL SYSTEM ARE TRANSFORMED FROM TIME DOMAIN TO FREQUENCY DOMAIN USING LAPLACE TRANSFORM.  THEN SYSTEM’S POLES AND ZEROS ARE ANALYZED IN THE COMPLEX PLANE USING TECHNIQUES SUCH AS ROOT LOCUS, NYQUIST PLOT.  IN THE ROOT LOCUS ,IT IS VERY IMPORTANT WHETHER THE POLES AND ZEROS ARE IN THE LEFT OR RIGHT HALF OF THE COMPLEX PLANE.  IF A SYSTEM HAS POLES THAT ARE :  IN THE LEFT ,IT WILL BE STABLE.  IN THE RIGHT , IT WILL BE UNSTABLE.  WHEREAS, ON THE IMAGINARY AXIS IT WILL HAVE MARGINAL STABILITY.  IF A SYSTEM HAS ZERO IN THE RIGHT HALF PLANE,IT IS A  NON-MINIMUM PHASE SYSTEM.

 2. SIGNAL ANALYSIS: 

THE COMPLEX NO ARE USED IN SIGNAL ANALYSIS FOR A CONVIENIENT DESCRIPTION FOR PERIODICALLY VARYING SIGNALS.

 3.IMPROPER INTEGRALS: 

IN APPLIED FIELDS, COMPLEX NOs ARE USED TO COMPUTE REAL VALUED IMPROPER INTEGRALS BY MEANS OF COMPLEX VALUED FUNCTIONS.

 4.QUANTUM MECHANICS:

 

THE COMPLEX NO FIELDS IS USED IN THE MATHEMATICAL FORMULATION OF QUANTUM MECHANICS. FOUNDATION FORMULAS OF QUANTUM MECHANICSTHE HEISENBERG’S MATRIX MECHANICS MAKES USE OF COMPLEX NOs.

 5. RELATIVITY:  IN RELATIVITY , FORMULAS FOR THE METRIC ON SPACE TIME BECOME SIMPLER IF ONE TAKES THE TIME VARIABLE TO BE IMAGINARY.  COMPLEX NOS ARE ESSENTIAL TO SPINORS WHICH ARE A GENERALIZATION OF THE TENSORS USED IN RELATIVITY.

 6. APPLIED MATHEMATICS:  IN DIFFERENTIAL EQUATIONS,IT IS COMMON TO FIND ALL COMPLEX ROOTS R OF THE CHARACTERISTIC EQUATION AND THEN ATTEMPT TO SOLVE THE SYSTEM IN TERMS OF BASE FUNCTIONS OF THE FORM F(T)=Ert.

 7. FLUID DYNAMICS:  IN FLUID DYNAMICS, COMPLEX FUNCTIONS ARE USED TO DESCRIBE POTENTIAL FLOW IN TWO DIMENSIONS.

 8. FRACTALS: 

FRACTALS IS A FRAGMENTAL GEOMETRIC SHAPE WHICH ARE PLOTTED IN THE COMPLEX PLANE, E.G. THE MANDELBROT SET .

***OPERATIONS*** • Complex numbers are added, subtracted, multiplied, and divided by formally applying the associative, commutative and distributive laws of algebra, together with the equation i 2 = −1: • 1.Addition: • 2.Subtraction: • 3.Multiplication: • 4.Division: frac(1 − i)(1 + i) • where c and d are not both zero. This is obtained by multiplying both the numerator and the denominator with the complex conjugate of the denominator. • Since the complex number a + bi is uniquely specified by the ordered pair (a, b), the complex numbers are in one-to-one correspondence with points on a plane. This complex plane is described below.